#### 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⊥ 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 centre 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

Is there an error in this question or solution?

Solution 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⊥ Ab and On ⊥ Cd; Show that Ompn is a Square. Concept: Chord Properties - Chords Equidistant from the Center Are Equal (Without Proof).