Question

Prove that "In two concentric circles, a chord of the bigger circle, that touches the smaller circle is bisected at the point of contact with the smaller circle."

Answer 1

Let $O$ be the centre of two concentric circles $C_1$and $C_2$

Let $AB$ is the chord of larger circle, $C_2$, which is a tangent to the smaller circle $C_1$ at point D.

Now, we have to prove that the chord $XY$ is bisected at $D$, that is $XD=DY$.
Join $OD$.

Now, since $OD$ is the radius of the circle $c_1$ and $XY$ is the tangent to $c_1$ at $D$.

So, $OP$ perpendicular $XY$ [ tangent at any point of circle perpendicular to radius at point of contact]

Since $XY$ is the chord of the circle $c_2$ and $OD$ perpendicular $XY$,

$\Rightarrow$ $XD=DY$ [perpendicular drawn from the centre to the chord always bisects.