Question

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.

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Answer 1

Answer:

Given:AB is the diameter of the circle with centre O.AP and AQ are two intersecting chords of the circle such that ZP AB = ZQAB

Proof:In ΔAOL and ΔΑΟΜ

ZOLA = ZOMB (each 90°)

OA OA (common line) =

Feedback

.. ZOAL = ZOAM (ZPAB = ZQAB) ... AAOL = AAOM by AAS congruence

criterion

⇒OL = OM by CPCT

⇒Chords AP and AQ are equidistant from

center O

⇒ AP = AQ(chords which are equidistant from the centre are equal.)

Answer 2

Answer:

Given:AB is the diameter of the circle with centre O.AP and AQ are two intersecting chords of the circle such that ∠PAB=∠QAB

Proof:In △AOL and △AOM

∠OLA=∠OMB (each 90

∘

)

OA=OA(common line)

∴∠OAL=∠OAM(∠PAB=∠QAB)

∴△AOL≅△AOM by AAS congruence criterion

⇒OL=OM by CPCT

⇒Chords AP and AQ are equidistant from centre O

⇒AP=AQ(chords which are equidistant from the centre are equal.)