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Question
In the given figure, AC is a diameter of circle, centre O. Chord BD is perpendicular to AC. Write down the angles p, q and r in terms of x.
Solution
∵ Arc subtends ∠AOB at the centre and ∠ACB at the remaining part of the circle.
∴ ∠AOB = 2∠ACB
`\implies` x = 2q
`\implies q = x/2`
But ∠ADB and ∠ACB are in the same segment
∴ ∠ADB = ∠ACB = q
Now in ΔAED.
p + q + 90° = 180° ...(Sum of angles of a Δ)
p + q = 90°
p = 90° – q
`p = 90^circ - x/2`
∵ Arc BC subtends ∠BOC at the centre and ∠ADC at the remaining part of the circle
∴ ∠BOC = 2∠BDC = 2r
∴ `r = 1/2 ∠BOC = 1/2 (180^circ - x)`
∵ (∠AOB + ∠BOC = 180°)
∴ `r = 90^circ - 1/2 x`
= `90^circ - x/2`
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