NCERT Class 9 Mathematics ( PDFDrive )

QUADRILATERALS 143

angle PAC and CD || AB (see Fig. 8.14). Show that

(i) ✁ DAC = ✁ BCA and (ii) ABCD is a parallelogram.

✂Solution : (i) ABC is isosceles in which AB = AC (Given)

So, ✁ ABC = ✁ ACB (Angles opposite to equal sides)

Also, ✁ PAC = ✁ ABC + ✁ ACB

(Exterior angle of a triangle)

or, ✁ PAC = 2✁ ACB (1)

Now, AD bisects ✁ PAC.

So, ✁ PAC = 2✁ DAC (2)

Therefore, [From (1) and (2)] Fig. 8.14

2✁ DAC = 2✁ ACB
or, ✁ DAC = ✁ ACB

(ii) Now, these equal angles form a pair of alternate angles when line segments BC
and AD are intersected by a transversal AC.

So, BC || AD

Also, BA || CD (Given)

Now, both pairs of opposite sides of quadrilateral ABCD are parallel.

So, ABCD is a parallelogram.

Example 4 : Two parallel lines l and m are intersected by a transversal p
(see Fig. 8.15). Show that the quadrilateral formed by the bisectors of interior angles
is a rectangle.

Solution : It is given that PS || QR and transversal p intersects them at points A and
C respectively.

The bisectors of ✁ PAC and ✁ ACQ intersect at B and bisectors of ✁ ACR and
✁ SAC intersect at D.

We are to show that quadrilateral ABCD is a
rectangle.

Now, ✁ PAC = ✁ ACR

(Alternate angles as l || m and p is a transversal)

So, 1 ✁ PAC = 1 ✁ ACR
2 2
i.e., ✁ BAC = ✁ ACD

Fig. 8.15

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144 MATHEMATICS

These form a pair of alternate angles for lines AB and DC with AC as transversal and
they are equal also.

So, AB || DC
Similarly,
BC || AD (Considering ✁ ACB and ✁ CAD)

Therefore, quadrilateral ABCD is a parallelogram.

Also, ✁ PAC + ✁ CAS = 180° (Linear pair)

So, 1 ✁ PAC + 1 ✁ CAS = 1 × 180° = 90°
22 2

or, ✁ BAC + ✁ CAD = 90°

or, ✁ BAD = 90°

So, ABCD is a parallelogram in which one angle is 90°.

Therefore, ABCD is a rectangle.

Example 5 : Show that the bisectors of angles of a parallelogram form a rectangle.

Solution : Let P, Q, R and S be the points of

intersection of the bisectors of ✁ A and ✁ B, ✁ B
and ✁ C, ✁ C and ✁ D, and ✁ D and ✁ A respectively

of parallelogram ABCD (see Fig. 8.16). Fig. 8.16

✂In ASD, what do you observe?
✁Since DS bisects D and AS bisects ✁ A, therefore,

✁ DAS + ✁ ADS = 1 ✁ A + 1 ✁ D
22

= 1 (✁ A + ✁ D)
2

1 (✁ A and ✁ D are interior angles
= 2 × 180°

on the same side of the transversal)

= 90°

Also, ✁ DAS + ✁ ADS + ✁ DSA = 180° (Angle sum property of a triangle)

or, 90° + ✁ DSA = 180°

or, ✁ DSA = 90°

So, ✁ PSR = 90° ✁(Being vertically opposite to DSA)

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QUADRILATERALS 145

✁ ✁Similarly, it can be shown that APB = 90° or SPQ = 90° (as it was shown for
✁ DSA). Similarly, ✁ PQR = 90° and ✁ SRQ = 90°.

So, PQRS is a quadrilateral in which all angles are right angles.

Can we conclude that it is a rectangle? Let us examine. We have shown that

✁ PSR = ✁ PQR = 90° and ✁ SPQ = ✁ SRQ = 90°. So both pairs of opposite angles

are equal.

Therefore, PQRS is a parallelogram in which one angle (in fact all angles) is 90° and
so, PQRS is a rectangle.

8.5 Another Condition for a Quadrilateral to be a Parallelogram

You have studied many properties of a parallelogram in this chapter and you have also
verified that if in a quadrilateral any one of those properties is satisfied, then it becomes
a parallelogram.

We now study yet another condition which is the least required condition for a
quadrilateral to be a parallelogram.

It is stated in the form of a theorem as given below:

Theorem 8.8 : A quadrilateral is a parallelogram if a pair of opposite sides is
equal and parallel.

Look at Fig 8.17 in which AB = CD and
AB || CD. Let us draw a diagonal AC. You can show

that ✂ ABC ✄ ✂ CDA by SAS congruence rule.

So, BC || AD (Why?)

Let us now take an example to apply this property Fig. 8.17
of a parallelogram.

Example 6 : ABCD is a parallelogram in which P
and Q are mid-points of opposite sides AB and CD
(see Fig. 8.18). If AQ intersects DP at S and BQ
intersects CP at R, show that:

(i) APCQ is a parallelogram.
(ii) DPBQ is a parallelogram.

(iii) PSQR is a parallelogram.

Fig. 8.18

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146 MATHEMATICS

Solution : (i) In quadrilateral APCQ,

AP || QC (Since AB || CD) (1)

Also, 11 (Given)
AP = AB, CQ = CD (Why?)

22
AB = CD

So, AP = QC (2)

Therefore, APCQ is a parallelogram [From (1) and (2) and Theorem 8.8]

(ii) Similarly, quadrilateral DPBQ is a parallelogram, because

DQ || PB and DQ = PB

(iii) In quadrilateral PSQR,

SP || QR (SP is a part of DP and QR is a part of QB)

Similarly, SQ || PR

So, PSQR is a parallelogram.

EXERCISE 8.1

1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the
quadrilateral.

2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.

3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it
is a rhombus.

4. Show that the diagonals of a square are equal and bisect each other at right angles.

5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right
angles, then it is a square.

6. Diagonal AC of a parallelogram ABCD bisects

✁ A (see Fig. 8.19). Show that
(i) it bisects ✁ C also,

(ii) ABCD is a rhombus.

7. ABCD is a rhombus. Show that diagonal AC Fig. 8.19

bisects ✁ A as well as ✁ C and diagonal BD
bisects ✁ B as well as ✁ D.

✁ ✁8. ABCD is a rectangle in which diagonal AC bisects A as well as C. Show that:
✁ ✁(i) ABCD is a square (ii) diagonal BD bisects B as well as D.

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QUADRILATERALS 147

9. In parallelogram ABCD, two points P and Q are Fig. 8.20
taken on diagonal BD such that DP = BQ
(see Fig. 8.20). Show that:

(i) ✂ APD ✄ ✂ CQB

(ii) AP = CQ

(iii) ✂ AQB ✄ ✂ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

10. ABCD is a parallelogram and AP and CQ are
perpendiculars from vertices A and C on diagonal
BD (see Fig. 8.21). Show that

(i) ✂ APB ✄ ✂ CQD

(ii) AP = CQ

11. In ✂ ABC and ✂ DEF,AB = DE,AB || DE, BC = EF Fig. 8.21
Fig. 8.22
and BC || EF. Vertices A, B and C are joined to Fig. 8.23
vertices D, E and F respectively (see Fig. 8.22).
Show that

(i) quadrilateral ABED is a parallelogram

(ii) quadrilateral BEFC is a parallelogram

(iii) AD || CF and AD = CF

(iv) quadrilateral ACFD is a parallelogram

(v) AC = DF

(vi) ✂ ABC ✄ ✂ DEF.

12. ABCD is a trapezium in which AB || CD and
AD = BC (see Fig. 8.23). Show that

(i) ✁ A = ✁ B
(ii) ✁ C = ✁ D
(iii) ✂ ABC ✄ ✂ BAD

(iv) diagonal AC = diagonal BD

[Hint : Extend AB and draw a line through C
parallel to DA intersecting AB produced at E.]

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148 MATHEMATICS

8.6 The Mid-point Theorem

You have studied many properties of a triangle as well as a quadrilateral. Now let us
study yet another result which is related to the mid-point of sides of a triangle. Perform
the following activity.

Draw a triangle and mark the mid-points E and F of two sides of the triangle. Join
the points E and F (see Fig. 8.24).

Measure EF and BC. Measure ✁ AEF and ✁ ABC.

What do you observe? You will find that :

EF = 1 BC and ✁ AEF = ✁ ABC Fig. 8.24
2
so, EF || BC
Repeat this activity with some more triangles.
So, you arrive at the following theorem:

Theorem 8.9 : The line segment joining the mid-points of two sides of a triangle
is parallel to the third side.

You can prove this theorem using the following
clue:

Observe Fig 8.25 in which E and F are mid-points
of AB and AC respectively and CD || BA.

✂ AEF ✄ ✂ CDF (ASA Rule)

So, EF = DF and BE = AE = DC (Why?)

Therefore, BCDE is a parallelogram. (Why?) Fig. 8.25
This gives EF || BC.

11
In this case, also note that EF = ED = BC.

22

Can you state the converse of Theorem 8.9? Is the converse true?
You will see that converse of the above theorem is also true which is stated as
below:

Theorem 8.10 : The line drawn through the mid-point of one side of a triangle,
parallel to another side bisects the third side.

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QUADRILATERALS 149

In Fig 8.26, observe that E is the mid-point of
AB, line l is passsing through E and is parallel to BC
and CM || BA.

Prove that AF = CF by using the congruence of

✂ AEF and ✂ CDF.

Example 7 : In ✂ ABC, D, E and F are respectively Fig. 8.26
Fig. 8.27
the mid-points of sides AB, BC and CA
Fig. 8.28
(see Fig. 8.27). Show that ✂ ABC is divided into four

congruent triangles by joining D, E and F.

Solution : As D and E are mid-points of sides AB
and BC of the triangle ABC, by Theorem 8.9,

DE || AC

Similarly, DF || BC and EF || AB

Therefore ADEF, BDFE and DFCE are all parallelograms.

Now DE is a diagonal of the parallelogram BDFE,

therefore, ✂ BDE ✄ ✂ FED
Similarly ✂ DAF ✄ ✂ FED

and ✂ EFC ✄ ✂ FED

So, all the four triangles are congruent.

Example 8 : l, m and n are three parallel lines
intersected by transversals p and q such that l, m
and n cut off equal intercepts AB and BC on p
(see Fig. 8.28). Show that l, m and n cut off equal
intercepts DE and EF on q also.

Solution : We are given that AB = BC and have
to prove that DE = EF.

Let us join A to F intersecting m at G..

The trapezium ACFD is divided into two triangles;

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150 MATHEMATICS

namely ✂ ACF and ✂ AFD.
In ✂ ACF, it is given that B is the mid-point of AC (AB = BC)

and BG || CF (since m || n).
So, G is the mid-point of AF (by using Theorem 8.10)

Now, in ✂ AFD, we can apply the same argument as G is the mid-point of AF,

GE || AD and so by Theorem 8.10, E is the mid-point of DF,
i.e., DE = EF.
In other words, l, m and n cut off equal intercepts on q also.

EXERCISE 8.2

1. ABCD is a quadrilateral in which P, Q, R and S are
mid-points of the sides AB, BC, CD and DA
(see Fig 8.29). AC is a diagonal. Show that :
1
(i) SR || AC and SR = AC
2
(ii) PQ = SR
(iii) PQRS is a parallelogram.
Fig. 8.29

2. ABCD is a rhombus and P, Q, R and S are ©wthe mid-points of the sides AB, BC, CD
and DA respectively. Show that the quadrilateral PQRS is a rectangle.

3. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA
respectively. Show that the quadrilateral PQRS is a rhombus.

4. ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD.
A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.30). Show that
F is the mid-point of BC.

Fig. 8.30

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QUADRILATERALS 151

5. In a parallelogram ABCD, E and F are the
mid-points of sides AB and CD respectively
(see Fig. 8.31). Show that the line segments AF
and EC trisect the diagonal BD.

Fig. 8.31

6. Show that the line segments joining the mid-points of the opposite sides of a
quadrilateral bisect each other.

7. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB

and parallel to BC intersects AC at D. Show that

(i) D is the mid-point of AC (ii) MD ☎ AC

1
(iii) CM = MA = AB

2

8.7 Summary

In this chapter, you have studied the following points :

1. Sum of the angles of a quadrilateral is 360°.

2. A diagonal of a parallelogram divides it into two congruent triangles.

3. In a parallelogram,

(i) opposite sides are equal (ii) opposite angles are equal

(iii) diagonals bisect each other

4. A quadrilateral is a parallelogram, if

(i) opposite sides are equal or (ii) opposite angles are equal

or (iii) diagonals bisect each other

or (iv)a pair of opposite sides is equal and parallel

5. Diagonals of a rectangle bisect each other and are equal and vice-versa.

6. Diagonals of a rhombus bisect each other at right angles and vice-versa.

7. Diagonals of a square bisect each other at right angles and are equal, and vice-versa.

8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the
third side and is half of it.

9. A line through the mid-point of a side of a triangle parallel to another side bisects the third
side.

10. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order,
is a parallelogram.

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CHAPTER 9

AREAS OF PARALLELOGRAMS AND TRIANGLES

9.1 Introduction

In Chapter 5, you have seen that the study of Geometry, originated with the
measurement of earth (lands) in the process of recasting boundaries of the fields and
dividing them into appropriate parts. For example, a farmer Budhia had a triangular
field and she wanted to divide it equally among her two daughters and one son. Without
actually calculating the area of the field, she just divided one side of the triangular field
into three equal parts and joined the two points of division to the opposite vertex. In
this way, the field was divided into three parts and she gave one part to each of her
children. Do you think that all the three parts so obtained by her were, in fact, equal in
area? To get answers to this type of questions and other related problems, there is a
need to have a relook at areas of plane figures, which you have already studied in
earlier classes.

You may recall that the part of the plane enclosed by a simple closed figure is
called a planar region corresponding to that figure. The magnitude or measure of this
planar region is called its area. This magnitude or measure is always expressed with
the help of a number (in some unit) such as 5 cm2, 8 m2, 3 hectares etc. So, we can say
that area of a figure is a number (in some unit) associated with the part of the plane
enclosed by the figure.

We are also familiar with the concept Fig. 9.1
of congruent figures from earlier classes
and from Chapter 7. Two figures are
called congruent, if they have the same
shape and the same size. In other words,
if two figures A and B are congruent
(see Fig. 9.1) , then using a tracing paper,

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AREAS OF PARALLELOGRAMS AND TRIANGLES 153

you can superpose one figure over the other such that it will cover the other completely.
So if two figures A and B are congruent, they must have equal areas. However,
the converse of this statement is not true. In other words, two figures having equal
areas need not be congruent. For example, in Fig. 9.2, rectangles ABCD and EFGH
have equal areas (9 × 4 cm2 and 6 × 6 cm2) but clearly they are not congruent. (Why?)

Fig. 9.2

Now let us look at Fig. 9.3 given below:

Fig. 9.3

You may observe that planar region formed by figure T is made up of two planar
regions formed by figures P and Q. You can easily see that

Area of figure T = Area of figure P + Area of figure Q.
You may denote the area of figure A as ar(A), area of figure B as ar(B), area of
figure T as ar(T), and so on. Now you can say that area of a figure is a number
(in some unit) associated with the part of the plane enclosed by the figure with
the following two properties:
(1) If A and B are two congruent figures, then ar(A) = ar(B);
and (2) if a planar region formed by a figure T is made up of two non-overlapping
planar regions formed by figures P and Q, then ar(T) = ar(P) + ar(Q).
You are also aware of some formulae for finding the areas of different figures
such as rectangle, square, parallelogram, triangle etc., from your earlier classes. In
this chapter, attempt shall be made to consolidate the knowledge about these formulae
by studying some relationship between the areas of these geometric figures under the

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154 MATHEMATICS

condition when they lie on the same base and between the same parallels. This study
will also be useful in the understanding of some results on ‘similarity of triangles’.

9.2 Figures on the Same Base and Between the Same Parallels

Look at the following figures:

Fig. 9.4

In Fig. 9.4(i), trapezium ABCD and parallelogram EFCD have a common side
DC. We say that trapezium ABCD and parallelogram EFCD are on the same base
DC. Similarly, in Fig. 9.4 (ii), parallelograms PQRS and MNRS are on the same base
SR; in Fig. 9.4(iii), triangles ABC and DBC are on the same base BC and in
Fig. 9.4(iv), parallelogram ABCD and triangle PDC are on the same base DC.

Now look at the following figures:

Fig. 9.5

In Fig. 9.5(i), clearly trapezium ABCD and parallelogram EFCD are on the same
base DC. In addition to the above, the vertices A and B (of trapezium ABCD) opposite
to base DC and the vertices E and F (of parallelogram EFCD) opposite to base DC lie
on a line AF parallel to DC. We say that trapezium ABCD and parallelogram EFCD
are on the same base DC and between the same parallels AF and DC. Similarly,
parallelograms PQRS and MNRS are on the same base SR and between the same
parallels PN and SR [see Fig.9.5 (ii)] as vertices P and Q of PQRS and vertices
M and N of MNRS lie on a line PN parallel to base SR.In the same way, triangles
ABC and DBC lie on the same base BC and between the same parallels AD and BC
[see Fig. 9.5 (iii)] and parallelogram ABCD and triangle PCD lie on the same base
DC and between the same parallels AP and DC [see Fig. 9.5(iv)].

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AREAS OF PARALLELOGRAMS AND TRIANGLES 155

So, two figures are said to be on the same base and between the same parallels,
if they have a common base (side) and the vertices (or the vertex) opposite to the
common base of each figure lie on a line parallel to the base.

Keeping in view the above statement, you cannot say that   PQR and   DQR of

Fig. 9.6(i) lie between the same parallels l and QR. Similarly, you cannot say that

Fig. 9.6

parallelograms EFGH and MNGH of Fig. 9.6(ii) lie between the same parallels EF

and HG and that parallelograms ABCD and EFCD of Fig. 9.6(iii) lie between the

same parallels AB and DC (even

though they have a common base DC

and lie between the parallels AD and

BC). So, it should clearly be noted

that out of the two parallels, one

must be the line containing the

common base.Note that ✁ABC and Fig. 9.7
✁DBE of Fig. 9.7(i) are not on the

common base. Similarly, ✁ABC and parallelogram PQRS of Fig. 9.7(ii) are also not on

the same base.

EXERCISE 9.1

1. Which of the following figures lie on the same base and between the same parallels.
In such a case, write the common base and the two parallels.

Fig. 9.8

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156 MATHEMATICS

9.3 Parallelograms on the same Base and Between the same Parallels

Now let us try to find a relation, if any, between the areas of two parallelograms on the
same base and between the same parallels. For this, let us perform the following
activities:
Activity 1 : Let us take a graph sheet and draw two parallelograms ABCD and
PQCD on it as shown in Fig. 9.9.

Fig. 9.9

The above two parallelograms are on the same base DC and between the same
parallels PB and DC. You may recall the method of finding the areas of these two
parallelograms by counting the squares.

In this method, the area is found by counting the number of complete squares
enclosed by the figure, the number of squares a having more than half their parts
enclosed by the figure and the number of squares having half their parts enclosed by
the figure. The squares whose less than half parts are enclosed by the figure are
ignored. You will find that areas of both the parallelograms are (approximately) 15cm2.
Repeat this activity* by drawing some more pairs of parallelograms on the graph
sheet. What do you observe? Are the areas of the two parallelograms different or
equal? If fact, they are equal. So, this may lead you to conclude that parallelograms
on the same base and between the same parallels are equal in area. However,
remember that this is just a verification.

Activity 2 : Draw a parallelogram ABCD on a thick
sheet of paper or on a cardboard sheet. Now, draw a
line-segment DE as shown in Fig. 9.10.

Fig. 9.10

*This activity can also be performed by using a Geoboard.

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AREAS OF PARALLELOGRAMS AND TRIANGLES 157

Next, cut a triangle A✂ D✂ E✂ congruent to

triangle ADE on a separate sheet with the help

of a tracing paper and place   A✂D✂E✂ in such
a way that A✂D✂ coincides with BC as shown

in Fig 9.11.Note that there are two Fig. 9.11

parallelograms ABCD and EE✂CD on the same
base DC and between the same parallels AE✂

and DC. What can you say about their areas?

As ✁ ADE ✄ ✁ A✂D✂E✂

Therefore ar (ADE) = ar (A✂D✂E✂)

Also ar (ABCD) = ar (ADE) + ar (EBCD)

= ar (A✂D✂E✂) + ar (EBCD)
= ar (EE✂CD)

So, the two parallelograms are equal in area.

Let us now try to prove this relation between the two such parallelograms.

Theorem 9.1 : Parallelograms on the same base and between the same parallels
are equal in area.

Proof : Two parallelograms ABCD and EFCD, on
the same base DC and between the same parallels
AF and DC are given (see Fig.9.12).

We need to prove that ar (ABCD) = ar (EFCD).

In   ADE and   BCF, Fig. 9.12

☎ DAE = ☎ CBF (Corresponding angles from AD || BC and transversal AF) (1)

☎ AED = ☎ BFC (Corresponding angles from ED || FC and transversal AF) (2)

Therefore, ☎ ADE = ☎ BCF (Angle sum property of a triangle) (3)

Also, AD = BC (Opposite sides of the parallelogram ABCD) (4)

So,   ADE ✆   BCF [By ASA rule, using (1), (3), and (4)]

Therefore, ar (ADE) = ar (BCF) (Congruent figures have equal areas) (5)

Now, ar (ABCD) = ar (ADE) + ar (EDCB)

= ar (BCF) + ar (EDCB) [From(5)]

= ar (EFCD)

So, parallelograms ABCD and EFCD are equal in area. ✝

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158 MATHEMATICS

Let us now take some examples to illustrate the use of the above theorem.

Example 1 : In Fig. 9.13, ABCD is a parallelogram
and EFCD is a rectangle.

Also, AL   DC. Prove that

(i) ar (ABCD) = ar (EFCD)

(ii) ar (ABCD) = DC × AL Fig. 9.13
Solution : (i) As a rectangle is also a parallelogram,

therefore, ar (ABCD) = ar (EFCD) (Theorem 9.1)

(ii) From above result,

ar (ABCD) = DC × FC (Area of the rectangle = length × breadth) (1)

As AL   DC, therefore, AFCL is also a rectangle

So, AL = FC (2)

Therefore, ar (ABCD) = DC × AL [From (1) and (2)]

Can you see from the Result (ii) above that area of a parallelogram is the product
of its any side and the coresponding altitude. Do you remember that you have
studied this formula for area of a parallelogram in Class VII. On the basis of this
formula, Theorem 9.1 can be rewritten as parallelograms on the same base or
equal bases and between the same parallels are equal in area.

Can you write the converse of the above statement? It is as follows: Parallelograms
on the same base (or equal bases) and having equal areas lie between the same
parallels. Is the converse true? Prove the converse using the formula for area of the
parallelogram.

Example 2 : If a triangle and a parallelogram are on the same base and between the
same parallels, then prove that the area of the triangle is equal to half the area of the
parallelogram.

Solution : Let ✁ ABP and parallelogram ABCD be

on the same base AB and between the same parallels

AB and PC (see Fig. 9.14).

You wish to prove that ar (PAB) = 1 ar (ABCD) Fig. 9.14
2

Draw BQ || AP to obtain another parallelogram ABQP. Now parallelograms ABQP

and ABCD are on the same base AB and between the same parallels AB and PC.

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AREAS OF PARALLELOGRAMS AND TRIANGLES 159

Therefore, ar (ABQP) = ar (ABCD) (By Theorem 9.1) (1)

But   PAB ✁   BQP (Diagonal PB divides parallelogram ABQP into two congruent

triangles.)

So, ar (PAB) = ar (BQP) (2)

Therefore, 1 (3)
ar (PAB) = ar (ABQP) [From (2)]

2

This gives ar (PAB) = 1 ar (ABCD) [From (1) and (3)]
2

EXERCISE 9.2

1. In Fig. 9.15, ABCD is a parallelogram, AE ✂ DC
and CF ✂ AD. If AB = 16 cm, AE = 8 cm and

CF = 10 cm, find AD.

2. If E,F,G and H are respectively the mid-points of
the sides of a parallelogram ABCD, show that

1 Fig. 9.15
ar (EFGH) = ar (ABCD) .

2

3. P and Q are any two points lying on the sides DC and AD respectively of a parallelogram
ABCD. Show that ar (APB) = ar (BQC).

4. In Fig. 9.16, P is a point in the interior of a
parallelogram ABCD. Show that

(i) ar (APB) + ar (PCD) = 1 ar (ABCD) Fig. 9.16
2

(ii) ar (APD) + ar (PBC) = ar (APB) + ar (PCD)
[Hint : Through P, draw a line parallel to AB.]

5. In Fig. 9.17, PQRS and ABRS are parallelograms
and X is any point on side BR. Show that
(i) ar (PQRS) = ar (ABRS)

1
(ii) ar (AX S) = ar (PQRS)

2

Fig. 9.17

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160 MATHEMATICS

6. A farmer was having a field in the form of a parallelogram PQRS. She took any point A
on RS and joined it to points P and Q. In how many parts the fields is divided? What
are the shapes of these parts? The farmer wants to sow wheat and pulses in equal
portions of the field separately. How should she do it?

9.4 Triangles on the same Base and between the same Parallels

Let us look at Fig. 9.18. In it, you have two triangles

ABC and PBC on the same base BC and between

the same parallels BC and AP. What can you say

about the areas of such triangles? To answer this

question, you may perform the activity of drawing

several pairs of triangles on the same base and

between the same parallels on the graph sheet and Fig. 9.18
find their areas by the method of counting the

squares. Each time, you will find that the areas of the two triangles are (approximately)

equal. This activity can be performed using a geoboard also. You will again find that

the two areas are (approximately) equal.

To obtain a logical answer to the above question,
you may proceed as follows:

In Fig. 9.18, draw CD || BA and CR || BP such
that D and R lie on line AP(see Fig.9.19).

From this, you obtain two parallelograms PBCR Fig. 9.19
and ABCD on the same base BC and between the
same parallels BC and AR.

Therefore, ar (ABCD) = ar (PBCR) (Why?)

Now   ABC ✁   CDA and   PBC ✁   CRP (Why?)

So, ar (ABC) = 1 ar (ABCD) and ar (PBC) = 1 ar (PBCR) (Why?)
2 2

Therefore, ar (ABC) = ar (PBC)

In this way, you have arrived at the following theorem:

Theorem 9.2 : Two triangles on the same base (or equal bases) and between the
same parallels are equal in area.

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AREAS OF PARALLELOGRAMS AND TRIANGLES 161

Now, suppose ABCD is a parallelogram whose one of the diagonals is AC

(see Fig. 9.20). Let AN   DC. Note that

✁ ADC ✂ ✁ CBA (Why?)

So, ar (ADC) = ar (CBA) (Why?)

Therefore, ar (ADC) = 1 ar (ABCD)
2

= 1 (DC ✄ AN) (Why?) Fig. 9.20
2

So, area of ✁ ADC = 1 × base DC × corresponding altitude AN
2

In other words, area of a triangle is half the product of its base (or any side) and

the corresponding altitude (or height). Do you remember that you have learnt this

formula for area of a triangle in Class VII ? From this formula, you can see that two

triangles with same base (or equal bases) and equal areas will have equal

corresponding altitudes.

For having equal corresponding altitudes, the triangles must lie between the same
parallels. From this, you arrive at the following converse of Theorem 9.2 .

Theorem 9.3 : Two triangles having the same base (or equal bases) and equal
areas lie between the same parallels.

Let us now take some examples to illustrate the use of the above results.

Example 3 : Show that a median of a triangle divides it into two triangles of equal
areas.

Solution : Let ABC be a triangle and let AD be one of its medians (see Fig. 9.21).

You wish to show that

ar (ABD) = ar (ACD).

Since the formula for area involves altitude, let us

draw AN   BC.

Now ar(ABD) = 1 × base × altitude (of ✁ ABD)
2

= 1 ✄ BD✄AN Fig. 9.21

2

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162 MATHEMATICS

= 1   CD AN (As BD = CD)

2

= 1 × base × altitude (of ✁ ACD)
2

= ar(ACD)

Example 4 : In Fig. 9.22, ABCD is a quadrilateral
and BE || AC and also BE meets DC produced at E.

Show that area of ✁ ADE is equal to the area of the

quadrilateral ABCD.

Solution : Observe the figure carefully .

✁ BAC and ✁ EAC lie on the same base AC and Fig. 9.22

between the same parallels AC and BE.

Therefore, ar(BAC) = ar(EAC) (By Theorem 9.2)

So, ar(BAC) + ar(ADC) = ar(EAC) + ar(ADC) (Adding same areas on both sides)

or ar(ABCD) = ar(ADE)

EXERCISE 9.3 Fig. 9.23

1. In Fig.9.23, E is any point on median AD of a

✁ ABC. Show that ar (ABE) = ar (ACE).

2. In a triangle ABC, E is the mid-point of median

AD. Show that ar (BED) = 1 ar(ABC) .
4

3. Show that the diagonals of a parallelogram divide
it into four triangles of equal area.

4. In Fig. 9.24, ABC and ABD are two triangles on
the same base AB. If line- segment CD is bisected
by AB at O, show that ar(ABC) = ar (ABD).

Fig. 9.24

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AREAS OF PARALLELOGRAMS AND TRIANGLES 163

5. D, E and F are respectively the mid-points of the sides BC, CA and AB of a   ABC.

Show that

(i) BDEF is a parallelogram. 1
(ii) ar (DEF) = ar (ABC)

4

1
(iii) ar (BDEF) = ar (ABC)

2

6. In Fig. 9.25, diagonals AC and BD of quadrilateral

ABCD intersect at O such that OB = OD.

If AB = CD, then show that:

(i) ar (DOC) = ar (AOB)

(ii) ar (DCB) = ar (ACB)

(iii) DA || CB or ABCD is a parallelogram.

[Hint : From D and B, draw perpendiculars to AC.] Fig. 9.25

7. D and E are points on sides AB and AC respectively of   ABC such that

ar (DBC) = ar (EBC). Prove that DE || BC.

8. XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E
and F respectively, show that

ar (ABE) = ar (ACF)

9. The sideAB of a parallelogramABCD is produced
to any point P. A line through A and parallel to CP
meets CB produced at Q and then parallelogram
PBQR is completed (see Fig. 9.26). Show that
ar (ABCD) = ar (PBQR).

[Hint : Join AC and PQ. Now compare ar (ACQ)
and ar (APQ).]

Fig. 9.26

10. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O.
Prove that ar (AOD) = ar (BOC).

11. In Fig. 9.27, ABCDE is a pentagon. A line through
B parallel to AC meets DC produced at F. Show
that

(i) ar (ACB) = ar (ACF)

(ii) ar (AEDF) = ar (ABCDE)

Fig. 9.27

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164 MATHEMATICS

12. A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat
of the village decided to take over some portion of his plot from one of the corners to
construct a Health Centre. Itwaari agrees to the above proposal with the condition
that he should be given equal amount of land in lieu of his land adjoining his plot so
as to form a triangular plot. Explain how this proposal will be implemented.

13. ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC
at Y. Prove that ar (ADX) = ar (ACY).

[Hint : Join CX.]

14. In Fig.9.28, AP || BQ || CR. Prove that

ar (AQC) = ar (PBR).

15. Diagonals AC and BD of a quadrilateral Fig. 9.28
ABCD intersect at O in such a way that
ar (AOD) = ar (BOC). Prove that ABCD is a
trapezium.

16. In Fig.9.29, ar (DRC) = ar (DPC) and
ar (BDP) = ar (ARC). Show that both the
quadrilaterals ABCD and DCPR are
trapeziums.

Fig. 9.29

EXERCISE 9.4 (Optional)*

1. Parallelogram ABCD and rectangle ABEF are on the same base AB and have equal
areas. Show that the perimeter of the parallelogram is greater than that of the rectangle.

2. In Fig. 9.30, D and E are two points on BC
such that BD = DE = EC. Show that
ar (ABD) = ar (ADE) = ar (AEC).
Can you now answer the question that you have
left in the ‘Introduction’ of this chapter, whether
the field of Budhia has been actually divided
into three parts of equal area?
Fig. 9.30

*These exercises are not from examination point of view.

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AREAS OF PARALLELOGRAMS AND TRIANGLES 165

[Remark: Note that by taking BD = DE = EC, the triangle ABC is divided into three
triangles ABD, ADE and AEC of equal areas. In the same way, by dividing BC into n
equal parts and joining the points of division so obtained to the opposite vertex of

BC, you can divide ✁ABC into n triangles of equal areas.]

3. In Fig. 9.31, ABCD, DCFE and ABFE are parallelograms. Show that
ar (ADE) = ar (BCF).

4. In Fig. 9.32, ABCD is a parallelogram and BC is produced to a point Q such that
AD = CQ. If AQ intersect DC at P, show that ar (BPC) = ar (DPQ).

[Hint : Join AC.]

Fig. 9.31 Fig. 9.32

5. In Fig.9.33, ABC and BDE are two equilateral
triangles such that D is the mid-point of BC. If AE
intersects BC at F, show that

1
(i) ar (BDE) = ar (ABC)

4

1
(ii) ar (BDE) = ar (BAE)

2

(iii) ar (ABC) = 2 ar (BEC) Fig. 9.33
(iv) ar (BFE) = ar (AFD)
(v) ar (BFE) = 2 ar (FED)

1
(vi) ar (FED) = 8 ar (AFC)

[Hint : Join EC and AD. Show that BE || AC and DE || AB, etc.]

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166 MATHEMATICS

6. Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that
ar (APB) × ar (CPD) = ar (APD) × ar (BPC).

[Hint : From A and C, draw perpendiculars to BD.]

7. P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R
is the mid-point of AP, show that

1 3
(i) ar (PRQ) = ar (ARC) (ii) ar (RQC) = 8 ar (ABC)

2

(iii) ar (PBQ) = ar (ARC)

8. In Fig. 9.34, ABC is a right triangle right angled at A. BCED, ACFG and ABMN are

squares on the sides BC, CA and AB respectively. Line segment AX   DE meets BC

at Y. Show that:

Fig. 9.34

(i) ✁ MBC ✂ ✁ ABD (ii) ar (BYXD) = 2 ar (MBC)

(iii) ar (BYXD) = ar (ABMN) (iv) ✁ FCB ✂ ✁ ACE

(v) ar (CYXE) = 2 ar (FCB) (vi) ar (CYXE) = ar (ACFG)

(vii) ar (BCED) = ar (ABMN) + ar (ACFG)

Note : Result (vii) is the famous Theorem of Pythagoras. You shall learn a simpler
proof of this theorem in Class X.

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AREAS OF PARALLELOGRAMS AND TRIANGLES 167

9.5 Summary

In this chapter, you have studied the following points :

1. Area of a figure is a number (in some unit) associated with the part of the plane enclosed
by that figure.

2. Two congruent figures have equal areas but the converse need not be true.

3. If a planar region formed by a figure T is made up of two non-overlapping planar regions
formed by figures P and Q, then ar (T) = ar (P) + ar (Q), where ar (X) denotes the area of
figure X.

4. Two figures are said to be on the same base and between the same parallels, if they have
a common base (side) and the vertices, (or the vertex) opposite to the common base of
each figure lie on a line parallel to the base.

5. Parallelograms on the same base (or equal bases) and between the same parallels are
equal in area.

6. Area of a parallelogram is the product of its base and the corresponding altitude.

7. Parallelograms on the same base (or equal bases) and having equal areas lie between the
same parallels.

8. If a parallelogram and a triangle are on the same base and between the same parallels, then
area of the triangle is half the area of the parallelogram.

9. Triangles on the same base (or equal bases) and between the same parallels are equal in
area.

10. Area of a triangle is half the product of its base and the corresponding altitude.

11. Triangles on the same base (or equal bases) and having equal areas lie between the same
parallels.

12. A median of a triangle divides it into two triangles of equal areas.

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CHAPTER 10

CIRCLES

10.1 Introduction

You may have come across many objects in daily life, which are round in shape, such
as wheels of a vehicle, bangles, dials of many clocks, coins of denominations 50 p,
Re 1 and Rs 5, key rings, buttons of shirts, etc. (see Fig.10.1). In a clock, you might
have observed that the second’s hand goes round the dial of the clock rapidly and its
tip moves in a round path. This path traced by the tip of the second’s hand is called a
circle. In this chapter, you will study about circles, other related terms and some
properties of a circle.

Fig. 10.1

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CIRCLES 169

10.2 Circles and Its Related Terms: A Review

Take a compass and fix a pencil in it. Put its pointed
leg on a point on a sheet of a paper. Open the other
leg to some distance. Keeping the pointed leg on the
same point, rotate the other leg through one revolution.
What is the closed figure traced by the pencil on
paper? As you know, it is a circle (see Fig.10.2). How
did you get a circle? You kept one point fixed (A in
Fig.10.2) and drew all the points that were at a fixed
distance from A. This gives us the following definition:

The collection of all the points in a plane, Fig. 10.2
which are at a fixed distance from a fixed point in
the plane, is called a circle.

The fixed point is called the centre of the circle
and the fixed distance is called the radius of the
circle. In Fig.10.3, O is the centre and the length OP
is the radius of the circle.

Remark : Note that the line segment joining the Fig. 10.3
centre and any point on the circle is also called a
radius of the circle. That is, ‘radius’ is used in two
senses-in the sense of a line segment and also in the
sense of its length.

You are already familiar with some of the
following concepts from Class VI. We are just
recalling them.

A circle divides the plane on which it lies into Fig. 10.4
three parts. They are: (i) inside the circle, which is
also called the interior of the circle; (ii) the circle
and (iii) outside the circle, which is also called the
exterior of the circle (see Fig.10.4). The circle and
its interior make up the circular region.

If you take two points P and Q on a circle, then the line segment PQ is called a
chord of the circle (see Fig. 10.5). The chord, which passes through the centre of the
circle, is called a diameter of the circle. As in the case of radius, the word ‘diameter’
is also used in two senses, that is, as a line segment and also as its length. Do you find
any other chord of the circle longer than a diameter? No, you see that a diameter is
the longest chord and all diameters have the same length, which is equal to two

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170 MATHEMATICS

times the radius. In Fig.10.5, AOB is a diameter of
the circle. How many diameters does a circle have?
Draw a circle and see how many diameters you can
find.

A piece of a circle between two points is called Fig. 10.5
an arc. Look at the pieces of the circle between two Fig. 10.6
points P and Q in Fig.10.6. You find that there are
two pieces, one longer and the other smaller
(see Fig.10.7). The longer one is called the major
arc PQ and the shorter one is called the minor arc

 PQ. The minor arc PQ is also denoted by PQ and
✁the major arc PQ by PRQ , where R is some point on
✂the arc between P and Q. Unless otherwise stated,

arc PQ or PQ stands for minor arc PQ. When P and

Q are ends of a diameter, then both arcs are equal
and each is called a semicircle.

The length of the complete circle is called its

circumference. The region between a chord and

either of its arcs is called a segment of the circular

region or simply a segment of the circle. You will find

that there are two types of segments also, which are

the major segment and the minor segment

(see Fig. 10.8). The region between an arc and the

two radii, joining the centre to the end points of the Fig. 10.7
arc is called a sector. Like segments, you find that

the minor arc corresponds to the minor sector and the major arc corresponds to the

major sector. In Fig. 10.9, the region OPQ is the minor sector and remaining part of

the circular region is the major sector. When two arcs are equal, that is, each is a

semicircle, then both segments and both sectors become the same and each is known

as a semicircular region.

Fig. 10.8 Fig. 10.9

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CIRCLES 171

EXERCISE 10.1

1. Fill in the blanks:

(i) The centre of a circle lies in of the circle. (exterior/ interior)

(ii) A point, whose distance from the centre of a circle is greater than its radius lies in
of the circle. (exterior/ interior)

(iii) The longest chord of a circle is a of the circle.

(iv) An arc is a when its ends are the ends of a diameter.

(v) Segment of a circle is the region between an arc and of the circle.

(vi) A circle divides the plane, on which it lies, in parts.

2. Write True or False: Give reasons for your answers.

(i) Line segment joining the centre to any point on the circle is a radius of the circle.

(ii) A circle has only finite number of equal chords.

(iii) If a circle is divided into three equal arcs, each is a major arc.

(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.

(v) Sector is the region between the chord and its corresponding arc.

(vi) A circle is a plane figure.

10.3 Angle Subtended by a Chord at a Point

Take a line segment PQ and a point R not on the line containing PQ. Join PR and QR

✁(see Fig. 10.10). Then PRQ is called the angle subtended by the line segment PQ
✁at the point R. What are angles POQ, PRQ and PSQ called in Fig. 10.11? POQ is
✁ ✁the angle subtended by the chord PQ at the centre O, PRQ and PSQ are

respectively the angles subtended by PQ at points R and S on the major and minor
arcs PQ.

Fig. 10.10 Fig. 10.11

Let us examine the relationship between the size of the chord and the angle
subtended by it at the centre. You may see by drawing different chords of a circle and

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172 MATHEMATICS

angles subtended by them at the centre that the longer Fig. 10.12
is the chord, the bigger will be the angle subtended
by it at the centre. What will happen if you take two
equal chords of a circle? Will the angles subtended at
the centre be the same or not?

Draw two or more equal chords of a circle and
measure the angles subtended by them at the centre
(see Fig.10.12). You will find that the angles subtended
by them at the centre are equal. Let us give a proof
of this fact.

Theorem 10.1 : Equal chords of a circle subtend equal angles at the centre.

Proof : You are given two equal chords AB and CD
of a circle with centre O (see Fig.10.13). You want

to prove that ✁ AOB = ✁ COD.

In triangles AOB and COD,

OA = OC (Radii of a circle)

OB = OD (Radii of a circle)

AB = CD (Given)

Therefore, ✂ AOB ✄ ✂ COD (SSS rule)

This gives ✁ AOB = ✁ COD Fig. 10.13

(Corresponding parts of congruent triangles)  

Remark : For convenience, the abbreviation CPCT will be used in place of
‘Corresponding parts of congruent triangles’, because we use this very frequently as
you will see.

Now if two chords of a circle subtend equal angles at the centre, what can you
say about the chords? Are they equal or not? Let us examine this by the following
activity:

Take a tracing paper and trace a circle on it. Cut Fig. 10.14
it along the circle to get a disc. At its centre O, draw
an angle AOB where A, B are points on the circle.
Make another angle POQ at the centre equal to

✁AOB. Cut the disc along AB and PQ

(see Fig. 10.14). You will get two segments ACB
and PRQ of the circle. If you put one on the other,
what do you observe? They cover each other, i.e.,
they are congruent. So AB = PQ.

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CIRCLES 173

Though you have seen it for this particular case, try it out for other equal angles
too. The chords will all turn out to be equal because of the following theorem:

Theorem 10.2 : If the angles subtended by the chords of a circle at the centre
are equal, then the chords are equal.

The above theorem is the converse of the Theorem 10.1. Note that in Fig. 10.13,

if you take ✁ AOB = ✁ COD, then
✥ AOB ✄ ✥ COD (Why?)

Can you now see that AB = CD?

EXERCISE 10.2

1. Recall that two circles are congruent if they have the same radii. Prove that equal
chords of congruent circles subtend equal angles at their centres.

2. Prove that if chords of congruent circles subtend equal angles at their centres, then
the chords are equal.

10.4 Perpendicular from the Centre to a Chord

Activity : Draw a circle on a tracing paper. Let O
be its centre. Draw a chord AB. Fold the paper along
a line through O so that a portion of the chord falls on
the other. Let the crease cut AB at the point M. Then,

✁ OMA = ✁ OMB = 90° or OM is perpendicular to

AB. Does the point B coincide with A (see Fig.10.15)?

Yes it will. So MA = MB. Fig. 10.15

Give a proof yourself by joining OA and OB and proving the right triangles OMA
and OMB to be congruent. This example is a particular instance of the following
result:

Theorem 10.3 : The perpendicular from the centre of a circle to a chord bisects
the chord.

What is the converse of this theorem? To write this, first let us be clear what is
assumed in Theorem 10.3 and what is proved. Given that the perpendicular from the
centre of a circle to a chord is drawn and to prove that it bisects the chord. Thus in the
converse, what the hypothesis is ‘if a line from the centre bisects a chord of a
circle’ and what is to be proved is ‘the line is perpendicular to the chord’. So the
converse is:

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174 MATHEMATICS

Theorem 10.4 : The line drawn through the centre of a circle to bisect a chord is
perpendicular to the chord.

Is this true? Try it for few cases and see. You will
see that it is true for these cases. See if it is true, in
general, by doing the following exercise. We will write
the stages and you give the reasons.

Let AB be a chord of a circle with centre O and

O is joined to the mid-point M of AB. You have to

☎prove that OM AB. Join OA and OB

(see Fig. 10.16). In triangles OAM and OBM,

OA = OB (Why ?) Fig. 10.16

AM = BM (Why ?)

OM = OM (Common)

Therefore, ✂OAM ✄ ✂OBM (How ?)

This gives ✁OMA = ✁OMB = 90° (Why ?)

10.5 Circle through Three Points

You have learnt in Chapter 6, that two points are sufficient to determine a line. That is,
there is one and only one line passing through two points. A natural question arises.
How many points are sufficient to determine a circle?

Take a point P. How many circles can be drawn through this point? You see that
there may be as many circles as you like passing through this point [see Fig. 10.17(i)].
Now take two points P and Q. You again see that there may be an infinite number of
circles passing through P and Q [see Fig.10.17(ii)]. What will happen when you take
three points A, B and C? Can you draw a circle passing through three collinear points?

Fig. 10. 17

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CIRCLES 175

No. If the points lie on a line, then the third point will
lie inside or outside the circle passing through two
points (see Fig 10.18).

Fig. 10.18

So, let us take three points A, B and C, which are not on the same line, or in other
words, they are not collinear [see Fig. 10.19(i)]. Draw perpendicular bisectors of AB
and BC say, PQ and RS respectively. Let these perpendicular bisectors intersect at
one point O. (Note that PQ and RS will intersect because they are not parallel) [see
Fig. 10.19(ii)].

(i) (ii)

Fig. 10.19

Now O lies on the perpendicular bisector PQ of AB, you have OA = OB, as every
point on the perpendicular bisector of a line segment is equidistant from its end points,
proved in Chapter 7.

Similarly, as O lies on the perpendicular bisector RS of BC, you get
OB = OC

So OA = OB = OC, which means that the points A, B and C are at equal distances
from the point O. So if you draw a circle with centre O and radius OA, it will also pass
through B and C. This shows that there is a circle passing through the three points A,
B and C. You know that two lines (perpendicular bisectors) can intersect at only one
point, so you can draw only one circle with radius OA. In other words, there is a
unique circle passing through A, B and C. You have now proved the following theorem:
Theorem 10.5 : There is one and only one circle passing through three given
non-collinear points.

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176 MATHEMATICS

Remark : If ABC is a triangle, then by Theorem 10.5, there is a unique circle passing

through the three vertices A, B and C of the triangle. This circle is called the

circumcircle of the ✂ ABC. Its centre and radius are called respectively the

circumcentre and the circumradius of the triangle.

Example 1 : Given an arc of a circle, complete the circle.

Solution : Let arc PQ of a circle be given. We have
to complete the circle, which means that we have to
find its centre and radius. Take a point R on the arc.
Join PR and RQ. Use the construction that has been
used in proving Theorem 10.5, to find the centre and
radius.

Taking the centre and the radius so obtained, we
can complete the circle (see Fig. 10.20).

Fig. 10.20

EXERCISE 10.3

1. Draw different pairs of circles. How many points does each pair have in common?
What is the maximum number of common points?

2. Suppose you are given a circle. Give a construction to find its centre.

3. If two circles intersect at two points, prove that their centres lie on the perpendicular
bisector of the common chord.

10.6 Equal Chords and Their Distances from the Centre

Let AB be a line and P be a point. Since there are Fig. 10.21
infinite numbers of points on a line, if you join these
points to P, you will get infinitely many line segments
PL1, PL2, PM, PL3, PL4, etc. Which of these is the
distance of AB from P? You may think a while and
get the answer. Out of these line segments, the
perpendicular from P to AB, namely PM in Fig. 10.21,
will be the least. In Mathematics, we define this least
length PM to be the distance of AB from P. So you
may say that:

The length of the perpendicular from a point to a line is the distance of the
line from the point.

Note that if the point lies on the line, the distance of the line from the point is zero.

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A circle can have infinitely many chords. You may observe by drawing chords of
a circle that longer chord is nearer to the centre than the smaller chord. You may
observe it by drawing several chords of a circle of different lengths and measuring
their distances from the centre. What is the distance of the diameter, which is the
longest chord from the centre? Since the centre lies on it, the distance is zero. Do you
think that there is some relationship between the length of chords and their distances
from the centre? Let us see if this is so.

Fig. 10.22

Activity : Draw a circle of any radius on a tracing paper. Draw two equal chords AB
and CD of it and also the perpendiculars OM and ON on them from the centre O. Fold
the figure so that D falls on B and C falls on A [see Fig.10.22 (i)]. You may observe
that O lies on the crease and N falls on M. Therefore, OM = ON. Repeat the activity
by drawing congruent circles with centres O and O✆ and taking equal chords AB and
CD one on each. Draw perpendiculars OM and O✆N on them [see Fig. 10.22(ii)]. Cut
one circular disc and put it on the other so that AB coincides with CD. Then you will
find that O coincides with O✆ and M coincides with N. In this way you verified the
following:

Theorem 10.6 : Equal chords of a circle (or of congruent circles) are equidistant
from the centre (or centres).

Next, it will be seen whether the converse of this theorem is true or not. For this,
draw a circle with centre O. From the centre O, draw two line segments OL and OM
of equal length and lying inside the circle [see Fig. 10.23(i)]. Then draw chords PQ
and RS of the circle perpendicular to OL and OM respectively [see Fig 10.23(ii)].
Measure the lengths of PQ and RS. Are these different? No, both are equal. Repeat
the activity for more equal line segments and drawing the chords perpendicular to

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178 MATHEMATICS

(i) (ii)

Fig. 10.23

them. This verifies the converse of the Theorem 10.6 which is stated as follows:

Theorem 10.7 : Chords equidistant from the centre of a circle are equal in
length.

We now take an example to illustrate the use of the above results:

Example 2 : If two intersecting chords of a circle make equal angles with the diameter
passing through their point of intersection, prove that the chords are equal.

Solution : Given that AB and CD are two chords of Fig. 10.24

a circle, with centre O intersecting at a point E. PQ (Why ?)
(Proved above)
is a diameter through E, such that ✁ AEQ = ✁ DEQ

(see Fig.10.24). You have to prove that AB = CD.

Draw perpendiculars OL and OM on chords AB and

CD respectively. Now

✁ LOE = 180° – 90° – ✁ LEO = 90° – ✁ LEO

(Angle sum property of a triangle)

= 90° – ✁ AEQ = 90° – ✁ DEQ
= 90° – ✁ MEO = ✁ MOE

In triangles OLE and OME,

✁ LEO = ✁ MEO
✁ LOE = ✁ MOE

Therefore, EO = EO (Common)
(Why ?)
✂ OLE ✄ ✂ OME

This gives OL = OM (CPCT)

So, AB = CD (Why ?)

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EXERCISE 10.4

1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between
their centres is 4 cm. Find the length of the common chord.

2. If two equal chords of a circle intersect within the circle, prove that the segments of
one chord are equal to corresponding segments of the other chord.

3. If two equal chords of a circle intersect within the circle, prove that the line
joining the point of intersection to the centre makes equal angles with the chords.

4. If a line intersects two concentric circles (circles
with the same centre) with centre O at A, B, C and
D, prove that AB = CD (see Fig. 10.25).

5. Three girls Reshma, Salma and Mandip are Fig. 10.25
playing a game by standing on a circle of radius
5m drawn in a park. Reshma throws a ball to
Salma, Salma to Mandip, Mandip to Reshma. If
the distance between Reshma and Salma and
between Salma and Mandip is 6m each, what is
the distance between Reshma and Mandip?

6. A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and
David are sitting at equal distance on its boundary each having a toy telephone in
his hands to talk each other. Find the length of the string of each phone.

10.7 Angle Subtended by an Arc of a Circle

You have seen that the end points of a chord other than diameter of a circle cuts it into
two arcs – one major and other minor. If you take two equal chords, what can you say
about the size of arcs? Is one arc made by first chord equal to the corresponding arc
made by another chord? In fact, they are more than just equal in length. They are
congruent in the sense that if one arc is put on the other, without bending or twisting,
one superimposes the other completely.

You can verify this fact by cutting the arc, Fig. 10.26
corresponding to the chord CD from the circle along
CD and put it on the corresponding arc made by equal
chord AB. You will find that the arc CD superimpose
the arc AB completely (see Fig. 10.26). This shows
that equal chords make congruent arcs and
conversely congruent arcs make equal chords of a
circle. You can state it as follows:

If two chords of a circle are equal, then their corresponding arcs are congruent
and conversely, if two arcs are congruent, then their corresponding chords are
equal.

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180 MATHEMATICS

Also the angle subtended by an arc at the centre
is defined to be angle subtended by the corresponding
chord at the centre in the sense that the minor arc
subtends the angle and the major arc subtends the
reflex angle. Therefore, in Fig 10.27, the angle

✁subtended by the minor arc PQ at O is POQ and

the angle subtended by the major arc PQ at O is
reflex angle POQ.

In view of the property above and Theorem 10.1, Fig. 10.27
the following result is true:

Congruent arcs (or equal arcs) of a circle subtend equal angles at the centre.

Therefore, the angle subtended by a chord of a circle at its centre is equal to the
angle subtended by the corresponding (minor) arc at the centre. The following theorem
gives the relationship between the angles subtended by an arc at the centre and at a
point on the circle.

Theorem 10.8 : The angle subtended by an arc at the centre is double the angle
subtended by it at any point on the remaining part of the circle.

Proof : Given an arc PQ of a circle subtending angles POQ at the centre O and
PAQ at a point A on the remaining part of the circle. We need to prove that

✁ POQ = 2 ✁ PAQ.

Fig. 10.28

Consider the three different cases as given in Fig. 10.28. In (i), arc PQ is minor; in (ii),
arc PQ is a semicircle and in (iii), arc PQ is major.

Let us begin by joining AO and extending it to a point B.

In all the cases, ✁ BOQ = ✁ OAQ + ✁ AQO

because an exterior angle of a triangle is equal to the sum of the two interior opposite
angles.

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CIRCLES 181

Also in ✂ OAQ,

OA = OQ (Radii of a circle)
(Theorem 7.5)
Therefore, ✁ OAQ = ✁ OQA (1)
(2)
This gives ✁ BOQ = 2 ✁ OAQ
(3)
Similarly, ✁ BOP = 2 ✁ OAP

From (1) and (2), ✁ BOP + ✁ BOQ = 2(✁ OAP + ✁ OAQ)

This is the same as ✁ POQ = 2 ✁ PAQ

For the case (iii), where PQ is the major arc, (3) is replaced by

reflex angle POQ = 2 ✁ PAQ  

Remark : Suppose we join points P and Q and

form a chord PQ in the above figures. Then

✁ PAQ is also called the angle formed in the

segment PAQP.

In Theorem 10.8, A can be any point on the

remaining part of the circle. So if you take any

other point C on the remaining part of the circle

(see Fig. 10.29), you have

Therefore, ✁ POQ = 2 ✁ PCQ = 2 ✁ PAQ Fig. 10.29
✁ PCQ = ✁ PAQ.

This proves the following:

Theorem 10.9 : Angles in the same segment of a circle are equal.

Again let us discuss the case (ii) of Theorem 10.8 separately. Here ✁PAQ is an angle
in the segment, which is a semicircle. Also, ✁ PAQ = 1 ✁ POQ = 1 × 180° = 90°.

22
If you take any other point C on the semicircle, again you get that

✁ PCQ = 90°

Therefore, you find another property of the circle as:

Angle in a semicircle is a right angle.

The converse of Theorem 10.9 is also true. It can be stated as:

Theorem 10.10 : If a line segment joining two points subtends equal angles at
two other points lying on the same side of the line containing the line segment,
the four points lie on a circle (i.e. they are concyclic).

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182 MATHEMATICS

You can see the truth of this result as follows:

In Fig. 10.30, AB is a line segment, which subtends equal angles at two points C and
D. That is

✁ ACB = ✁ ADB

To show that the points A, B, C and D lie on a circle
let us draw a circle through the points A, C and B.
Suppose it does not pass through the point D. Then it
will intersect AD (or extended AD) at a point, say E

(or E✆).

If points A, C, E and B lie on a circle,

✁ ACB = ✁ AEB (Why?)

But it is given that ✁ ACB = ✁ ADB. Fig. 10.30

Therefore, ✁ AEB = ✁ ADB.

This is not possible unless E coincides with D. (Why?)

✆Similarly, E should also coincide with D.

10.8 Cyclic Quadrilaterals Fig. 10.31

A quadrilateral ABCD is called cyclic if all the four vertices
of it lie on a circle (see Fig 10.31). You will find a peculiar
property in such quadrilaterals. Draw several cyclic
quadrilaterals of different sides and name each of these
as ABCD. (This can be done by drawing several circles
of different radii and taking four points on each of them.)
Measure the opposite angles and write your observations
in the following table.

S.No. of Quadrilateral ✁A ✁B ✁C ✁D ✁ A +✁ C ✁ B +✁ D
1.
2.
3.
4.
5.
6.

What do you infer from the table?

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CIRCLES 183

You find that ✁A + ✁C = 180° and ✁B + ✁D = 180°, neglecting the error in

measurements. This verifies the following:

Theorem 10.11 : The sum of either pair of opposite angles of a cyclic
quadrilateral is 180º.

In fact, the converse of this theorem, which is stated below is also true.

Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is
180º, the quadrilateral is cyclic.

You can see the truth of this theorem by following a method similar to the method
adopted for Theorem 10.10.

Example 3 : In Fig. 10.32, AB is a diameter of the circle, CD is a chord equal to the
radius of the circle. AC and BD when extended intersect at a point E. Prove that

✁ AEB = 60°.

Solution : Join OC, OD and BC.

Triangle ODC is equilateral (Why?)

Therefore, ✁ COD = 60°

Now, ✁ CBD = 1 ✁ COD (Theorem 10.8)
This gives 2

✁ CBD = 30°

Again, ✁ ACB = 90° (Why ?)

So, ✁ BCE = 180° – ✁ ACB = 90°

Which gives ✁ CEB = 90° – 30° = 60°, i.e. ✁ AEB = 60° Fig. 10.32

Example 4 : In Fig 10.33, ABCD is a cyclic

quadrilateral in which AC and BD are its diagonals.

If ✁ DBC = 55° and ✁ BAC = 45°, find ✁ BCD.

Solution : ✁ CAD = ✁ DBC = 55°

(Angles in the same segment)

Therefore, ✁ DAB = ✁ CAD + ✁ BAC

= 55° + 45° = 100°

But ✁ DAB + ✁ BCD = 180°

(Opposite angles of a cyclic quadrilateral) Fig. 10.33

So, ✁ BCD = 180° – 100° = 80°

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184 MATHEMATICS

Example 5 : Two circles intersect at two points A
and B. AD and AC are diameters to the two circles
(see Fig.10.34). Prove that B lies on the line segment
DC.

Solution : Join AB. Fig. 10.34

✁ ABD = 90° (Angle in a semicircle)
✁ ABC = 90° (Angle in a semicircle)
So, ✁ ABD + ✁ ABC = 90° + 90° = 180°

Therefore, DBC is a line. That is B lies on the line segment DC.

Example 6 : Prove that the quadrilateral formed (if possible) by the internal angle
bisectors of any quadrilateral is cyclic.

Solution : In Fig. 10.35, ABCD is a quadrilateral in

which the angle bisectors AH, BF, CF and DH of
internal angles A, B, C and D respectively form a

quadrilateral EFGH.

Now, ✁ FEH = ✁ AEB = 180° – ✁ EAB – ✁ EBA (Why ?)

= 180° – 1 (✁ A + ✁ B) Fig. 10.35

2

and ✁ FGH = ✁ CGD = 180° – ✁ GCD – ✁ GDC (Why ?)

= 180° – 1 (✁ C + ✁ D)
2

Therefore, ✁ FEH + ✁ FGH = 180° – 1 (✁ A + ✁ B) + 180° – 1 (✁ C + ✁ D)

22

= 360° – 1 (✁ A+ ✁ B +✁ C +✁ D) = 360° – 1 × 360°
2 2

= 360° – 180° = 180°

Therefore, by Theorem 10.12, the quadrilateral EFGH is cyclic.

EXERCISE 10.5

1. In Fig. 10.36, A,B and C are three points on a

 circle with centre O such that BOC = 30° and
  AOB = 60°. If D is a point on the circle other
 than the arc ABC, find ADC.

Fig. 10.36

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CIRCLES 185

2. A chord of a circle is equal to the radius of the
circle. Find the angle subtended by the chord at
a point on the minor arc and also at a point on the
major arc.

✁3. In Fig. 10.37, PQR = 100°, where P, Q and R are
✁points on a circle with centre O. Find OPR.

Fig. 10.37

4. In Fig. 10.38, ✁ ABC = 69°, ✁ ACB = 31°, find
✁ BDC.

Fig. 10.38

5. In Fig. 10.39, A, B, C and D are four points on a
circle. AC and BD intersect at a point E such

that ✁ BEC = 130° and ✁ ECD = 20°. Find
✁ BAC.

Fig. 10.39

✁6. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If DBC = 70°,
✁ BAC is 30°, find ✁ BCD. Further, if AB = BC, find ✁ ECD.

7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of
the quadrilateral, prove that it is a rectangle.

8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

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186 MATHEMATICS

9. Two circles intersect at two points B and C.
Through B, two line segments ABD and PBQ
are drawn to intersect the circles at A, D and P,
Q respectively (see Fig. 10.40). Prove that

✁ ACP = ✁ QCD.

Fig. 10.40

10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of
intersection of these circles lie on the third side.

11. ABC and ADC are two right triangles with common hypotenuse AC. Prove that

✁ CAD = ✁ CBD.

12. Prove that a cyclic parallelogram is a rectangle.

EXERCISE 10.6 (Optional)*

1. Prove that the line of centres of two intersecting circles subtends equal angles at the
two points of intersection.

2. Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel
to each other and are on opposite sides of its centre. If the distance between AB and
CD is 6 cm, find the radius of the circle.

3. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is
at distance 4 cm from the centre, what is the distance of the other chord from the
centre?

4. Let the vertex of an angle ABC be located outside a circle and let the sides of the angle

✁intersect equal chords AD and CE with the circle. Prove that ABC is equal to half the

difference of the angles subtended by the chords AC and DE at the centre.
5. Prove that the circle drawn with any side of a rhombus as diameter, passes through

the point of intersection of its diagonals.
6. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if

necessary) at E. Prove that AE = AD.
7. AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are

diameters, (ii) ABCD is a rectangle.
8. Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and

11
F respectively. Prove that the angles of the triangle DEF are 90° – A, 90° – B and

1 22
90° – C.

2

*These exercises are not from examination point of view.

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CIRCLES 187

9. Two congruent circles intersect each other at points A and B. Through A any line

segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

✁10. In any triangle ABC, if the angle bisector of A and perpendicular bisector of BC

intersect, prove that they intersect on the circumcircle of the triangle ABC.

10.9 Summary

In this chapter, you have studied the following points:

1. A circle is the collection of all points in a plane, which are equidistant from a fixed point in
the plane.

2. Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.
3. If the angles subtended by two chords of a circle (or of congruent circles) at the centre

(corresponding centres) are equal, the chords are equal.

4. The perpendicular from the centre of a circle to a chord bisects the chord.

5. The line drawn through the centre of a circle to bisect a chord is perpendicular to the
chord.

6. There is one and only one circle passing through three non-collinear points.

7. Equal chords of a circle (or of congruent circles) are equidistant from the centre (or
corresponding centres).

8. Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent
circles) are equal.

9. If two arcs of a circle are congruent, then their corresponding chords are equal and
conversely if two chords of a circle are equal, then their corresponding arcs (minor, major)
are congruent.

10. Congruent arcs of a circle subtend equal angles at the centre.

11. The angle subtended by an arc at the centre is double the angle subtended by it at any
point on the remaining part of the circle.

12. Angles in the same segment of a circle are equal.

13. Angle in a semicircle is a right angle.

14. If a line segment joining two points subtends equal angles at two other points lying on
the same side of the line containing the line segment, the four points lie on a circle.

15. The sum of either pair of opposite angles of a cyclic quadrilateral is 1800.
16. If sum of a pair of opposite angles of a quadrilateral is 1800, the quadrilateral is cyclic.

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CHAPTER 11

CONSTRUCTIONS

11.1 Introduction

In earlier chapters, the diagrams, which were necessary to prove a theorem or solving
exercises were not necessarily precise. They were drawn only to give you a feeling for
the situation and as an aid for proper reasoning. However, sometimes one needs an
accurate figure, for example - to draw a map of a building to be constructed, to design
tools, and various parts of a machine, to draw road maps etc. To draw such figures
some basic geometrical instruments are needed. You must be having a geometry box
which contains the following:

(i) A graduated scale, on one side of which centimetres and millimetres are
marked off and on the other side inches and their parts are marked off.

(ii) A pair of set - squares, one with angles 90°, 60° and 30° and other with angles
90°, 45° and 45°.

(iii) A pair of dividers (or a divider) with adjustments.

(iv) A pair of compasses (or a compass) with provision of fitting a pencil at one
end.

(v) A protractor.

Normally, all these instruments are needed in drawing a geometrical figure, such
as a triangle, a circle, a quadrilateral, a polygon, etc. with given measurements. But a
geometrical construction is the process of drawing a geometrical figure using only two
instruments – an ungraduated ruler, also called a straight edge and a compass. In
construction where measurements are also required, you may use a graduated scale
and protractor also. In this chapter, some basic constructions will be considered. These
will then be used to construct certain kinds of triangles.

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CONSTRUCTIONS 189

11.2 Basic Constructions

In Class VI, you have learnt how to construct a circle, the perpendicular bisector of a
line segment, angles of 30°, 45°, 60°, 90° and 120°, and the bisector of a given angle,
without giving any justification for these constructions. In this section, you will construct
some of these, with reasoning behind, why these constructions are valid.

Construction 11.1 : To construct the bisector of a given angle.

Given an angle ABC, we want to construct its bisector.

Steps of Construction :

1. Taking B as centre and any radius, draw an arc to intersect the rays BA and BC,
say at E and D respectively [see Fig.11.1(i)].

1
2. Next, taking D and E as centres and with the radius more than 2 DE, draw arcs to

intersect each other, say at F.

3. Draw the ray BF [see Fig.11.1(ii)]. This ray BF is the required bisector of the angle
ABC.

Fig. 11.1

Let us see how this method gives us the required angle bisector.

Join DF and EF.

In triangles BEF and BDF,

BE = BD (Radii of the same arc)

EF = DF (Arcs of equal radii)

BF = BF (Common)
(SSS rule)
Therefore, ✁BEF ✂ ✁BDF
This gives ✄EBF = ✄ DBF (CPCT)

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190 MATHEMATICS

Construction 11.2 : To construct the perpendicular bisector of a given line
segment.

Given a line segment AB, we want to construct its perpendicular bisector.

Steps of Construction :

1. Taking A and B as centres and radius more than

1
AB, draw arcs on both sides of the line segment

2
AB (to intersect each other).

2. Let these arcs intersect each other at P and Q.
Join PQ (see Fig.11.2).

3. Let PQ intersect AB at the point M. Then line
PMQ is the required perpendicular bisector ofAB.

Let us see how this method gives us the
perpendicular bisector of AB.

Join A and B to both P and Q to form AP, AQ, BP Fig. 11.2
and BQ.

In triangles PAQ and PBQ,

AP = BP (Arcs of equal radii)

AQ = BQ (Arcs of equal radii)

Therefore, PQ = PQ (Common)
So, (SSS rule)
✁ PAQ ✂ ✁ PBQ
✄ APM = ✄ BPM (CPCT)

Now in triangles PMA and PMB,

AP = BP (As before)

Therefore, PM = PM (Common)
So, (Proved above)
As ✄ APM = ✄ BPM
✁ PMA ✂ ✁ PMB (SAS rule)
AM = BM and ✄ PMA = ✄ PMB (CPCT)
✄ PMA + ✄ PMB = 180°
(Linear pair axiom),

we get

✄ PMA = ✄ PMB = 90°.

Therefore, PM, that is, PMQ is the perpendicular bisector of AB.

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CONSTRUCTIONS 191

Construction 11.3 : To construct an angle of 600 at the initial point of a given
ray.

Let us take a ray AB with initial point A [see Fig. 11.3(i)]. We want to construct a ray

AC such that ✄ CAB = 60°. One way of doing so is given below.

Steps of Construction :

1. Taking A as centre and some radius, draw an arc
of a circle, which intersects AB, say at a point D.

2. Taking D as centre and with the same radius as
before, draw an arc intersecting the previously
drawn arc, say at a point E.

3. Draw the ray AC passing through E [see Fig 11.3 (ii)].

Then ✄ CAB is the required angle of 60°. Now,

let us see how this method gives us the required
angle of 60°.

Join DE. Fig. 11.3
Then, AE = AD = DE (By construction)

Therefore, ✁ EAD is an equilateral triangle and the ✄ EAD, which is the same as
✄ CAB is equal to 60°.

EXERCISE 11.1

1. Construct an angle of 900 at the initial point of a given ray and justify the construction.
2. Construct an angle of 450 at the initial point of a given ray and justify the construction.
3. Construct the angles of the following measurements:

(i) 30° 1° (iii) 15°
(ii) 22

2

4. Construct the following angles and verify by measuring them by a protractor:

(i) 75° (ii) 105° (iii) 135°

5. Construct an equilateral triangle, given its side and justify the construction.

11.3 Some Constructions of Triangles

So far, some basic constructions have been considered. Next, some constructions of
triangles will be done by using the constructions given in earlier classes and given
above. Recall from the Chapter 7 that SAS, SSS, ASA and RHS rules give the
congruency of two triangles. Therefore, a triangle is unique if : (i) two sides and the

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192 MATHEMATICS

included angle is given, (ii) three sides are given, (iii) two angles and the included side
is given and, (iv) in a right triangle, hypotenuse and one side is given. You have already
learnt how to construct such triangles in Class VII. Now, let us consider some more
constructions of triangles. You may have noted that at least three parts of a triangle
have to be given for constructing it but not all combinations of three parts are sufficient
for the purpose. For example, if two sides and an angle (not the included angle) are
given, then it is not always possible to construct such a triangle uniquely.

Construction 11.4 : To construct a triangle, given its base, a base angle and sum

of other two sides.

✄Given the base BC, a base angle, say B and the sum AB + AC of the other two sides

of a triangle ABC, you are required to construct it.

Steps of Construction :

1. Draw the base BC and at the point B make an
angle, say XBC equal to the given angle.

2. Cut a line segment BD equal to AB + AC from
the ray BX.

✄3. Join DC and make an angle DCY equal to BDC.

4. Let CY intersect BX at A (see Fig. 11.4).

Then, ABC is the required triangle.

Let us see how you get the required triangle.

✄Base BC and B are drawn as given. Next in triangle Fig. 11.4

ACD,

✄ACD = ✄ ADC (By construction)

Therefore, AC = AD and then

AB = BD – AD = BD – AC

AB + AC = BD

Alternative method :

Follow the first two steps as above. Then draw
perpendicular bisector PQ of CD to intersect BD at
a point A (see Fig 11.5). Join AC. Then ABC is the
required triangle. Note that A lies on the perpendicular
bisector of CD, therefore AD = AC.

Remark : The construction of the triangle is not Fig. 11.5

☎possible if the sum AB + AC BC.

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