Mathematics
In the figure, given below, O is the center of the circumcircle of triangle XYZ. Tangents at X and Y intersect at point T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY.
Circles
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Answer
YT and XT are tangents to the circle.
∴ ∠OYT = 90° and ∠OXT = 90°.
In quadrilateral OYTX,
⇒ ∠XOY + ∠OYT + ∠OXT + ∠XTY = 360°
⇒ ∠XOY + 90° + 90° + 80° = 360°
⇒ ∠XOY = 360° - 260° = 100°.
From figure,
⇒ ∠XOZ + ∠YOZ + ∠XOY = 360°
⇒ 140° + ∠YOZ + 100° = 360°
⇒ ∠YOZ = 360° - 240° = 120°.
We know that,
When two angles are subtended by the same arc, the angle at the centre of a circle is twice the angle at the circumference.
∴ ∠YOZ = 2∠ZXY
⇒ ∠ZXY = ∠YOZ = = 60°.
Hence, ∠ZXY = 60°.
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