Question

Find the value of `angleAPQ`?

The tangents drawn at points `P and Q` on a circle intersect at `A`. If `anglePAQ = 60 degree`, find the value of `angleAPQ`.

Answer 1

`60^@`

Explanation:

First draw the image as instructed below. You will immediately understand how to do this.
In the circle, let the centre be O.
Join O, P and O, Q.
It is given that the tangents meet at A and `angle APQ = 60^@`
Now join O, A.

Now, In right`triangle OPA` and right`triangle OAQ`,
OP = OQ (Radius of the same circle)
OA is the common side.

So, According to the RHS Criterion,

`triangle OPA cong triangle OAQ`

That means, AP = AQ, as they were the corresponding side of the congruent triangles (c.p.c.t)

Now, join P,Q.

Then, in `triangle APQ`,
AP = AQ (Previously Proven)

Therefore, `angle APQ = angle AQP`

As, `angle APQ + angle AQP + angle PAQ = 180^@`,

Therefore, `2angle APQ + 60^@ = 180^@` [As `angle APQ = angle AQP` and `angle PAQ = 60^@`]

`rArr angle APQ = (120/2)^@ = 60^@`

Hence, explained.