In the given figure A C is a tangent to the circle with centre O . If ∠ A D B = 55 ° , find x a n d y . Give reasons for your answers.

Given, We know that angle between the radius and the tangent at the point of contact is the right angle. ∠ A = 90 ° Also, in ∆ O B E , O B = O E = r a d i u s r In Δ A B D , ∠ A + ∠ B + ∠ A D B = 180 ° 90 ° + ∠ B + 55 ° = 180 ° ∠ B = 180 ° − 90 ° − 55 ° = 35 ° ∠ B = ∠ O E B = 35 ° a n d ∠ O E B = ∠ D E C = 35 ° (vertically opposite angles) ∠ E D C + ∠ A D E = 180 ° ∠ E D C + 55 ° = 180 ° ∠ E D C = 180 ° − 55 ° = 125 ° I n Δ E D C , ∠ D E C + ∠ E D C + x = 180 ° 35 ° + 125 ° + x = 180 ° x = 180 ° − 35 ° − 125 ° = 20 ° I n Δ A O C , ∠ A + ∠ y + ∠ x = 180 0 90 ° + y + 20 ° = 180 ° y = 180 ° − 90 ° − 20 ° = 70 ° Hence, the value of x = 20 ° a n d y = 70 ° .

In the figure given below O is the centre of the circle. If the Q R = O P and angle ∠ O R P = 20 ° . Find the value of x giving reasons.

Given, O P = Q R and O P = O T = O Q = Q R So in ∆ R Q P R Q = Q O So ∠ Q R O = ∠ Q O R = 20 ° (the same side having same opposite angle) In ∆ R Q P ∠ R Q P + ∠ Q P R + ∠ P R Q = 180 ° ∠ R Q P = 180 - 20 - 20 = 140 ° Now ∠ R Q P + O Q P = 180 ° (linear pair) ∠ O Q P = 180 - 140 = 40 ° Now in ∆ P O Q O Q = P O ( Both are radius) ∠ Q P O = ∠ O Q P = 40 ° So ∠ P O Q = 180 - ∠ O P Q - ∠ O Q P ∠ P O Q = 180 - 40 - 40 = 100 ° Now ∠ X + ∠ P O Q + ∠ R O Q = 180 ° X = 180 - 100 - 20 X = 60 ° Hence, this is the required answer.

E M B I B E

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 11

HARD

10th ICSE

IMPORTANT

Earn 100

$ABC$ is a triangle with $AB = 10 cm, BC = 8 cm$ and $AC = 6 cm$ . Three circles are drawn touching each other with the vertices as their centres. Find the radii of the three circles.

Important Questions on Circle

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 12

In the figure given below, $P Q = 24 cm, Q R = 7 cm$ and $∠ P Q R = 90 °$ . Find the radius of the inscribed circle of $∆ P Q R$ .

Question Image

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 12

In the given figure, there are two concentric circles with centre

O

of radii

5 c m

and

3 c m .

From an external point

P

, tangents

PA

and

PB

are drawn to these circles. If

A P = 12 c m

, find the length of

BP

.

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 13

In the figure given below,

AB = 8 cm

and

M

is mid-point of

AB

. Semicircles are drawn on

AB

,

AM

and

MB

as diameters. A circle with centre

C

touches all three semicircles as shown, find its radius.
Question Image

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 14

The length of the direct common tangent to two circles of radii

12 cm

and

4 cm

is

15 cm

. Calculate the distance between their centres.

EASY

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 15

The length of the radii of two circles are

8 cm

and

3 cm

, the distance between the centres is

13 cm

. What is the length of the direct common tangent of the two circles?

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 16

In the figure, given below, $AC$ is a transverse common tangent to two circles with centres $P$ and $Q$ and of radii $6 cm$ and $3 cm$ respectively.

Question Image

GIven that $AB = 8 cm,$ calculate $PQ .$

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 17

Two circles with centres

A, B

are of radii

6 cm

and

3 cm

respectively. If

A B = 15 cm

, find the length of a transverse common tangent to these circles.

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 15 - Circle>Exercise 15.3>Q 18

In the figure given below, two circles touch internally at $P$ from an external point $Q$ on the common tangent at $P$ , two tangents $Q A$ and $Q B$ are drawn to the two circles. Prove that $Q A = Q B$ .