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Question
AB and CD are two equal chords of a circle with center O which intersect each other at a right angle at point P.
If OM ⊥ AB and ON ⊥ CD;
show that OMPN is a square.
Solution
Clearly , all the angles of OMPN are 90°.
OM ⊥ AB and ON ⊥ CD
∴ BM = `1/2"AB" = 1/2`CD = CN ....(i) ...[ perpendicular drawn from the center of a circle to a chord bisects it ]
As the two equal chords, AB and CD intersect at point P inside the circle,
∴ AP = DP and CP = BP .....(ii)
Now, CN - CP = BM - BP ...[ by (i) and (ii) ]
⇒ PN = MP
∴ Quadrilateral OMPN is A square.
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