In Fig. 16.197, AOC is a diameter if the circle and arc AXB= 1/2 arc BYC. Find ∠BOC.
Answers
Given : AOC is a diameter if the circle and arc AXB = 1/2 arc BYC.
To Find : ∠BOC
Proof :
We have,
Arc AXB = Arc BYC …………...(i)
Since, Arc AXBYC is the arc equal to half the circumference
And,
Angle subtended by half circumference at the centre is 180°
Arc AXBYC = Arc AXB + Arc BYC
Arc AXBYC = ½ Arc BYC + 1 Arc BYC
[From eq 1]
Arc AXBYC = Arc BYC (½ + 1)
Arc AXBYC = Arc BYC (2 + 1)/2
Arc AXBYC = 3/2 Arc BYC
Arc BYC = ⅔ Arc AXBYC
Now,
Since Arc BYC subtends ∠BOC at the centre and Arc AXBYC subtends ∠AOC at the centre .
∠BOC = ⅔ ∠AOC
∠BOC = ⅔ × 180°
∠BOC = 2 × 60°
∠BOC = 120°
Hence the value of ∠BOC is 120°.
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Answer:
Step-by-step explanation:Arc AXB = Arc BYC …………...(i)
Since, Arc AXBYC is the arc equal to half the circumference
And,
Angle subtended by half circumference at the centre is 180°
Arc AXBYC = Arc AXB + Arc BYC
Arc AXBYC = ½ Arc BYC + 1 Arc BYC
[From eq 1]
Arc AXBYC = Arc BYC (½ + 1)
Arc AXBYC = Arc BYC (2 + 1)/2
Arc AXBYC = 3/2 Arc BYC
Arc BYC = ⅔ Arc AXBYC
Now,
Since Arc BYC subtends ∠BOC at the centre and Arc AXBYC subtends ∠AOC at the centre .
∠BOC = ⅔ ∠AOC
∠BOC = ⅔ × 180°
∠BOC = 2 × 60°
∠BOC = 120°