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.

Answer 1

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.