Question

In the given figure, XY is the diameter of the circle and PQ is a tangent to the circle at Y.

If ∠AXB = 50° and ∠ABX = 70°, find ∠BAY and ∠APY.

Answer 1

In the above figure,

XY is a diameter of the circle PQ is tangent to the circle at Y.

∠AXB = 50° and ∠ABX = 70°

i. In ΔAXB,

∠XAB + ∠ABX + ∠AXB = 180°  ...(Angles of a triangle)

`=>` ∠XAB + 70° + 50° = 180°

`=>` ∠XAB + 120° = 180°

`=>` ∠XAB = 180° – 120° = 60°

But, ∠XAY = 90°  ...(Angle in a semicircle)

∴ ∠BAY = ∠XAY – ∠XAB

= 90° – 60°

= 30°

ii. Similarly ∠XBY = 90°  ...(Angle in a semicircle)

And ∠CXB = 70°

∴ ∠PBY = ∠XBY – ∠XBA

= 90° – 70°

= 20°

∴ ∠BYA = 180° – ∠AXB  ...(∵ ∠BYA + ∠AYB = 180°)

= 180° – 50°

= 130°

∠PYA = ∠ABY  ...(Angles in the alternate segment)

∠PBY = 20°

And ∠PYB = ∠PYA + ∠AYB

= 20° + 130°

= 150°

∴ ∠APY = 180° – (∠PYA + ∠ABY)

= 180° – (150° + 20°)

= 180° – 170°

= 10°