AB and CD are two equal chords of a circle with centre O which intersect each other at right angle - Maths -

Question

AB and CD are two equal chords of a circle with centre O which
intersect each other at right angle at point P If OM is
perpendicular to AB and ON perpendicular to CD; show that OMPN is a
square

Ajanta Trivedi · Accepted Answer

Given : AB and CD are two equal chords of a circle
with centre O which intersect at right angle at point
P

OM âŠ¥ AB and ON
âŠ¥ CD

TPT: quadrilateral OMPN is a
square

Proof:

in quadrilateral OMPN

âˆ OMP =
âˆ ONP =âˆ MPN=
90Â°(given)

â‡’ âˆ MON
= 90Â°

therefore OMPN is a rectangle
(1)

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

â‡’ OM = ON
(2)

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

since adjacent sides of a rectangle are equal
therefore all the sides are equal

hence OMPN is a square

AB and CD are two equal chords of a circle with centre O which intersect each other at right angle at point P. If OM is perpendicular to AB and ON perpendicular to CD; show that OMPN is a square.