Question

In the given figure, $AB,BC$ and $AC$ are tangents to the circle at $P,Q$ and $R$. If $AB=AC$, show that $Q$ is the mid-point of $BC$.

Answer 1

Given that:

$AB=AC$

To show:

$Q$ is mid-point of $BC.$

Solution:

$AP=AR$

$BP=BQ$ (tangents drawn from an external point to the circle are equal.)

$CQ=CR$

Now,

$AB=AC$ (Given)

or, $AP+BP=AR+CR$

or, $AP+BP=AP+CR$ ($\therefore AP=AR$)

or, $BP=CR$

or, $BQ=CQ$ $(\therefore BP=BQ,CQ=CR)$

Since, $BQ=CQ$ so, $Q$ is the mid-point of $BC.$