IMC 2006 Q19
19. The diagram shows a regular pentagon and a regular hexagon which overlap. What is the value of \(x\) ?
- A 82
- B 84
- C 85
- D 87
- E 91
Intermediate Mathematical Challenge 2006, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 84
Each interior angle of a regular pentagon is 108°, whilst each interior angle of a regular hexagon is 120°. The non-regular pentagon in the centre of the diagram contains two angles which are interior angles of the regular hexagon, two angles which are interior angles of the regular pentagon and a fifth angle, the one marked \(x\)°. So \(x + 2 \times 120 + 2 \times 108 = 5 \times 108 = 540\). Hence \(x = 84\).
Solution © UK Mathematics Trust, from the IMC 2006 solutions.