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In the given figure, O is the center of the circumcircle of ∆ XYZ. Tangents at X and Y intersect at T. ∠ XTY = 80°, what is the value of ∠ZXY.

(a) 20° 

(b) 40° 

(c) 60°

(d) 80°

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Answer : (d) 80º 

Given, 

∠XTY = 80° 

TX = TY (Tangents from the same external point are equal)

⇒ ∠TXY = ∠TYX 

= \(\frac{1}{2}\) (180° – ∠XTY) 

\(\frac{1}{2}\) (180° – 80°) = 50° 

OX ⊥ XT (radii ⊥ tangent at point of contact) 

⇒ ∠OXT = 90° ⇒ ∠OXY = ∠OXT – ∠TXY = 90° – 50° = 40° 

Also, OM ⊥ ZY 

∴ In Δ XMY, ∠XYM = 180° – (∠XMY + ∠MXY) 

= 180° – (90° + 40°) = 50° 

Also, by alternate segment theorem,

∠XZY = ∠TXY = 50° 

∴ In Δ XZY, ∠X = 180° – (∠XZY + ∠XYZ) 

= 180° – (50° + 50°) = 80°.

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