sheniblog-EM-SSLC Worksheets - Questions and Answers - John P A

a) Diagram

b) △ABD is a 45◦ − 45◦ − 90◦ triangle. AD = 100m
△ABC is a 30◦ − 60◦ = 90◦ triangle. AC = 1√00 = 57.80m

3

CD = 100 − 57.80 = 42.2m
c) Speed = distance ÷ time = 42.2 ÷ 2 = 21.1 metre per minute .

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2020-21 Academic year Worksheets

Mathematics X
Trigonometry

52

Concepts √
2
⋆ The sides of a 45◦ − 45◦ − 90◦ triangle are in the ratio 1:1 :

⋆ The sides of a 30◦ − 60◦ − 90◦ triangle are in the ratio √ : 2
1: 3

⋆ The angle between the direction of vision and horizontal is called angle of elevation or angle
of depression.

Worksheet 52
1) In the figure AB = BC = 12cm , ∠B = 120◦, ∠ACD = 45◦, CD = 8cm

a) What is the length of AC?
b) What are the altitudes from B and D to AC
c) Calculate the area of ABCD.

1

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a) Since AB = BC,△ABC is an isosceles triangle .

Draw BP perpendicular to AC.
△ABP i√s a 30◦ − 60◦ −√90◦ triangle.
AP = 6 √3cm , P C = 6 3cm
AC = 12 3cm.

b) P B = 6cm , △AQC is a 45◦ − 45◦ − 90◦ triangle. DC = 8cm DQ =

√8 cm
2

c) Area = 1 × √ × 6 + 1 × √ × (6 + √ = √ + √ sq.cm
2 12 3 2 12 3 4 2, 6 3(6 4 2)

2) In the figure AB is the diametre of the seimicircle. P C is perpendicular to AB.
If BC = 12cm , ∠P CB = 30◦then

a) Find P Band P C
b) What is the length of AP
c) What is the radius of the semicircle.

a) △CP B √is a 30◦ − 60◦ − 90◦ triangle.Since BC = 12cm, P B = 6cm ,
P C = 6 3cm

b) P A × P B =√P C2,
P A × 6 = (6 3)2
6 × P A = 36 × 3, P A = 18cm

c) Radius 12cm

2

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3) In triangle ABC, ∠B = 90◦ ,BC = 7, AC − AB = 1 .

a) Find the length of other two sides
b) Find sin A + cos A

a) AC2 − AB2 = 72
(AC − AB)(AC + AB) = 49
1 × (AC + AB) = 49, AC + AB = 49
AC + AB = 49, AC − AB = 1 → 2AC = 50, AC = 25cm , AB =
24cm

b) sin A = 7 , cos A = 24
25 25
31
sin A + cos A = 25

4) ABCD is a trapezium . AB = 18cm, CD = 12cm , BC = 6cm , ∠B = 40◦

a) What is the diatance between the parallel sides AB and CD
b) Find the area of the trapezium

3

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a) sin 40 = QC , 0.64 = QC , QC = 6 × 0.64 = 3.84 cm
6 6

b) Area = 1 × h(a + b) = 1 × 3.84 × (12 + 18) = 57.6 sq.cm
2 2

5) One side of a triangle is 18cm ,angle at the ends are 40◦, 30◦ cm .

a) Draw a rough figure .
b) What is the altitude to this side
c) Calculate the area of the triangle

a) Figure

b) △AP C√ is a 30◦ − 60◦ − 90◦ triangle.
x=h 3= 1.732hcm ,h=
x
h h 1.73
18−x 18−x
tan 40 = , 0.83 =

0.83 × (18 − x) = h

0.83(18 − x) = x , 0.83 × 1.73(18 − x) = x,
1.73
25.74
1.43 × (18 − x) = x, 25.74 − 1.43x = x, 25.74 = 2.43x, x = 2.43 =

10.59cm

h = x = 10.59 = 6.12cm
1.73 1.73

c) Area = 1 × 18 × 6.12 = 55.08 sq.cm
2

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06

Coordinates

സൂചകസംഖ്യകൾ

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2020-21 Academic year Works

Mathematics X
Coordinates

45
Concepts

⋆ Two perpendicular straight lines divide the plane into four parts .The intersecting point of the lines
is called origin.

⋆ The position of a point can be determined by a pair of real numbers .In P (x, y) x is called x
coordinate and y is called y coordinate

⋆ The coordinates of the origin is (0, 0)
⋆ y coordinates of all points on x axis is 0.y coodinates of all points on a line parallel to y axis are

equal.
⋆ x coordinates of all points on y are is 0. x coordinates of all points on a line parallel to y axis are

equal.
⋆ The distance between two points on x axis or on a line parallel to x axis is the absolute value of

the difference between their x coordinates
⋆ The distance between two points on y axis or on a line parallel to y axis is the absolute value of

the difference between their y coordinates

Worksheet 45
1) Draw coordinate axes and mark A(−2, −2)

a) Write the coordinates of B which is 4 unit away parallel to y axis in the upward direction.
b) Write the coordinates of C which is 6 unit in the right of B parallel to x axis
c) Write the coordinates of D which is 4 unit above C on the line parallel to y axis
d) What is the distance between A and D?

a) B(−2, −2 + 4) = B(−2, 2)
b) C(−2 + 6, 2) = C(4, 2)
c) D(4, 2 + 4) = D(4, 6)

√√
d) AD = AP 2 + P D2 = 62 + 82 1= 10

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2) A(1, 1), B(−3, 1), C(−3, −4), D(1, −4) are the oordintes of the vertices of a rectangle.
a) What is the length of the side AB?
b) What is the length of the side AD?
c) Calcualte the perimetre and area of the rectangle.
a) AB =| 1 −− 3 |= 4
b) AD =| 1 −− 4 |= 5
c) Perimetre= 2(4 + 5) = 18
Area = 4 × 5 = 20

3) There is a circle with centre at the origin . The circle passes through (5, 0)
a) What is the radius of the circle?
b) What are the coordinates of the points where the circle cut the axes?
c) Is (3, 4) a point on the circle? How can we realize it?

a) 5
b) A(5, 0), B(0, 5).C(−5, 0), D(0, −5)

√
c) The diatance from origin to the point (3, 4) is = 32 + 42 = 5, the radius of the circle.

This point is on this circle.
4) The line passing through (0, 4) parallel to x axis and the line passing through (4, 0) parallel to y axis meet

at a point.
a) Write the coordinates of the intersecting point.
b) What is the distance from origin to the intersecting point.
c) A circle is drawn with the origin as the centre and distance from origin to the intersecting point as
radius. What are the points where the circle cut the axes.

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a) (4, 4)
√
b) 4 2
√ √√ √
c) A(4 2, 0), B(0, 4 2), C(−4 2, 0), D(0, −4 2)

5) The vertices of a right triangle are A(1, 1), B(4, 1), C(1, 5).
a) Name the vertex at which 90◦ angle is taken
b) What is the length of perpendicular sides?
c) What is the length of its hypotenuse?
d) What is the radius of its circumcircle?

a) A(1, 1)

b) AB =| 4 − 1 |= 3, AC =| 5 − 1 |= 4
√

c) BC = 32 + 42 = 5

d) Circumradius = 5 = 2.5
2

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2020-21 Academic year Works

Mathematics X
Coordinates

46
Concepts

⋆ Two perpendicular straight lines divide the plane into four parts .The intersecting point of the lines
is called origin.

⋆ The position of a point can be determined by a pair of real numbers .In P (x, y) x is called x
coordinate and y is called y coordinate

⋆ The coordinates of the origin is (0, 0)

⋆ y coordinates of all points on x axis is 0.y coodinates of all points on a line parallel to y axis are
equal.

⋆ x coordinates of all points on y are is 0. x coordinates of all points on a line parallel to y axis are
equal.

⋆ The distance between two points on x axis or on a line parallel to x axis is the absolute value of
the difference between their x coordinates

⋆ The distance between two points on y axis or on a line parallel to y axis is the absolute value of
the difference between their y coordinates

Worksheet 46

1) In △ABC , A(1, 3), B(7, 3), C(4, 11) are the vertices

a) What is the length of AB?
b) What is the altitude to AB
c) Calculate the area of △ABC

a) AB =| 7 − 1 |= 6

b) h =| 11 − 3 |= 8

c) Area = 1 × 6 × 8 = 24 sq.cm
2

2) △ABC is an equilateral triangle. Side ABcoincides xaxis. If A(−1, 0), B(5, 0)then

a) What is the length of AB?
b) What is the altitude of the triangle?
c) What are the coodinate pairs of C?

a) AB =| 5 −− 1 |= 6 1
√

b) Altitude = 3 3
√√

c) C(2, 3 3), (C(2, −3 3)

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3) Three vertices of ABCDare A(0, 0), B(8, 0)C(8, 4)
a) Write the coordiantes of D
b) Find the perimetre of the rectangle.
c) Calculate the area of the rectangle.

a) D(0, 4)
b) AB = CD = 8, BC = AD = 4

Perimetre= 2(8 + 4) = 24
c) Area = 8 × 4 = 32sq.unit

4) A(4, 0), B(0, 4), C(−4, 0), D(0, −4)are the vertices of a quadrilateral
a) Suggest a suitable name to ABCD
b) Find the length of a side?
c) Calcualte the area and perimetre

a) Square
√

b) 4 2
√ √√

c) Area (4 2)2 = 32 sq.unit, Perimetre = 4 × 4 2 = 16 2

5) In triangle ABC, A(1, 2), B(7, 2)are two vertices.

a) What is the length of the side AB
b) In triangle ABC, ∠A = 90◦.Write a pair of coordinates of C
c) What is the length of side AC?
d) Calculate the area of the triangle.

a) AB =| 7 − 1 |= 6

b) C(1, 5) or any other pair with x coordinate 1

c) If C(1, 5), AC =| 5 − 2 |= 3

d) In the right triangle ABC with A(1, 2), B(7, 2) and C(1, 5)

Area = 1 × 6× 3 = 9 sq.unit
2

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2020-21 Academic year Works

Mathematics X
Coordinates

46
Concepts

⋆ Two perpendicular straight lines divide the plane into four parts .The intersecting point of the lines
is called origin.

⋆ The position of a point can be determined by a pair of real numbers .In P (x, y) x is called x
coordinate and y is called y coordinate

⋆ The coordinates of the origin is (0, 0)
⋆ y coordinates of all points on x axis is 0.y coodinates of all points on a line parallel to y axis are

equal.
⋆ x coordinates of all points on y are is 0. x coordinates of all points on a line parallel to y axis are

equal.
⋆ The distance between two points on x axis or on a line parallel to x axis is the absolute value of

the difference between their x coordinates
⋆ The distance between two points on y axis or on a line parallel to y axis is the absolute value of

the difference between their y coordinates

Worksheet 46
1) P (3, 4) is a point on a circle with centre at the origin

a) What is the radius of the circle?
b) P QRS is a rectangle with its vertices are on this circle, sides are parallel to the axes . Write the

coordinates of its vertices.
c) What are the points where the circle cut the axes
d) Calculate perimetre and area of the rectangle.

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√
a) Radius OP = 32 + 42 = 5
b) Q(−3, 4), R(−3, −4), S(3, −4)
c) (5, 0), (0, 5), (−5, 0), (0, −5)
d) Length QR = P S = 8, breadth P Q = RS = 6

perimetre = 2(8 + 6) = 28,Area = 48
2) OABC is a parallelogram ,O(0, 0), A(4, 0), B(6, 5)

a) Write the coordinates of C
b) Write the length of OA and BC
c) What is the diatance between the parallel sides OA and BC
d) Calculate area and perimetre of the parallelogram

2

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a) OA = 4 → BC = 4, C(6 − 4, 5) = C(2, 5)
b) OA = 4, BC = 4
c) The distance between parallel sides OA and BC is 5
d) From the figure,the √perpendicular√from B to x axis , AP and AB form a right triangle.

Hypotenuse AB =√ 52 + 22 = 29
Perimetre = 8 + 2 29
Area of the parallelogram = 4 × 5 = 20

3) P is a point on the circle with centre at the origin and radius 5.If OP makes an angle 30◦ with the positive
side pf x axis ,

a) What are the points where the circle cut the axes?

b) Write the coordinates of P

c) The vertices of the rectangle P QRS, with the sides parallel to the axes are on the circle.Write the
coordinates of the vertices.

a) (5, 0), (0, 5), (−5, 0), (0, −5)

b) Draw a line perpendicular to x axis .Let it be P N .△ON P is a 30◦ − 60◦ − 90◦ triangle.

Side √o=pp25o,sOiteNto=9025◦√is35.
PN

P ( 5 3, 5 ).
2 2
√ √ √
c) Q(− 5 3, 5 ), R(− 5 3, − 5 ), S( 5 3, − 5 )
2 2 2 2 2 2

4) ABCD is a rectangle ,sides are parallel to the axes .If A(2, 3), AB = 6, BC = 5then

3

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a) Write the coordinates of B, C, D
b) Find the perimetre of the rectangle.
c) Calculate the area of the rectangle.

a) B(2 − 6, 3) = B(−4, 3)
C(−4, −2), D(2, −2)

b) AB = 6, BC = 5
Perimetre = 22

c) Area = 6 × 5 = 30

5) The perpendicular sides of the right triangle coincides the axes,right angled is at the origin . The mid point
of the hypotenuse is (6, 8).If the sum of the perpendicular sides is 28
a) What is the radius of the circumcircle.
b) What is the length of its hypotenuse?
c) Find the area of the triangle.

√
a) Radius of the circumcircle is 62 + 82 = 10

b) Length of the hypotenuse is 20

c) Draw P M perpendicular to x axis and P N perpendicular to y axis.

OP = P A = 10,△P OA is an isosceles triangle.Since OM = 6, OA = 12

Similarly , △OP B is isosceles triangle,ON = 8, OB = 16

Area= 1 × 12 × 16 = 96 .
2

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2020-21 Academic year Works

Mathematics X
Coordinates

46
Concepts

⋆ Two perpendicular straight lines divide the plane into four parts .The intersecting point of the lines
is called origin.

⋆ The position of a point can be determined by a pair of real numbers .In P (x, y) x is called x
coordinate and y is called y coordinate

⋆ The coordinates of the origin is (0, 0)
⋆ y coordinates of all points on x axis is 0.y coodinates of all points on a line parallel to y axis are

equal.
⋆ x coordinates of all points on y are is 0. x coordinates of all points on a line parallel to y axis are

equal.
⋆ The distance between two points on x axis or on a line parallel to x axis is the absolute value of

the difference between their x coordinates
⋆ The distance between two points on y axis or on a line parallel to y axis is the absolute value of

the difference between their y coordinates

Worksheet 46
1) P (3, 4) is a point in a circle with centre at the origin.

Q(x, y) is another point on this circle ,∠AOQ = 30◦then

a) What is the radius of this circle?
b) What are the points where the circle cut the axes ?
c) Write the coordinates of Q
d) Write the coordinates of three more points1on this circle.

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√√
a) OP = OM 2 + P M 2 = 32 + 42 = 5

b) (5, 0), (0, 5), (−5, 0), (0, −5)

c) △ON Q is a 30◦ − 60◦ − 90◦ t√riangle
OQQ(−=25 √5,3∴, 52Q) N = 5 , ON = 5 3
2 2

d) (−3, 4), (−3, −4), (3, −4)

2) ABCD is an isosceles trapezium.A(1, 1), B(8, 1), AB is parallel to CD.If AD = 4, ∠A = 30◦then

a) What is the length AB?
b) Write the coordinates of D
c) Write the coordinates of C
d) Calculate the area of the trapezium.

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a) AB =| 8 − 1 |= 7

b) Draw DM perpendicular to AB

△AM D is a 30◦ −√60◦ − 90◦ triangle
DP = 2√, AM = 2 3
D(1 + 2 3, 3)

√
c) Draw CN√perpendicular to AB, CN = 2, AN = 8 − 2 3

C(8 − 2 3, 3)
√√
d) Area 1 × 2 × (7 − 4 3 + 7) = 14 − 4 3
2

3) ABC is an equilateral triangle. If A(1, 1), B(7, 1)then

a) What is the length of one side?
b) What is the altitude of this triangle?
c) Write the coordinates of C
d) Calculate the area of the triangle.

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a) AB =| 7 − 1 |= 6
√
b) 3 3
√
c) C(1 + 3, 1 + 3 3)
√√
d) Area = 1 ×6×3 3=9 3
2

4) (2, 1) is a point on the circle with centre at the origin.

a) What is the radius of the circle?
b) What are the points where the circle cut the axes?
c) Write the coordinates of 7 more points on this circle.

√√
a) Radius 12 + 22 = 5
√ √√ √
b) ( 5, 0), (0, 5), (− 5, 0), (0, − 5)

c) (−1, 2), (−1, −2), (1, −2), (2, 1), (−2, 1), (−2, −1), (2, −1)

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5) In the figure ABCD is a square. OD = 10, ∠AOD = 30◦.

a) Write the coordinates of A
b) What is the length of one side of the square?
c) Write the coordinates of the vertices of the square.

√
a) OA√= 5 3

A(5 3, 0)
b) AD = 5, Side is 5 unit

√ √ √√
c) A(5 3, 0), B(5 + 5 3, 0), C(5 + 5 3, 5), D(5 3, 5)

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2020-21 Academic year Works

Mathematics X
Coordinates

46
Concepts

• If P , Q are two points on a line parallel to x axis ,their y coordinates are equal.In general
P (x1, y1), Q(x2, y1) can be considered as two points.

• If P , Q are two points on the line parallel to y axis, their x coordinates are equal.In general
P (x1, y1), Q(x1, y2) can be taken as tow points

• x coordinates and y coordinates of points on inclined line are different. P (x1, y1), Q(x2, y2)can
be taken as the points.
√

• The distance between P (x1, y1), Q(x2, y2) is = (x2 − x1)2 + (y2 − y1)2.

Worksheet 46
1) Complete the following activities

a) Draw coordinate axes and mark the points P (x1, y1), Q(x2, y2)
b) Draw a line through P parallel to xaxes, a line passing through Qparallel to yaxis
c) Mark the intersecting point as R
d) Calcualte the length√P R and QR
e) Prove that P Q = (x2 − x1)2 + (y2 − y1)2

a),b),c) see the figute

d) P R =| x2 − x1 |, QR =| y2 − y1 |

e) P Q2 = P R2 + QR2, P Q2 =| x2 − x1 |2 + | y2 − y1 |2

Note | a√|2= a2 1
P Q = (x2 − x1)2 + (y2 − y1)2

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2) Using the distance formula calculate the following.

a) The distance between P (−6, 7) and Q(−1, −5)
b) What is the distance from origin to (−5, 12)
c) Find the distance between P (−7, −3) and ,Q(−5, −11)

√
a) P Q = √(x2 − x1)2 + (y2 − y1)2

P Q = √(−1 −− 6)2 + (√−5 − 7)2
P Q = 52 + (−12)2 = 169 = 13

b) O(0, 0)√, A(−5, 12) √√
OA = (−5 − 0)2 + (12 − 0)2 = 25 + 144 = 169 = 13

√
c) P Q = √(x2 − x1)2 + (y2 − y1)2
√ √
P Q = (−5 −− 7)2 + (−11 −− 3)2 = 22 + 82 = 68

3) The distance between A(2, y)and B(−4, 3) is 10unit

a) Form an equation using the diatance formula
b) What are the real numbers suitable for y?
c) Write the coordinates of these points .

√
a) AB =√ (x2 − x1)2 + (y2 − y1)2.

10 = (−4 − 2)2 + (3 − y)2

y2 − 6y − 55 = 0

√
−b± b2 −4ac
b) y= 2a

y = 11 or −5

c) A(2, 11), B(−4, 3)
A(2, −5), B(−4, 3)

4) Consider the points A(1, −1), B(5, 2), C(9, 5)
a) Find the distance AB ,BC and AC
b) Prove that these points are on a line.
c) What is the mid point of AC?

√ √√
a) AB = √(x2 − x1)2 + (y2 − y1)2,AB = √(5 − 1)2 + (2 −− 1)2 =√ 16 + 9 = 5

BC = √(x2 − x1)2 + (y2 − y1)2,BC = √ (9 − 5)2 + (5 − 2)2 = √16 + 9 = 5
AC = (x2 − x1)2 + (y2 − y1)2,AC = (9 − 1)2 + (5 −− 1)2 = 64 + 36 = 10

b) AB + BC = 10, AC = 10 → AB + BC = AC
A, B, C are on a line

c) AB = 5, BC = 5. Therefore B is the mid point of AC

5) P (x, y) is equidistant from A(5, 1) and ,B(1, 5)
a) What is the relation between x and y
b) How many triangles are there with AB as the base and satisfying this condition.
c) What is the altitude if ABP is an equilateral triangle.

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√√
a) P A = (x − 5)2 + (y − 1)2,P B = (x − 1)2 + (y − 5)2

P A = P B → x2 − 10x + 25 + y2 − 2y + 1 = x2 − 2x + 1 + y2 − 10y + 25
8x = 8y → x = y
b) Infinit number of triangles are possible. P will be a point on the perpendicular bisector of
AB.It is an isosceles traingle.

√ √ √√
c) AB = (x√2 − y1)2 + (y2 − y1)2, AB = (1 − 5)2 + (5 − 1)2 = 32 = 4 2

Altitude= 2 6
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2020-21 Academic year Worksheets

Mathematics X
Coordinates

46

Concepts

• If P , Q are two points on a line parallel to x axis ,their y coordinates are equal.In general
P (x1, y1), Q(x2, y1) can be considered as two points.

• If P , Q are two points on the line parallel to y axis, their x coordinates are equal.In general
P (x1, y1), Q(x1, y2) can be taken as two points

• x coordinates and y coordinates of points on inclined line are different. P (x1, y1), Q(x2, y2)can
be taken as the points.
√

• The distance between P (x1, y1), Q(x2, y2) is (x2 − x1)2 + (y2 − y1)2.

Worksheet 46
1) The distance from a point P on x axis to A(7, 6) and B(−3, 4) are equal

a) What is the y coordinate of P
b) Form an equation using the distance formula.
c) Write the coordinates of P
d) Find the sides of △ABP .

a) 0
√

b) Distance from P (x, 0) to A(7, 6) is = √(x − 7)2 + (0 − 6)2
Distance from P (x, 0) to B(−3, 4) is = (x −− 3)2 + (0 − 4)2
√√

c) Since P A = P B then (x − 7)2 + (0 − 6)2 = (x −− 3)2 + (0 − 4)2
squaring on both sides ,
(x − 7)2 + 36 = (x + 3)2 + 16, x2 − 14x + 49 + 36 = x2 + 6x + 9 + 16
x = 3, P (3, 0) ,
√ √√

d) P A = √ 7 − 3)2 + (√6 − 0)2 = 42 + 62 = 5√2
P B = 52 , AB = 7 −− 3)2 + (6 − 4)2 = 104
Since P A2 + P B2 = AB2 , we can say ,this is an isosceles right angled triangle.

2) Consider the points A(4, 2), B(7, 5), C(9, 7)
a) Find the distances AB, BC and AC
b) Can we construct △ABC ? why?
c) Write the property of these points.

1

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√ √ √√
a) AB = √(7 − 4)2 + (5 − 2)2 = √32 + 32 = √18 = 3√ 2

BC = √(9 − 7)2 + (7 − 5)2 = √22 + 22 = √8 = 2 √2
AC = (9 − 4)2 + (7 − 2)2 = 52 + 52 = 50 = 5 2
b) The sum of two sides is not greater than the third side. Triangle cannot be constructed.
c) AB + BC = AC. So, the points are on a line.

√
3) The distance from x axis to (7, −4) is 2 5.

a) Take a point on x axis and form an equation.
b) How many points are there on x axis satisfying this condition.
c) What is the distance between these points.

√
a) √Distance from P (x, 0) to (7, −√4) is = 2 5

(x − 7)2 + (0 −− 4)2 = 2 5
Squaring on both sides (x − 7)2 + 42 = 20, x2 − 14x + 49 + 16 = 20, x2 − 14x + 45 = 0

b) x= √ = √ = 9, 5
−b± b2−4ac
−−14± (−14)2−4×1×45
2a 2×1

There are two points satisying this condition.Points are (9, 0), (5, 0)

c) The distance between these points is | 9 − 5 |= 4

4) Consider the points A(0, 1), B(1, 4), C(4, 3), D(3, 0)

a) Find the sides of ABCD
b) Find the length of diagonals.
c) Suggest a suitable name to this quadrilateral.

√√
a) AB = √(1 − 0)2 + (4 − 1)2 = √10
√
BC = √(4 − 1)2 + (3 − 4)2 = √32 + (−1)2 = 10√
CD = √(3 − 4)2 + (0 − 3)2 = √(−1)2 + (−3)2√= 10
AD = (0 − 3)2 + (1 − 0)2 = (−3)2 + 12 = 10

√ √√
b) AC = √(4 − 0)2 + (3 − 1)2 = √42 + 22 = 20√
√
BD = (3 − 1)2 + (0 − 4)2 = 22 + (−4)2 = 4 + 16 = 20

c) Sides are equal. Diagonals are equal .ABCD is a square .

5) Consider the points A(2, −2), B(14, 10), C(11, 13), D(−1, 1)

a) Find the sides of ABCD
b) Find the length of the diagonals.
c) Suggest a suitable name to this quadrilateral.

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√ √√
a) AB = √(14 − 2)2 + (10 −− 2)2 = √122 + 122 = 12 √2

BC = √(11 − 14)2 + (13 − 10)2 = √ (−3)2 + 32 = 3 2 √
CD = √(−1 − 11)2 + (1 − 13)2 = √(−12)2 + (−12)2√= 12 2
AD = (2 −− 1)2 + (−2 −− 1)2 = 32 + (−3)2 = 3 2

√ √√
b) AC = √(11 − 2)2 + (13 −− 2)2 = √92 + 152 = √306

BD = (14 −− 1)2 + (10 − 1)2 = 152 + 92 = 306
c) AB = CD, BC = AD opposite Sides are equal.

AC = BDDiagonals are equal. ABCD is a rectangle.
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2020-21 Academic year Worksheets

Mathematics X
Coordinates

46

Concepts

• If P , Q are two points on a line parallel to x axis ,their y coordinates are equal.In general
P (x1, y1), Q(x2, y1) can be considered as two points.

• If P , Q are two points on the line parallel to y axis, their x coordinates are equal.In general
P (x1, y1), Q(x1, y2) can be taken as two points

• x coordinates and y coordinates of points on inclined line are different. P (x1, y1), Q(x2, y2)can
be taken as the points.
√

• The distance between P (x1, y1), Q(x2, y2) is (x2 − x1)2 + (y2 − y1)2.

Worksheet 46
1) Consider the points A(2, 3), B(3, 4), C(5, 6), D(4, 5)

a) Calculate the AB and CD
b) Calcualte AD and BC
c) Find the length of diagonals ABCD
d) Suggest a suitable name to ABCD.

√
a) AB = √(x2 − x1)2 + (y2 − y1)√2.

AB = √(3 − 2)2 + (4 − 3)2 = √2.
CD = (4 − 5)2 + (5 − 6)2 = 2

√ √√
b) AD = √(4 − 2)2 + (5 − 3)2 = √8 = 2√2

BC = (5 − 3)2 + (6 − 4)2 = 8 = 2 2
√ √√

c) AC = √(5 − 2)2 + (6 − 3)2 = √18 = 3 2
BD = (4 − 3)2 + (5 − 4)2 = 2

d) AB = CD, AD = BC opposite sides are equal
AC ̸= BD diagonals are not equal . ABCDis a parallelogram .

2) △OAB is an equilateral triangle. If O(0, 0), A(0, 6)then
a) Draw a rough diagram
b) Find the length of one side .
c) Write a pair of coordinates of B
d) How many equailateral triangles are there satisfying this condition.

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a) Look at the picture

b) OA =| 6 − 0 |= 6

c) Mid point of OA is P (0, 3) √

△BAP is a 30◦ − 60◦ − 90◦triangle .P A = 3, P B = 3 3

√
Coordinates of B are B(3 3, 3)

√
d) Two eqilateral triangles are possible. Triangle wit√h vertices O(0, 0), B(0, 6), C(3 3, 3)

and triangle with vertices O(0, 0), B(0, 6), C(−3 3, 3)
3) Look at the triangle drawn on a graph paper.

a) Draw perpendiculars AL, BM and CN from A, B and C to x axis .Write the coordiantes of L, M
and N

b) Find the area of BM LA
c) Find the area of ALN C

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d) Find the area of △ABC
a) Look at the picture

L(2, 0), M (1, 0), N (5, 0)

b) BM LA is a trapezium. BM = 3, AL = 1, M L = 1

Area = 1 × 1× (3 + 1) = 2
2

c) ALN C is a trapezium AL = 1, N C = 3, LN = 3

Area = 1 × 3× (1 + 3) = 6
2

d) BCN M is a rectangle . Area = M N × M B = 4 × 3 = 12
Area of triangle = 12 − (2 + 6) = 4

4) Vertices of a triangle are A(8, 6), B(8, −2), C(2, −2)

a) Find the centre of its circumcircle.
b) What is the radius of the circumcircle.

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a) √Let P (x, y) be the circumc√entre.P A = √(x − 8)2 + (y − 6)2, P B =

(x − 8)2 + (y + 2)2, P C = (x − 2)2 + (y + 2)2

PA = PB = PC

∴ P A2 = P B2 = P C2

(x − 8)2 + (y − 6)2 = (x − 8)2 + (y + 2)2

x2 + y2 − 16x − 12y + 100 = x2 + y2 − 16x + 4y + 68, 16y = 32, y = 2

P B2 = P C2, (x − 8)2 + (y + 2)2 = (x − 2)2 + (y + 2)2, 12x = 60, x = 5

P (5, 2)
√

b) Radius = (5 − 8)2 + (2 − 6)2 = 5

See the picture.

5) A(−3, 0), B(1, −3), C(4, 1) are the vertices of a triangle.

a) Find the length of its sides
b) Prove that △ABCis an isosceles right triangle.
c) calculate the area of this triangle.

√√
a) AB = √(1 −− 3)2 + (−3 − 0)2 = 42 + (−3)2 = 5

BC = √(4 − 1)2 + (1 + 3)2 = 5
AC = 5 2

b) AB√ = BC This is an isosceles triangle. AB2 + BC2 = 25 + 25 = 50, AC2 =
(5 2)2 = 50
AB2 + BC2 = AC2

This is a right triangle . Isoscelest right traingle.

c) Area = 1 ×5×5= 25
2 2

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2020-21 Academic year Worksheets

Mathematics X
Coordinates

46
Concepts

• If P , Q are two points on a line parallel to x axis ,their y coordinates are equal.In general
P (x1, y1), Q(x2, y1) can be considered as two points.

• If P , Q are two points on the line parallel to y axis, their x coordinates are equal.In general
P (x1, y1), Q(x1, y2) can be taken as two points

• x coordinates and y coordinates of points on inclined line are different. P (x1, y1), Q(x2, y2)can
be taken as the points.
√

• The distance between P (x1, y1), Q(x2, y2) is (x2 − x1)2 + (y2 − y1)2.

Worksheet 46
1) OABCis a parallelogram . If O(0, 0), A(5, 0), B(7, 4)then

a) Draw a rough diagram
b) Write the coordinates of C
c) Calcualte the area of the parallelogram.

a) Look at the picture

b) Side OAis parallel to BC.Therefore the difference of x coordinates of O, A is same as
the difference of x coordinates of B and C
Similarly in the case of y coordinates
C(7 − 5, 4) = C(2, 4)

c) Area = 5 × 4 = 20

2) In the trapezium ABCD, A(8, 5), B(−8, 5), C(−5, −3), D(5, −3)then

a) Find the length of parallel sides 1

b) What is the diatance between parallel sides ?

c) Calculate the area of the trapezium

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a) y coodinates of A and B are equal. Line AB is parallel to x axis. y coordinates of C and

Dare equal. CD is parallel to x.
That is AB is parallel to CD. AB =| 8 −− 8 |= 16, CD =| 5 −− 5 |= 10

b) Distance between AB and CD is | 5 −− 3 |= 8

c) Area = 1 × 8(16 + 10) = 4 × 26 = 104
2

3) Draw a line parallel to x axis passing through (0, 6).Draw another line parallel to y axis passing through
(8, 0).

a) Find the coordinates of the intersecting point P
c) What is the diatance from origin to P .
d) Write the coordinates of one more point on this line other than origin.

a) P (8, 6) Look at the picture

√
b) OP = 82 + 62 = 10
c) Q(−8, −6)

4) ABC is an equilateral triangle. If A(3, 2), B(7, 2)then

a) Find the length of its sides.
b) What is the altitude of the triangle?
c) Find the suitable coordinate pairs of C
d) Calculate the area of the triangle.

a) AB =| 7 − 3 |= 4
√
b) Altitude= 2 3
√√
c) C(3 + 2,√2 + 2 3) or C(√3 + 2, −(2 3 − 2))
(5, 2 = 2 3)or (5, 2 − 2 3
√√
d) Area = 1 ×4×2 3=4 3
2

5) P (2, −1), Q(3, 4), R(−2, 3), S(−3, −2) are the vertices of a quadrilateral.

a) Find the length of sides .
b) What is the length of its diagonals?

2

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c) Suggest a suitable name to this quadrilateral.
d) Calculate the area .

√√
a) P Q = √(3 − 2)2 + (4 + 1)2 = √26

QR = √ (−2 − 3)2 + (3 − 4)2 = √26
RS = √(−3 + 2)2 + (−2 − 3)2 = √26
SP = (−3 − 2)2 + (−2 + 1)2 = 26

√√
b) Diagona√ls P R = (−2 − 2)2 + (3 + √1)2 = 4 2

QS = (−3 − 3)2 + (−2 − 4)2 = 6 2

c) P Q = QR = RS = SP
P R ≠ QS .This is a rhombus

d) Area = 1 × d1 × d2 = 24 sq.unit
2

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2020-21 Academic year Worksheets

Mathematics X
Coordinates

46
Concepts

• If P , Q are two points on a line parallel to x axis ,their y coordinates are equal.In general
P (x1, y1), Q(x2, y1) can be considered as two points.

• If P , Q are two points on the line parallel to y axis, their x coordinates are equal.In general
P (x1, y1), Q(x1, y2) can be taken as two points

• x coordinates and y coordinates of points on inclined line are different. P (x1, y1), Q(x2, y2)can
be taken as the points.
√

• The distance between P (x1, y1), Q(x2, y2) is (x2 − x1)2 + (y2 − y1)2.

Worksheet 46
1) In the figure ABCD is a parallelogram.If A(2, 1), B(5, 1), D(3, 3)then

a) Write the coordinates of C
b) Find the length of side AB and the distance between the parallel sides AB and CD
c) Calculate the area of the parallelogram.

a) AB is parallel to CD
The change in the x coordinates of A and B is same as the change in the x coordinates of
C and D.
Change in the y coordinates of A and B is same as the change in the y coodrinates of C
and D
C(3 + 3, 3) = C(6, 3)

b) AB =| 5 − 2 |= 3
Distance between the parallel sides =| 3 − 1 |= 2

c) Area = 3 × 2 = 6

1

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2) In the parallelogram ABCD , if A(0, 1), B(5, 3), D(0, 7)then

a) Write the coordinates of C
b) What is the diatance between the sides AD and BC
c) Calculate the area of the parallelogram

a) AD is parallel to BC
The difference in the x coordinates of A and D is same as that of B and C. It is zero.
The difference in the y coordinates of A and D is same as that of B and C. It is 6.
C(5, 3 + 6) = C(5, 9)

b) Distance =| 5 − 0 |= 5
c) Area = 6 × 5 = 30
3) OABC is a rhombus. If O(0, 0), A(a, 0), C(b, c)then
a) Draw the diagram
c) Write the coordinates of B
d) Prove that the diagonals are perpendicular to eachother

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a) see the picture

b) B(a + b, c)
c) see diagram

OB2 = (a + b)2 + c2,AC2 = (a − b)2 + c2

Consider the triangle formed by the verices O, A and the intersectimg point M of the

diagonals(Mark M in the figure)

OM 2 + AM 2 = ( OB )2 + ( AC )2 = OB 2 +AC 2
2 2 4
(a+b)2 +c2 +(a−b)2 +c2 2a2 +2(b2 +c2 )
4 = 2a2 +2b2 +2c2 = 4
4
2a2 +2a2
= 4 = a2

Note : In the figure ,in triangle AP B, a2 = b2 + c2

That is △OM A is a right triangle.Diagonals are perpendicular.

4) ABCDEF is a regular hexagon. If A(−4, 0), B(4, 0)then

a) Draw the diagram
b) Find the length of one side
c) Write the coordiantes of other vertices
d) Calcualte the area of the hexagon.

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a) see the picture

b) AB =| 4 −− 4 |= 8

c) △BP C is a 30◦ −60◦ −90◦ triangle.(Mark P in the figure . It is the foot of the perpendicular

from C to x axis) √

BC =√8, BP = 4, P√C = 4 3 √ √
C(8, 4 3), F (−8, 4 3), D(4, 8 3), E(−4, 8 3)

5) ABCD is a rectangle.

a) Draw coordinate azes with A as the origin
b) If ais the length and b is the breadth , write the coordiante of the vertices
c) If P is a point inside the rectangle then prove that P A2 + P C2 = P B2 + P D2.

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a,b) see the picture

c) OP 2 + CP 2 = x2 + y2 + (x − a)2 + (y − b)2
= 2x2 + 2y2 − 2ax − 2by + a2 + b2
P A2 + P B2 = (x − a)2 + y2 + (x − a)2 + (y − b)2
= 2x2 + 2y2 − 2ax − 2by + a2 + b2
OP 2 + CP 2 = P A2 + P B2

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07

Tangents

തൊടുവരകൾ

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts

⋆ If a line touches only one point on a circle then, the line will be a tangent to the circle.
⋆ Tangent is perpendicular to the radius to the point where the line touches the circle
⋆ Radius , tangent and the line joining center of the circle to a point on the tangent form a right angled

triangle.

1) Construct a tangent to a circle by the steps given below
a) Draw a circle of radius 3cm and mark a point P on the circle.
b) Mark O as the centre of the circle and draw the radius OP
c) Draw the tangent to the circle at P
d) Draw another tangent to this circle parallel to the first tangent.
Answers

Extent the radius to the diametre. Draw tangent at the other end of the diametre also
2) Draw suitable figure find the lengths asked in the quaestion.

a) A tangent of length 12cm is drawn to a circle from a point outside the circle.If the radius of the circle
is 5cm find the distance from centre to the exterior point from which the tangent is drawn.

b) What is the length of tangent drawn from a point at the distance 10 cm away from centre of a circle
of radius 6cm

c) A tangent is drawn from a point at the distance 26 cm away from the centre of a circle. If the length
of the tangent is 24cm find the radius of the circle.
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Answers

Draw suitable figuresIf l is the length of tangent, r is the radius of the circle and d is the

distance from the center to the outer point

d2 = l2 + r2 √

d2 = 122 + 52 = 144 + 25 = 169, d = 169 = 13cm If l is the length of tangent, r is

the radius of the circle and d is the distance from the center to the outer point

d2 = l2 + r2 √

102 = l2 + 62 , l2 = 100 − 36 = 64, l = 64 = 8cm If l is the length of tangent, r is

the radius of the circle and d is the distance from the center to the outer point

d2 = l2 + r2 √

262 = 242 + r2, r2 = 262 − 242 = 676 − 576 = 100, r = 100 = 10cm

3bac) O is the center of the circle, ∠OP A = 30◦, OP = 16 , P A is a tangent from the outer point P ,then
a) Draw a rough diagram
b) What are the angles of △OAP
c) What is the radius of the circle?
d) What is the length of the tangent?

Answers
a) see the diagarm

b) ∠OP A = 30◦, ∠OAP = 90◦, ∠AOP = 60◦

c) This is a 30◦ − 60◦ − 90◦ triangle.

Side opp√osite to 90◦ is 16cm . Therefore the side opposite to 30◦ is 8cm, side opposite to
60◦ is 8 3cm √

Length of tangent P A = 8 3cm, Radius OA = 8cm

4) In the figure O is the centre of the circle. A tangent P A is drawn from P outside the circle at the distance
12cm from the centre. If the length of the tangent and radius are equal then

a) Draw a rough diagram
b) What are the angles of △OAP ?
c) What is the length of tangent and radius?

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Answers

a)

b) ∠OAP = 90◦, OA = P A. The angles opposite to equal sides are equal. Each of them is

45◦

△OAP is a 45◦ − 45◦ − 90◦ triangle.
√√
c) Length of tangent = √12 = 6 2cm, Radius = 6 2cm
2

5) O is the centre of a circle.A tangent P A is drawn from the outer point P to the circle at A

a) Draw a rough diagram .
b) If ∠P OA = 60◦then what are the other angles of △OAP
c) If ∠P OA = 60◦, and the radius of the circle is 10cm find the length of tangent.
d) What is the length of the line OP

Answers
a) see the diagram

b) Other angles: ∠OAP = 90◦, ∠P OA = 60◦,∠OP A = 30◦

c) △OP A is a 30◦ − 60◦ − 90◦ triangle.
Side opposite to 30◦ is 10cm
Side opposite to 90◦ is 20c√m.
Length of the tangent is 10 3cm

d) OP = 20cm

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts

⋆ If a line touches only one point on a circle, the line will be a tangent to the circle.
⋆ Tangent is perpendicular to the radius to the point where the line touches the circle
⋆ Radius , tangent and the line joining center to a point on the tangent form a right angled triangle.

1) In the figure ∠OP A = 40◦, OP = 18cm then

a) What is the measure of ∠AOP ?

b) What is the radius of the circle?

c) What is the length of the tangent?
[sin 40 = 0.6428, cos 40◦ = 0.7660, tan 40 = 0.8391]

Answers

a) ∠AOP = 90 − 40 = 50◦

b) sin 40◦ = OA = OA
OP 18
OA = 18 × 0.6428 = 11.57 cm

c) cos 40◦ = PA
18
P A = 0.7660 × 18 = 13.788cm

2) In the figure ∠P OB = 120◦, OP = 24cm , ABis the diametre of the circle.

1

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a) What are the angles of △P OA?
b) What is the diametre of the circle?
c) What is the length of the tangent from P

Answers

a) In △AOP , ∠P AO = 90◦, ∠P OA = 180 − 120 = 60◦,∠OP A = 30◦

b) Side opposite to 90◦ is 24cm ,Side opposite to 30◦ is 12 cm

AB = 24 cm √
is√12 3cm
c) side opposite to 60◦

Length of tangent is 12 3cm.

3) The length of tangent drawn from a point at a distance 8 cm from the centre to a circle is 4cm. Construct
the tangent. Measure the radius of the circle and write aside.
Answers

a) Draw a line P Aof 4cm
b) Draw a line perpendicular to P A at A
c) Draw an arc with centre at P and radius 7cm which cut the perpendicular line at O.
d) Take O as the centre of the circle and radius OA which completes the construction.

4) In the figure the length of tangent P A is 12cm and P B = 8cm . what is the radius of the circle?

Answers
⋆ OA = OB = r
OA2 + P A2 = OP 2,
⋆ r2 + 122 = (r + 8)2, r2 + 144 = r2 + 16r + 64, 80 = 16r, r = 5 cm

5) In the figure O is the centre of the circle and P A is a tangent. If the area of the triangle is OP Ais 6 sq.cm
and OP = 5cm

2

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a) What is the radius of the circle?
b) What is the length of tangent?

Answers

a) Let OA = r, P A = x . 1 rx = 6, rx = 12
2
r2 + x2 = 52

(r + x)2 = r2 + x2 + 2rx, (r + x)2 = 25 + 24 = 49, r + x = 7

(r − x)2 = (r + x)2 − 4rx = 49 − 48 = 1

r + x = 7, r − x = 1 → 2r = 8, r = 4
√

b) Length of tangent x = 52 − 42 = 3

Note :If radius is 3 cm then length of tangent is 4cm . If radius is 4then the length of tangent

will be 3cm . Discuss this possiblity with your friends and teachers.

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
⋆ Two tangents from an outer point and two radii to the point where the tangents touch the circle form

a cyclic quadrilateral.
1) In the figure P A, P B are tangents . Ois the centre of the circle.

a) What are the measures of ∠OAP, ∠OBP ?
b) If ∠AP B = 40◦ then what is the measure of∠AOB
c) The lines AB and CDintersect at C .What is the relation between the length of lines CO, CP, CA

and CB?
Answers

a) Tangent is perpendicular to the radius .
∠OAP = ∠OBP = 90◦

b) OAP B is a cyclic quadrilateral .∠AOB = 180 − 40 = 140◦
c) OAP B is a cyclic quadrilateral.A Circle passes through the vertices . The lines OP and

AB are the chords of the circle. They intersect at C
CO × CP = CA × CB
2) In the figure P A and P B are tangents Ois the centre of the circle ,∠AQB = 50◦then

1

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a) What is the measure of ∠AOB?
b) What is the measure of angle ∠ARB, ∠AP B?

Answers
a) ∠AOB = 2 × 50◦ = 100◦
b) QARB is cyclic. ∠ARB = 180 − 50 = 130◦
c) OAP Bis cyclic. ∠AP B = 180 − 100 = 80◦

3) In the figure Ois the centre of the circle, P A, P B are tangents . If ∠OAB = 20◦then

a) What is the measure of ∠AOB and , ∠AQB?
b) What is the measure of ∠ARB?
c) What is the measure of ∠AP B?

Answers

a) OA = OB, ∴ ∠OBA = 20◦

∠AOB = 180 − (20 + 20) = 140◦

∠AQB = 1 × ∠AOB = 70◦
2

b) AQBR is cyclic. ∠ARB = 180 − 70 = 110◦

c) OAP B is cyclic .∠AP B = 180 − 140 = 40◦

4) Draw two tangents from an outer point of a circle of radius 3cm such that the angle between the tangents
is 60◦

a) What is the distance from centre to the outer point?
b) What is the length of tangents െതാ വര െട (െതാ വരക െട)നീളം എ ?

Answers

⋆ Draw a circle of radius 3cm. Draw two radii such that the angle between them is 180 − 60 =
120◦.Draw radii OA, OB

⋆ Draw tangents at A and B. They meet at P

⋆ ∠AP B = 60◦

a) Triangle OAP is a 30◦ − 60◦ − 90◦triangle. Side opposite to 30◦ 3 cm , Side opposite to

90◦ is 6 cm to 60◦ is √
The side opposite 3 3cm

OP = 6cm
√√

b) Length of tangent is 3 3cm ,P A = P B = 3 3cm

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5) Two angles of a trinagle are 40◦, 60◦.The sides of the triangle touches a circle of radius 3 cm
Answers
⋆ Draw a circle of radius 3 cm
⋆ Since two angles are 40◦, 60◦ their supplementary angles are 180−40 = 140◦, 180−60 =
120◦. Draw radii such that it divide the angle around the centre as 140◦, 120◦, 100◦
⋆ Draw tangents to the circle at the ends of the radii.
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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts

⋆ Two tangents from an outer point and two radii to the point where the tangents touch the
circle form a cyclic quadrilateral.

1) The sides of an equilateral triangle touches the a circle of radius 3cm .Construct the triangle.
Answers
⋆ Draw a circle of radius 3cm . Mark the centre as O
⋆ Since the angles of an equilateral triangle are 60◦ , divide the angle around the centre
as three equal parts of 180 − 60 = 120◦

⋆ For this draw the radii OA, OB, OC

⋆ Draw tangents at A, B, C to the circle.The tangents make the triangle P QR

2) In the figure P A and P B are the tangents to the circle . ∠AC B = 1 × ∠AP B
3

a) If ∠AP B = xthen find ∠ACB, ∠AOB, ∠ADB
b) Find x
c) Find the measure of ∠ACB, ∠AOB, ∠ADB

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Answers

a) ∠AC B = 1 x
3
2
∠AOB = 3 x,

∠ADB = 180 − 1 x
3

b) 2 x + x = 180, 5x = 180,x = 180×3 = 108
3 3 5

c) ∠AC B = 108 = 36◦, ∠AOB = 72◦, ∠ADB = 180 − 36 = 144◦
3

3) One angle of a rhombus is 60◦. The sides touches a circle of diametre 5cm . Construct the rhombus.

Answers
a) Draw a line of length 5cm . Draw a circle with this line as diametre . Mark the ends
of the diametre A and B

b) Draw another diametre CD such that the angle between the diametres 180 − 60 =
120◦

c) Draw tangents at A, B, C, D to the circle. This makes the rhombus.

4) In the figure O is the centre of the circle. P A and P B are the tangents. If ∠ADB = 110◦then

a) Find the measure of ∠ACB
b) Find the measure of ∠AOB
c) Find the measure of ∠AP B
Answers

a) ∠ACB = 180 − 110 = 70◦
b) ∠AOB = 2 × 70 = 140◦
c) ∠AP B = 180 − 140 = 40◦

5) Two angles of a triangle are 120◦, 40◦.The sides touches a circle of radius 3cm . Construct the
triangle.

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