Q. 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.
Class 9th
NCERT Mathematics
Circles
Exercise 10.4
Question no. 2
Answers
Answer:
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. ... Its two equal chords AB and CD intersect at E. To prove: AE = DE and CE = BE. Construction: Draw OM ⊥ AB and ON ⊥ CD.
Step-by-step explanation:
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.
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Answer:
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