# CBSE Class 10-Mathematics: Questions and Answers Chapter – 10 Circles Part 16 (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Question 14**:

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

**Answer**:

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 ,

[Each radius r]

Hypotenuse. Hypotenuse. OP [common]

**Question 15**:

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

**Answer**:

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 ] .

**Answer**:

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) ]