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

${\displaystyle \Rightarrow }$ x = 2q

${\displaystyle \Rightarrow q=\frac{x}{2}}$

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

${\displaystyle p={90}^{\circ }-\frac{x}{2}}$

∵ Arc BC subtends ∠BOC at the centre and ∠ADC at the remaining part of the circle

∴ ∠BOC = 2∠BDC = 2r

∴ ${\displaystyle r=\frac{1}{2}\angle BOC=\frac{1}{2}({180}^{\circ }-x)}$

∵ (∠AOB + ∠BOC = 180°)

∴ ${\displaystyle r={90}^{\circ }-\frac{1}{2}x}$

= ${\displaystyle {90}^{\circ }-\frac{x}{2}}$