IMC 2016 Q20
20. Two semicircles are drawn in a rectangle as shown.
What is the width of the overlap of the two semicircles?
- A 3 cm
- B 4 cm
- C 5 cm
- D 6 cm
- E 7 cm
Intermediate Mathematical Challenge 2016, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 6 cm
In the diagram, \(O\) is the centre of the lower semicircle, \(A\) and \(C\) are the points of intersection of the two semicircles and \(B\) is the point at the centre of the rectangle and also of the overlap. Now \(OA\) is a radius of the semicircle so \(OA\) has length 5 cm. Also \(OB\) is half the height of the rectangle so has length 4 cm. Angle \(ABO\) is a right angle. So triangle \(ABO\) is a (3, 4, 5) triangle and hence \(AC = 2 \times 3\) cm = 6 cm.
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).