In the given figure, AC is the diameter of circle, centre O. Chord BD is perpendicular to AC. Write down the angles p,q and r in terms of x.

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Solution

$\angle AOB = 2 \angle ACB = 2 \angle ADB$ (angles subtended by an chord on the center is double that subtended by the same chord on the circumference)

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

( Angle subtended by chord AB at the circumference is half of the angle subtended by it at the centre.)

$\angle ADB = \frac{x}{2}$

$∠ A D B = ∠ A C B$ ( angle subtended by the same chord in the same sector)

$\angle ADC = 90^o$ (angle subtended by the diameter is always $90^{0}$

$r +\frac{x}{2} = 90^o \Rightarrow r = 90-\frac{x}{2}$

Now $\angle DAC = \angle DBC$ (angles in the same segment)

$p = 90^o-q \Rightarrow p = 90^o-\frac{x}{2}$

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