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.

Circle is inscribed — Circle is Inscribed

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.