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