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

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Solution

$O$ is the common centre of two concentric circles. Line $P Q$ is the tangent to the smaller circle at $R$ .
Draw seg $O R$ .
$O R$ is the radius of smaller circle and $P Q$ is a tangent to the circle at point $R$ .
$∴$ seg $O R ⊥$ line $P Q$ .
Now, seg $P Q$ is a chord of larger circle and $O R$ is perpendicular to it.
We know that, a perpendicular from the centre of a circle to any of its chord, bisects the chord.
$∴ P R = R Q$
$∴ R$ is the midpoint of seg $P Q$ .

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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.

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