Mathematics
In a triangle ABC, the incircle (centre O) touches BC, CA and AB at P, Q and R respectively. Calculate :
(i) ∠QOR
(ii) ∠QPR, given that ∠A = 60°.
Circles
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Answer
(i) From figure,
OQ ⊥ AC and OR ⊥ AB (∵ OQ and OR are the radii and AC and AB are tangents.)
Now in quadrilateral AROQ,
∠A = 60°, ∠ORA = 90° and ∠OQA = 90°.
∠A + ∠ORA + ∠OQA + ∠QOR = 360°
60° + 90° + 90° + ∠QOR = 360°
240° + ∠QOR = 360°
∠QOR = 360° - 240°
∠QOR = 120°.
Hence, the value of ∠QOR = 120°.
(ii) Arc QR subtends ∠QOR at the centre and ∠QPR at the remaining part of the circle.
∴ ∠QOR = 2∠QPR (∵ angle subtended at centre by an arc is double the angle subtended at remaining part of circle)
120° = 2∠QPR
∠QPR = = 60°.
Hence, the value of ∠QPR = 60°.
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