IMC 2015 Q4
4. The diagram shows a regular pentagon inside a square.
What is the value of \(x\)?
- A 48
- B 51
- C 54
- D 60
- E 72
Intermediate Mathematical Challenge 2015, Question 4. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 54
The sum of the exterior angles of a convex polygon equals 360°. The angle marked \(p\)° is the exterior angle of a regular pentagon. So \(p = 360 \div 5 = 72\). The angle sum of a triangle equals 180°, so \(q = 180 - 90 - 72 = 18\). The angle marked \(r\)° is the interior angle of a regular pentagon, so \(r = 180 - 72 = 108\). The angles marked \(q\)°, \(r\)° and \(x\)° lie along a straight line, so \(x = 180 - (q + r)\) \(= 180 - (18 + 108) = 54\).
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).