Question

) Prove that isosceles trapezium is cyclic OR If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Answer 1

Answer:

Answer

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

o

△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.