NCERT Class 10 Chapter 11 Circles CBSE Board Sample Problems Short Answer (For CBSE, ICSE, IAS, NET, NRA 2023)

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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.