in the given circle o is the centre and two chords pq and rs of the circle intersect within the circle at a point t such that angle otp is equal to angle otr prove that chords are equal
Answers
Given : O is the centre and two chords PQ and RS of the circle
PQ & RS two chords intersect within the circle at a point T such that ∠OTP = ∠OTR
To Find : Prove that the chords are equal.
Solution:
Draw OM ⊥ PQ and ON ⊥ RS
=> M & N are mid point of PQ and RS
in ΔOMT and Δ ONT
OT = OT Common
∠OMT = ∠ONT = 90°
∠OTM = ∠OTN (∵ ∠OTM = ∠OTP and ∠OTN = ∠OTR )
=> ΔOMT ≅ Δ ONT ( AAS)
=> OM = ON
PM² = OP² - OM²
RN² = OR² - ON²
OP = OR = radius
OM = ON shown above
Hence PM² = RN²
=> PM = RN
M & N are mid point of PQ and RS
=> PQ/2 = RS/2
=> PQ = RS
Hence Chords are equal
QED
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