# NCERT Class 10 Chapter 11 Circles CBSE Board Sample Problems Short Answer

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## Question

**In figure, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O. in such a way that the sides AB. BC. CD and DA touch the circle at the points P. Q, R and S respectively. Prove that**

### Solution

We know that tangents drawn to a circle from an outer points are equal.

So,

CR = CQ and DR=DS.

Now, consider

Hence proved.

## Question

**If given figure, AP and BP are tangents to a circle with centre O. such that AP = 5 cm and** **Find the length of chord AB.**

### Solution

In we have AP = BP

[Tangents from an external point arc equally inclined lo segment joining centre to point]

Let

Then in

As all three angles of APB are . So APH is an equilateral triangle.

Hence

## Question

**In given figure, a circle is inscribed in a** **. Such that it touches the sides AB, BC and CA at points D. E and F respectively. If the lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm respectively, find the lengths of AD, BE and CF**

### Solution

Given.

Let (Tangent drawn from external point to circle arc equal)

## Question

**In figure given, AOB is a diameter of a circle with centre O and AC is a tangent to the circle at A. If .****, then find****.**

### Solution

(Linear Pair Axiom)

Now, (angIe between radius OA and tangent AC is )

Now, in,

( sum of angles in triangle is )

## Question

**In given figure. PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and** **, find**

### Solution

Construction: Join AO.

Given: PQ is tangent. AB is diameter

To find:

Solution: In (Equal radii)

(Angles opposite to equal sides are equal)

But, (i)

Since, OC PO (Tangent is perpendicular to radius at point of contact)

## Question

**From an external point P, tangents PA and PB are drawn to a circle with centre O. If** **, then find** **.**

### Solution

Given,

(angle between radius OA and tangent PA is )

Now, Tangents from an external point are same)

Angle between radius OB and tangent PB is )

Now in AOB we have

PA = PB

(sum of angles in triangle is )

## Question

**The in circle of an isosceles triangle ABC, with AB = AC, touches the sides AB, BC and CA at D, E and F respectively. Prove that E bisects BC.**

### Solution

We have ……………………….1

……………………………………….2

(Tangents drawn from an external point are equal)

On subtracting 2 from 1 we get

………………………………………3

Also………………………………4

From 3 and 4 we get

………………………………………5

Also, ……………………………6

From 5 and 6

E bisects BC

## Question

**In given fig. from an external point P, two tangents PT and PS are drawn to a circle center O and radius r. If PO = 2r, show that**

### Solution

Let

In right triangle OTP we have

Hence

, wehave OT = OS

In,

Hence proved

## Question

**In given fig. a circle inscribed in** **touches its sides BC, CA and AB at the points P, Q, R respectively. If AB = AC then prove that BP = CP.**

### Solution

( , equal tangents)

(lengths of equal tangents)

## Question

**Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.**

### Solution

Given: A circle with center O.

With tangent XV at point of contact P.

Top rove:

Proof: Let Q be point on XV

Connect OQ

Suppose it touches the circle at R

Hence,

(As OP=OR radius)

Same will be the case with all other points on circle

Hence, OP is the smallest line that connects XV

## Question

**A circle is inscribed in the quadrilateral ABCD Given BC=38cm, BQ=27cm. CD =25 cm** **find the radius of the circle**

### Solution

Say the side BC touches the circle at point, the side CD touches the circle at point R and the side DA touches the circle at point S and side AB touches the circle at point P. Also let the center of circle be O:

And hence

, hence

and is a square.

Radius of the circle

## Question

**Find the value of x.**

### Solution

Length of tangents from external point are equal.