# CBSE Class 10-Mathematics: Questions and Answers Chapter – 10 Circles Part 14 (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Question 7**:

A circle is touching the side BC of at P and touching AB and AC produced at Q and R, respectively. Prove that (perimeter of ) .

**Answer**:

We know that the two tangents drawn to a circle from an external point are equal.

Perimeter of

**Question 8**:

If PA and PB are two tangents drawn from a point P to a circle with center O touching it at A and B. Prove that OP is the perpendicular bisector of AB.

**Answer**:

Let intersect AB at a point C, we have to prove that and

are two tangents from a point P to the circle with center

In two ACP and BCP, we have

[ tangents from P to the circle are equal]

[Common]

And

But

Hence, is perpendicular bisector of .

**Question 9**:

In the given figure, PQ is tangent at point R of the circle with center O. If , find .

**Answer**:

Given is tangent at point R and

[ Tangent of a circle is perpendicular to Radius]

**Question 10**:

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

**Answer**:

Let the circle touch the sides AB, BC, CD and DA at the points P, Q, R, and S, respectively.

Join OP, OQ, OR and OS.

Join OA, OB, OC, and OD.

Since the two tangents drawn from an external point subtend equal angles at the center.

But

And

And