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 perpendicular to AB and ON perpendicular ti CD. Show that OMPN is a square.​

Answer 1

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Answer 2

Chords of the circle = AB and CD (Given)

Centre of the circle = O (Given)

Intersection point at right angle of the circle = P.

Since all the angles of OMPN are 90°, thus

OM ⊥ AB and ON ⊥ CD

in quadrilateral OMPN

∠OMP = ∠ONP =∠MPN =  90°(given)

= ∠MON = 90°

Therefore OMPN is a rectangle --- eq 1

Since, the perpendicular distance of equal chords from the centre of the circle are always equal, hence

= OM = ON --- eq 2

From equation (1) and (2) it can be concluded that  the adjacent sides of a rectangle are equal and thus all sides are equal

Hence OMPN is a square