Question

if two equal chord of a circle intresects within the circle , prove that the segments of the others chord

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Answer 1

Drop a perpendicular from O to both chords AB and CD

In △OMP and △ONP

As chords are equal, perpendicular from centre would also be equal.

OM=ON

OP is common.

∠OMP=∠ONP=90

△OMP ≅ △ONP (RHS Congruence)

PM=PN ......................(1)

AM=BM (Perpendicular from centre bisects the chord)

Similarly ,CN=DN

As AB=CD

AB−AM=CD−DN

BM=CN ........................(2)

From (1) and (2)

BM−PM=CN−PN

PB=PC

AM=DN (Half the length of equal chords are equal)

AM+PM=DN+PN

AP=PD

Therefore , PB=PC and AP=PD is proved.