2017 · Q58
Two straight lines AB and AC include an angle. A circle is drawn in this angle which touches both these lines. One more circle is drawn which touches both these lines as well as the previous circle. If the area of the bigger circle is 9 times the area of the smaller circle, then what must be the angle A ?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A45°
- B60°
- C75°
- D90°
Explanation
With A/2 = φ and radii R > r: the centres lie on the bisector at vertex-distances R/sin φ and r/sin φ, and are R+r apart, so (R−r)/sin φ = R+r, i.e. sin φ = (R−r)/(R+r). Area ratio 9 gives R/r = 3, so sin φ = 1/2, φ = 30°, and A = 60°. Hence option (B) 60°.