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Question
Two congruent drdes have their centres at 0 and P. Mis the midpoint of the line segment OP. A straight line is drawn through M cutting the two circles at the points A, B, C and D. Prove that the chords AB and CD are equal.
Solution
Given: Two congruent circles with centre 0 and P. Mis the mid-point of OP
To prove: Chord AB and CD are equal.
Construction: Draw OQ ⊥ AB and PR ⊥ CD.
Proof: In Δ OQM and Δ PRM
∠ OQM = ∠ PRC ...(Each 90°)
OM =MP ....(As M is the mid-point)
∠OMQ = ∠ PMR ...(Verically opposite angles)
Therefore, Δ OQM ≅ ΔPRM
⇒ OQ = PR ...(By CPCT)
Now the perpendicular distances of two chords 1n two congruent circles are equal, therefore chords are also equal.
⇒ AB = CD
Hence proved.
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