# CBSE Class 10-Mathematics: Chapter – 11 Constructions Part 12 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 20**:

Draw a pair of tangents to a circle of radius which are inclined to each other at an angle of

**Answer**:

Steps of construction:

1. Draw a circle with center O and radius .

2. Extend OA to B such that

3. With A as center draw a circle of radius . Suppose it intersect the circle drawn in step 1 at the points P and Q.

4. Join BP and BQ. Then BP and BQ are the required tangents.

Justification: In ,

Also,

is an equilateral triangle

In ,

Similarly,

**Question 21**:

From a point P two tangents are drawn to a circle with center . If diameter of the circle, show that is equilateral

**Answer**:

Join OP.

Suppose meets the circle at Q. Join AQ.

We have

i.e.. , diameter

diameter

Diameter – radius

Thus, OQ = PQ = radius

Thus, OP is the hypotenuse of right triangle

OAP and Q is the mid-point of OP

OA = AQ = OQ

[ mid-point of hypotenuse of a right triangle is equidistant from the vertices]

is equilateral

So,

Also

But

Hence, is equilateral.