AB AND CD ARE TWO EQUAL CHORDS OF A CIRCLE WITH CENTRE O WHICH INRESECTS EACH OTHER AT RIGHT ANGLE AT RIGHT ANGLE AT P . IF OM PERPENDICULAR TO AND ON PERPENDICULAR TO CD , SHOW THAT OMPN IS A SQUARE

Given : AB and CD are two equal chords which intersect at right angle and OM ⊥ AB and ON ⊥ CD

Now in quadrilateral OMPN

∠OMP = ∠ONP = ∠MOPN = 90°

⇒ ∠MON = 90°

Hence OMPN is a rectangle  ...  (1)

but we know that perpendicular distance of equal chords from the centre of the circle are equal.

⇒ OM = ON  ...  (2)

from (1) and (2) we can conclude that

OMPN is a square  (∵ adjacent side of a rectangle are equal)