Question

in two concentric circles prove that all chords of the outer circle which touch the inner are of equal length.

Answer 1

Let PQ and RS be the chords of the circle that touch the inner circle at M and N respectively.

PQ and RS are the tangents to the inner circle, and OM and ON are the radii of the smaller circle.

OM = ON

Thus PQ and RS are equidistant from the centre, therefore they are equal.

Hence PQ = RS.

Answer 2

Let PQ and RS be the chords of the circle that touch the inner circle at M and N respectively.  

PQ and RS are the tangents to the inner circle, and OM and ON are the radii of the smaller circle.

OM = ON  

Thus PQ and RS are equidistant from the centre, therefore they are equal.

Hence PQ = RS.