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प्रश्न
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.
उत्तर
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°
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