Question

in the figure, AB, BC, and AC are tangents to the circle at P, Q and R. if AB=AC, show that Q is the midpoint of BC ​

Answer 1

Answer:

Step-by-step explanation:

In the attached figure (Figure-1), AB, BC, and AC are tangents to the circle at P, Q and R. and AB=AC.

Proved that, Q is the midpoint of BC ​or BQ=CQ

If we draw two tangents on a circle, from one external point, then they have equal tangent segments. That means the distance between the external point and the point of tangency are the equal.

Therefore,

AP and AR are two tangents from point A so, AP=AR

BP and BQ are two tangents from point B so, BP=BQ

CR and CQ are two tangents from point B so, CR=CQ

As,

$AB=AC\\\\\therefore AP+BP=AR+CR\\\\\therefore AP+BP=AP+CR\text{ [AP=AR]}\\\\\therefore BP=CR$

As BP=BQ and CR=CQ

So it was proved that

BQ=CQ

Hence it was proved that Q is the midpoint of BC.​