Question

Two concentric circles are given. Prove that all chords of the circle with larger radius which touch the circle with smaller radius are congruent.

Answer 1

Given are two concentric circles.
Let two chords PQ and RS of the circle with larger radius touch the circle with smaller radius.
We have to prove that PQ ≅ RS.

Proof: Let two chords PQ and RS of the circle with larger radius touch the circle with smaller radius at points M and N respectively.
PQ and RS are tangents to the circle with smaller radius,
∴ OM = ON = radius of smaller circle.
Thus, chords AB and CD are equidistant from the centre of the circle with larger radius.
∴ PQ = RS
∴ PQ ≅ RS
[ note :- PQ = 2√(PO² - OM²) , and RS = 2√(RO² - ON²) but PO = RO = radius of larger circle and OM = ON = radius of smaller circle. so, PQ = RS]
∴All chords of the circle with larger radius which touch the circle with smaller radius are congruent.
Hence proved.