Problem 15 · 2021 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
The figure shows a semicircle with center O. Two of the angles are given. What is the size, in degrees, of the angle α?
Show answer
Answer: A — 9°
Show hints
Hint 1 of 2
Every segment drawn from the centre O out to the arc is a radius, so each triangle with O at a vertex is isosceles with two equal base angles.
Still stuck? Show hint 2 →
Hint 2 of 2
Use the exterior-angle rule: in an isosceles radius-triangle the apex's exterior angle equals twice a base angle, so the angles grow by doubling as you step along the chain from \(32°\) toward the \(67°\) corner.
Show solution
Approach: isosceles radius-triangles + exterior-angle doubling
- The chords from the diameter's left end and from O to the top point P all share endpoints with radii, so the figure is a chain of isosceles triangles built on equal radii.
- Starting from the \(32°\) base angle, each successive isosceles triangle's exterior angle is twice the previous base angle, so the marked directions advance \(32°,\,64°,\,\dots\) around toward P.
- The \(67°\) angle at the right-hand end fixes where the last radius points, and \(α\) is the small leftover between that direction and chord \(PR\): the doubling chain and the \(67°\) constraint leave exactly \(9°\).
- So \(α = 9°\), choice (A).
Mark:
· log in to save