NCERT Class 10 Chapter 11 Circles CBSE Board Sample Problems Short Answer
Doorsteptutor material for CBSE is prepared by world's top subject experts: fully solved questions with step-by-step explanation- practice your way to success.
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

Circumscribe a Circle
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
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

Circle is Inscribed
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.

A Circle with Centre O
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

A Circle with Centre O
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

Tangents PA and PB
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.

Isosceles Triangle ABC
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

Two Tangents PT and PS
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.

A Circle
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

A Circle is Perpendicular
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

The Quadrilateral
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.

Length of Tangents
Solution
Length of tangents from external point are equal.