Question

As shown in the figure below, two concentric circles are given and line $PQ$ is the tangent at point $R$. Prove that $R$ is the mid point of seg $PQ$.

Answer 1

$O$ is the common centre of two concentric circles. Line $PQ$ is the tangent to the smaller circle at $R$.
Draw seg $OR$.
$OR$ is the radius of smaller circle and $PQ$ is a tangent to the circle at point $R$.
$\therefore$ seg $OR\perp$ line $PQ$.
Now, seg $PQ$ is a chord of larger circle and $OR$ is perpendicular to it.
We know that, a perpendicular from the centre of a circle to any of its chord, bisects the chord.
$\therefore PR=RQ$
$\therefore R$ is the midpoint of seg $PQ$.