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.

Answer 1

∵ 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`