EM-SSLC Worksheets - Questions - John P A

3) O is the centre of the circle.If ∠ABC = 120◦,AC = 10then

a) The diametre from Ameet the circle at P . Mark the point P on the circle.
b) Draw triangle AP C ,write the measure of ∠AP C
c) What is the diametre and radius of the circle.
d) Calcualte the area of the triangle.
4) In the figure AP is perpendicular to BC, ∠ACD = 150◦,∠BAP = 60◦

a) What is the length of AP and AB.
b) What is the measure of angle ACP ?
c) What is the length of P C?
d) What is the area of the triangle ABC
5) In the figure ABCDis a square .∠DP A = 30◦, ∠CP B = 45◦, P B = 4 cm then

a) What is the length of BC? [email protected]
b) What is the length of AP ? 9847307721
c) Find the area of the rectangle ?
d) Find the length of P Dand P C

2

2020-21 Academic year Works

Mathematics X
Trigonometry

47

Concepts

a) A triangle can be scaled without altering its angles.While doing so length of the sides changes
keeping the ratio of the sides constant.This is what we have studied in similar triangles.

b) Unchaging angles and unchanging ratio of the sides make a special type of angle
measurements.These are known as trigonometric measurement of the angles.

c) We define trigonometric measurement of angles on the acute angles of a right triangle.

I√n 45◦ − 45◦ − 90◦right triangle the sides opposite to these angles are in the ratio1 : 1 :
2.Whatever be the size of the triangle this ratio remains unchanged.
√
d) The sides opposite to 30◦ − 60◦ − 90◦ angles of a right triangle are in the ratio1 : 3 : 2.

This ratio is independent of the size of the triangle.This leads to the measurement of angles,known

as sin, cos, tan.

e) In triangle ABC, A, B, Care the angles and a, b, c are the opposite sides. IfB = 90◦then

sin A = a , cos A = c , tan A = a
b b c

sin 30◦ = 21√, sin 60◦ = √ sin 45◦
3 √1 .
f) 2 , =
2

cos 30◦ = 3 , cos 60◦ = 12√, cos 45◦ = √1 .
2
2

tan 30◦ = √1 , tan 60◦ = 3, tan 45◦ = 1
3

Worksheet 47

1) In triangle ABC,if ∠B = 90◦ ,sin A = 3 then
5

a) Draw a rough diagram

b) Write cos A and tan A

c) Write cos C and tan C

2) If in triangle ABC, ∠B = 90◦ ,sin A = 0.8then

a) Draw a rough diagram.
b) Write cos A, tan A
c) Write cos C, tan C

1

3) In triangle ABC , ∠B = 90◦ ,cos C = x
a) Draw a rough diagram.
a) Write sin C, tan C
b) Write cos A, sin A tan A

4) In the figure AB = 10cm, the altitue from Ato BC is 8cm ,∠C = 45◦.

a) What is the length of BD?
b) Write sin B, cos B, tan B
c) Write the length of BC
d) Find the area of triangle ABC
e) Find the perimetre of triangle ABC.
5) In the figure the side AB is perpendicular to BC, BP = 1, ∠BAP = ∠P AC = 30◦ then

a) What is the length of AB
b) What is the length of BC and P C

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2020-21 Academic year Worksheets

Mathematics X
Trigonometry

48

Concepts

a) A triangle can be scaled without altering its angles.While doing so length of the sides changes
keeping the ratio of the sides constant.This is what we have studied in similar triangles.

b) Unchaging angles and unchanging ratio of the sides make a special type of angle
measurements.These are known as trigonometric measurement of the angles.

c) We define trigonometric measurement of angles on the acute angles of a right triangle.

I√n 45◦ − 45◦ − 90◦right triangle the sides opposite to these angles are in the ratio1 : 1 :
2.Whatever be the size of the triangle this ratio remains unchanged.
√
d) The sides opposite to 30◦ − 60◦ − 90◦ angles of a right triangle are in the ratio1 : 3 : 2.

This ratio is independent of the size of the triangle.This leads to the measurement of angles,known

as sin, cos, tan.

e) In triangle ABC, A, B, Care the angles and a, b, c are the opposite sides. IfB = 90◦then

sin A = a , cos A = c , tan A = a
b b c

sin 30◦ = 12√, sin 60◦ = √ sin 45◦
3 √1
f) 2 , = 2 .

cos 30◦ = 3 , cos 60◦ = 12√, 3c,otsa4n5◦45=◦ √1 .
tan 30◦ = 2 2

√1 , tan 60◦ = =1
3

Worksheet 48
1) In triangle ABC , AB = 8cm , BC = 12cm ,∠B = 120◦

a) What is the altitude from C to AB?
b) Find the area of triangle ABC.

1

2) In triangle ABC, ∠B = 90◦, ∠A = 40◦

a) What is the measure of ∠C
b) Compare sin A and cos C
c) Compare sin C and cos A?
d) Write the conclusion on comparing sin measure and cos measure of acute angles of a right triangle.

3) In triangle ABC, ∠B = 90◦ , tan A = 12 then
5

a) Find the tan measure of angle C

b) What is sin A

c) Find 1+sin A
1−sin A

4) In triangle ABC, ∠B = 30◦, ∠C = 60◦, BD = 12cm

a) BC is perpendicular to DA,If DB = x then what is DC?
b) Write equations connecting △BDA and △CDA
c) Find x
d) What is the perpendicular distance from A to BC
e) Find the area of triangle ABC.

5) ABCD is a trapezium .∠A = 60◦, ∠B = 30◦, AB = 14cm , AD = 4cm

a) What is the diatance between parallel sides [email protected]
b) What is the length of CD 9847307721
c) Find the perimetre of the trapezium
d) Claculate the area of the trapezium.

2

2020-21 Academic year Worksheets

Mathematics X
Trigonometry

49
Concepts
a) We can see a table of all trigonometric measures of angles ranging from 0◦ to 90◦in the text
book.You can learn a detailed table in the higher
b) When the angle measures increases from 0◦ to 90◦, sin measure increases from 0 to 1.
c) When angle measures increases from 0 to 90 cos measure decreases from 1 to 0
d) Maximum value of sin and cos is 1.
e) sinmeasure and cosmeasure become equal at the acute angle 45◦.
f) sin 0 = cos 90 = 0, sin 90 = cos 0 = 1
g) If the sum of two angles is 90◦then sin of one angle is equal to cos of other angle.
Example,sin 40◦ = cos 50◦

Worksheet 49
1) There is a regular octagon of side 4cm . A quadrilateral is shaded .

a) What is the measure of ∠C?
b) What is the length the rectangleABCF
c) Calculate the area of the coloured region.

1

2) In the parallelogram ABCD, ∠A = 40◦, AB = 10cm , AD = 8cm
a) Draw a rough diagram
b) What is the diatance between the parallel sides AB, CD
c) Calculate the area of the parallelogram.

[sin 40◦ = 0.6428, cos 40◦ = 0.7660, tan 40◦ = 0.8391]
3) In triangle ABC, ∠A = 25◦, AB = 14cm , AC = 18cm .

a) Draw a rough diagram
b) What is the altitude from C to AB
c) Calculate the area of the triangle ABC.
[sin 25◦ = 0.64226, cos 25◦ = 0.9063, tan 25◦ = 0.4553]
4) In the figure AB = AC = 12cm , ∠A = 140◦

a) Write the measure of ∠B and ∠C
b) What is the altitude fron C to AB?
c) Find the area of triangle ABC
[sin 40◦ = 0.6428, cos 40◦ = 0.7660, tan 40◦ = 0.8391]
5) In triangle ABC, O is the centre of its circumcircle.One side AB = 24cm , opposite angle ∠C = 140◦
a) Draw a rough diagram .Draw the diametre from A which cut the circle at P . Join P B
b) What is the measure of ∠AP B?
c) Find the radius of the circle.
[sin 50◦ = 0.7660, cos 50◦ = 0.6420, tan 50◦ = 1.1918]

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2

2020-21 Academic year Worksheets

Mathematics X
Trigonometry

49
Concepts
a) We can see a table of all trigonometric measures of angles ranging from 0◦ to 90◦in the text
book.You can learn a detailed table in the higher
b) When the angle measures increases from 0◦ to 90◦, sin measure increases from 0 to 1.
c) When angle measures increases from 0 to 90 cos measure decreases from 1 to 0
d) Maximum value of sin and cos is 1.
e) sinmeasure and cosmeasure become equal at the acute angle 45◦.
f) sin 0 = cos 90 = 0, sin 90 = cos 0 = 1
g) If the sum of two angles is 90◦then sin of one angle is equal to cos of other angle.
Example,sin 40◦ = cos 50◦

Worksheet 49
1) Consider the following trigonometric measures of angles

sin 42◦, cos 78◦, sin 70◦, cos 14◦

a) Rewrite all these into equivalent sin measures
b) Find the largest and smallest among these
c) Write these in the ascending order.
2) In the concentric circles, O is the centre and the radius of the outer circle is 12cm .The line AB , the chord
of the big circle touches the small circle at P .Also, OP is perpendicular to AB. If ∠P BO = 40◦ then

a) What is the radius of the small circle?
b) What is the length of the chord AB?
c) Find the area in between these two circles?

1

3) ABCD is a trapezium.The line AB parallel to CD.Also, AB = 18cm , AD = 6cm , ∠A = 50◦then

a) What is the distance between the parallel lines AB and CD
b) What is the length of the side CD
c) Calcualte the area of the trapezium.

4) 40◦ angle is drawn with the line AB of length 8cm as an arm and the vertex at A. A line is drawn from B
perpendicular to the arm ,which is marked as BP .

a) What are the angle measures of the triangle AP B?
b) Find the length of the sidesAP and BP
c) Find the area of the triangle.

5) a, b, c are the sides opposite to the angles A, B, C of triangle ABC.

a) Draw a rough diagram

b) Prove that area of the triangle is 1 bc sin A.
2

c) If two sides are 16cm and 10cm, the angle between them is 50◦, then find the area of the triangle.

[email protected]
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2

2020-21 Academic year Worksheets

Mathematics X
Trigonometry

50

Concepts
a) We can see a table of all trigonometric measures of angles ranging from 0◦ to 90◦in the text

book.You can learn a detailed table in the higher
b) When the angle measures increases from 0◦ to 90◦, sin measure increases from 0 to 1.

c) When angle measures increases from 0 to 90 cos measure decreases from 1 to 0

d) Maximum value of sin and cos is 1.
e) sinmeasure and cosmeasure become equal at the acute angle 45◦.

f) sin 0 = cos 90 = 0, sin 90 = cos 0 = 1
g) If the sum of two angles is 90◦then sin of one angle is equal to cos of other angle.

Example,sin 40◦ = cos 50◦

Worksheet 50

1) Find the value of the following without using trigonometric table.

a) sin 18◦
cos 72◦

b) cos 48◦ − sin 42◦

c) cos 38◦ cos 52◦ − sin 38◦ sin 52◦

2) sin3 A = sin A × sin A × sin A.
If sin A + sin B + sin C = 3then

a) What is the value of sin3 A + sin3 B + sin3 C?
b) What is the value of cos3 A + cos3 B + cos3 C?

c) What is the value of cos A + cos B + cos C?

3) If BD = a, BC = b, AB = h, ∠ADB = 30◦, ∠ACB = 60◦then

√
Prove that h = ab

1

4) ABCD is a trapezium .ABis parallel to CD, AD = BC = 12cm , CD = 5cm .
If ∠A = 40◦then

a) What is ∠B?
b) What is the length of AB
c) What is the diatance between the parallel sides ?
d) Find the are of the trapezium.

5) The diagonals of the rhombus ABCD intersect at O. One side is 10 cm ,
If ∠OAB = 20◦then

a) What is the measure of ∠AOB?
b) What is the length of the diagonal AC?
c) What is the length of the diagonal BD?
d) Calculate the area of the rhombus ?

[email protected]
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2

2020-21 Academic year Worksheets

Mathematics X
Trigonometry

50

Concepts
a) We can see a table of all trigonometric measures of angles ranging from 0◦ to 90◦in the text

book.You can learn a detailed table in the higher
b) When the angle measures increases from 0◦ to 90◦, sin measure increases from 0 to 1.

c) When angle measures increases from 0 to 90 cos measure decreases from 1 to 0

d) Maximum value of sin and cos is 1.
e) sinmeasure and cosmeasure become equal at the acute angle 45◦.

f) sin 0 = cos 90 = 0, sin 90 = cos 0 = 1
g) If the sum of two angles is 90◦then sin of one angle is equal to cos of other angle.

Example,sin 40◦ = cos 50◦

Worksheet 50

1) a, b, c are the sides opposite to the angles ∠A, ∠B, ∠C of triangle ABC.

a) Draw a rough diagram of triangle its circumcircle and the diametre from B

b) If R is the radius of the circumcircle then prove that a = 2R
sin A

c) If ∠A = 40◦, a = 12cm then find the radius of the circumcircle.

2) Radius of the circumcircle of a regular pentagon is 6cm.Consider the triangle formed by a side and two radii
at its ends.

a) Find the side of the pentagon.
b) What is the perpendicular distance from the circumcentre to the side.
c) Calcualte the area of the pentagon.

3 a, b, c are the sides of a triangle and its opposite angles A, B and C

a) Prove that a = b cos C + c cos B

b) If R is the radius of the circumcircle then prove that area of the triangle A = abc
4R

4) The chord of a circle with radius 7cm makes an angle 110◦ at its centre.

a) What is the length of the chord?
b) What is the perpendicular distance from centre to the chord.
c) Calculate the area of the triangle formed by the chord and the radii at its ends.

5) There is a stetched string from the top of a flag post to a point on the ground.The point on the ground is at
the diatance 50m away from the foot of the flag-post.The string makes an angle 40◦ with the ground.

a) Draw a rough diagram.
b) Calculate the height of the flag post.
c) Find the length of the string.

1 [email protected]
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2020-21 Academic year Worksheets

Mathematics X
Trigonometry

51
Concepts
a) In the first figure the top of a flagpost is observed at an angle of elevation x.Angle of
elevation is the angle between the ray of vision and horizontal ground
b) When the object is viewed from the top of a tower, the angle become the angle of
depression. It is denoted by y in the second figure.

Worksheet 51
√
1) The height of a building is 100 3metre.The top of this building is observed from a point at the
distance 100m from the foot of the building.
a) Draw a rough diagram
b) What is the angle of elevation?
c) If the angle of elevation is 45◦ what is the distance from the foot of the building to the point
on the ground?
2) The string of a kite is 100m long and it makes an angle of 60◦ with the horizontal.
a) Draw a rough diagram
b) Find the height of the kite assumimg that there is no slack in the string.
3) The top of building is observed from a point at the diatance a and b from the foot of a the building
on either side .The angle of elevation are 30◦, 60◦.

1

a) Draw a rough diagram.
√

b) If h is the height of the building , prove that h = ab
4) The top of a building can be observed at an angle of elevation 45◦ from a point at some diatance

away from the foot of the building.When moved a diatance 20m towards the tower the angle of
elevation becomes 60◦.

a) Draw a rough diagram.
b) Form equations using the given conditions.
c) What is the distance from the foot of the tower to the point the top is observed.
d) Calculate the height of the tower.
5) The top and bottom of a 7 metre tall building can be seen at the angle of depression 45◦ and 60◦
from the top of a tower.
a) Draw a rough diagram.
b) What is the diatance between the tower and the building.
c) Calculate the height of the tower.

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2

2020-21 Academic year Worksheets

Mathematics X
Trigonometry

52
Concepts
a) In the first figure the top of a flagpost is observed at an angle of elevation x.Angle of
elevation is the angle between the ray of vision and horizontal ground
b) When the object is viewed from the top of a tower, the angle become the angle of
depression. It is denoted by y in the second figure.

Worksheet 52

1) From a point on the ground 40 metre away from the foot of the tower sees the top of the tower at
an angle of elevation 30◦ and sees the top of the water tank on the top of the tower at an angle
of elevation 45◦.

a) Draw a rough diagram. 1
b) Find the height of the tower.
c) Find the height of the water tank

2) A man standing on the bank of a river sees the top of the tree at an angle of elevation 50◦.Steppimg
20 m back finds the angle to sees at an angle of elevation 30◦
a) Draw a rough diagram.
b) Find the width of the river. ഴ െട വീതി കണ ാ ക
c) Calcualte the height of the tree.

3) A tree of height 12m is broken by the wind . The top struck the ground at an angle of 35◦ .
a) Draw a rough diagram
b) At what height from the bottom of the tree is broken by the wind?
c) Calculate the distance from the foot of the tree to the its tip on the on the ground.

4) The shadow of a vertical tower on level ground increases by 10m when the altitude of the Sun
changes from the angle of elevation 45◦ to 30◦.
a) Draw a rough diagram
b) Calculate the height of the building.

5) A tall building and a short building are standing on a level ground.The angle of elevation of the top
of the short building from the foot of the tall building is 30◦.
The angle of elevation of the top of the tall building from the foot of the short building is 60◦. The
tall building has height 50m.
a) Draw a rough diagram
b) Whatb is the diatance between the buildings.
c) Calculate the height of the short building.
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2

2020-21 Academic year Worksheets

Mathematics X
Trigonometry

52
Concepts
a) In the first figure the top of a flagpost is observed at an angle of elevation x.Angle of
elevation is the angle between the ray of vision and horizontal ground
b) When the object is viewed from the top of a tower, the angle become the angle of
depression. It is denoted by y in the second figure.

Worksheet 52
1) The diatance between two buildings is 100metre.The height of one building is double the height of

other building.The top of the buildings can be seen at the angle of elevations 60◦ and 30◦ from a
point in between the buildings.

a) Draw a rough diagram
b) What is the diatance from the foot of the tall tower to the point of observation.
c) Claculate the height of the buildings. 1
2) The top of a 30 high building can be seen from a point at some diatance from the foot of the
building is at an angle of elevation 30◦. When the poit of observation is some distance closer to
the building the angle become 60◦
a) Draw a rough diagram

3) A baloon , moving horizontally at the height 88.2m from the ground sees at an angle of elevation
60◦ from a point on the ground. After some time it can be seen at the angle 40◦ from the same
point
a) Draw a rough diagram.
b) Calculate the diatance moved by the baloon during this time.

4) A man sees the top and bottom of a hill at the angle of elevation 70◦ and angle of depression 30◦
from the top of a tower of height 10m
a) Draw a rough diagarm.
b) What is the distance between the hill and the building
c) Calculate the height of the hill

5) A man sees a car moving towards a building at uniform speed at an angle of depression 30◦ from
the top of the building. After 6 seconds the angle of drepression becomes 60◦.
a) Draw a rough diagram.
b) How long the car take to reach near the building?
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2

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) A quadrilateral is colouded inside an equilateral triangle of side 12cm .The sides of the equilateral
triangles are divided into three parts by dots.

a) What is the area of △QRC?
b) What is the area of △P BS?
c) What is the area of △AP Q?
d) Calculate the area of the coloured quadilateral.

1

2) Ois the centre of a circle of radius 4cm. ∠P AB = 28◦

a) What is the length of the side P B in triangle AP B?
b) What is the length of the side P A?
c) Calculate the area of △P AB
3) A man sees a boat approaching the shore at the angle of depression 30◦ from the top of a light
house . After 6 seconds the angle of depression becomes 60◦.
a) Draw a rough diagram
b) How long will the boat takes to reach the shore .
c) If the speed of the boat is 25kilometre per hour, what is the distance from the shore to the

second point of observation.
4) A man sees the top of a tower at angle of elevation 40◦ from the top of a building of height

15metre. He sees the top of the tower at the angle 70◦ from the foot of the building.
a) Draw a rough diagram .
b) Calculate the height of the tower.
c) What is the distance between tower and building?

5) A man sees the top of a light house of height 100m at the angle of elevation 60◦. After 2 minutes
the angle becomes 45◦.
a) Draw a rough diagram.
b) Calculate the distance between the positions of obaservation.
c) Calculate the speed of the boat.
[email protected]
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2

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?
s from B and D to AC Calculate the area of ABCD.

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.

1

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

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

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

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06

Coordinates

സൂചകസംഖ്യകൾ

2020-21 Academic year Works

Mathematics X
ചകസംഖ കൾ

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 axis

⋆ 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 and 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 and 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?
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.
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?
4) The line passing through (0, 4) parallel to x axis and the line passing through (4, 0) and parallel to y axis
meet at a point.
a) Write the coordinates of the intersecting point.
b) What is the diatance from origin to the intersecting point.
c) A circle is drawn with the origin as the cen1tre and distance from origin to the intersecting point as

radius. What are the points where the circle cut the axes.
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?

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

2) △ABC is an equilateral triangle. Side ABcoincides xaxi. 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?

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.

1

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

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.

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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 axes
d) Calculate perimetre and area of the rectangle.

1

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

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

a) Write the coordinates of B, C, D
b) Find the perimetre of the rectangle.
c) Calculate the area of the rectangle.
5) The perpendicular sides of the right triangle coincides the axes,right angle is at the origin . The mid point of
the hypotenuse is (6, 8).If the sum of the perpendicular sides is 28

2

a) What is the radius of the circumcircle.
b) What is the length of its hypotenuse?
c) Find the area of the triangle.

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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 opoints where the circle cut the axes ?
c) Write the coordinates of Q
d) Write the coordinates of three more points on this circle.

1

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.
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 two pair of the coordinates of C
d) Calculate the area of the triangle.
4) (1, 2) 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.
5) In the figure ABCD is a square. OD = 10, ∠AOD = 30◦.

2

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.

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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
2) Using the diatance 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)
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 .
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?
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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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 .
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

√
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.

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.
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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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.
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.

1

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
d) Find the area of △ABC
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.
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.

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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.
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
b) What is the diatance between parallel sides ?
c) Calculate the area of the trapezium
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.

1

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

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2020-21 Academic year Worksheets

Mathematics X
ചകസംഖ കൾ

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

a) Write the coordinates of C 1

b) What is the diatance between the sides AD and BC

c) Calculate the area of the parallelogram

3) OABC is a rhombus .If O(0, 0), A(a, 0), D(b, c)then

a) Draw a diagram
c) write the coordinates of C
d) Prove that the diagonals are perpendicular to eachother.
4) ABCDEF is aregular hexagon.If A(−4, 0), B(4, 0)then

a) Draw a diagarm
b) Find the length of a side.
c) Write the coordinates of the vertices
d) Calculate the area of the hexagon.

5) ABCD is a rectangle.

a) Draw coordinate axes with A as the origin.
b) If a is the length and bis the breadth write the verices.
c) If P is a point inside the rectangle prove that P A2 + P C2 = P B2 + P D2

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07

Tangents

തൊടുവരകൾ

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) 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.

2) Draw suitable figure and 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.

3) In the figure O is the center of the circle, ∠OP A = 30◦, OP = 16 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?

1

4) In the figure O is the centre of the circle. A tangent P A is drawn from P outside the circle at the diatance
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?

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

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

1

3) The length of tangent drawn from a point at a distance 7 cm from the centre to a circle is 4cm. Construct
the tangent. Measure the radius of the circle and write aside.

4) In the figure the length of tangent P A is 12cm and P B = 7cm . what is the radius of the circle?

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

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

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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?
2) In the figure P A and P B are tangents Ois the centre of the circle ,∠AQB = 50◦then

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

1

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?
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 ?
5) Two angles of a trinagle are 40◦, 60◦.The sides of the triangle touches a circle of radius 3 cm

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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.

2) In the figure P A and P B are the tangents to the circle . ∠ACB =

1 × ∠AP B
3

a) If ∠AP B = xthen find ∠ACB, ∠AOB, ∠ADB
b) Find x
c) Find the measure of ∠ACB, ∠AOB, ∠ADB
3) One angle of a rhombus is 60◦. The sides touches a circle of diametre 5cm
. Construct the rhombus.

1

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

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