Two identical circles touch each other externally at point Z. XY is a direct common tangent, which touches the circles at X and Y respectively. What is the ∠XYZ ?
Answer & explanation
Correct answer: option 1
Let P be a point on XY such that, PZ is at right angles to the Line Joining the centers of the circles.
Note that, PZ is a common tangent to both circles.
This is because tangent is perpendicular to radius at point of contact for any circle.
let ∠PXZ= α and ∠PYZ = β.
PX = PZ [lengths of the tangents from an external point Z]
In a triangle ZXP, ∠PXZ = ∠XZP = α
similarly PY = ZP and ∠PZY = ∠ZYP = β
now in the triangle XZB,
∠ZXY + ∠ZYX + ∠XZY = 180° [sum of the interior angles in a triangle]
α + β + (α + β) = 180° (Since ∠XZY = ∠XZP + ∠PZY = α + β.
2α + 2β = 180°
α + β = 90°
Therefore, ∠XZY = α + β = 90°