# CBSE Class 10-Mathematics: Questions and Answers Chapter –10 Circles Part 16

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Question 14:

Prove that the lengths of two tangents drawn from an external point to a circle are equal.

Given: P is an external point to the circle .

and PR are two tangents from P to the circle.

To Prove:

Construction: Join

Proof:

A tangent to a circle is perpendicular to the radius through the point of contact

Now in right triangles and ,

Hypotenuse. Hypotenuse. OP [common]

Question 15:

The circle of touches the sides BC, CA, and AB at D, and F, respectively. If , prove that .

Tangents drawn from an external point to a circle are equal in length

[Tangents from A] …(i)

[Tangents from B] …(ii)

[Tangents from C] …(iii)

Adding (i), (ii)and (iii), we get

But

Question 16:

A circle touches the side BC of a at point P and touches AB and AC when produced at Q and R respectively, show that AQ [Perimeter of ].

Since the tangents from an external point to a circle are equal in length,

…(i) [from point B]

…(ii) [from point C]

And …(iii) [From point A]

From (iii), we have

[Using (i) and (ii)]

Now perimeter of

[using (iv)]

[using (i)]