Question

X, Y, Z and C are the points on the circumference of a circle with centre ‘O’. AB is a tangent to the circle at ‘X’ and ZY = XY.

Given ∠OBX = 32° and ∠AXZ = 66°. Find:

1. ∠BOX
2. ∠CYZ
3. ∠ZYX
4. ∠OXY

Answer 1

(a) ∠OXB = 90°   (∵ Radius and tangent and perpendicular)

In ΔOXB,

By the angle sum property,

∠OXB + ∠OBX + ∠BOX = 180°

90° + 32° + ∠BOX = 180°

∠BOX = 180° – 122°

= 58°

(b) Since the angle in the circle is half of the angle at the centre

∠CYX = `(∠BOX)/2`

= `(58°)/2`

= 29°

(c) ∠AXZ = ∠ZYX = 66°

Angle in the alternate segment.

(d) In ΔZXY,

∠YZX = ∠YXZ     ...(angles opposite to equal sides are equal)

∠YZX + ∠YXZ + ∠ZYX = 180°    (By angle sum property)

2∠YZX + 66° = 180°

2∠YZX = 66° − 180°

∠YZX = 57°

and ∠YXZ = 57°

∠YZX = ∠YXB = 57°   (∵ angle in the alternate segment)

∠OXB = 90°

Then, ∠OXY + ∠YXB = ∠OXВ

∠OXY + 57° = 90°

∠OXY = 90° − 57°

= 33°