IMC 2017 Q17
17. The diagram shows two rectangles and a regular pentagon. One side of each rectangle has been extended to meet at \(X\).
What is the value of \(x\)?
- A 52
- B 54
- C 56
- D 58
- E 60
Intermediate Mathematical Challenge 2017, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 54
Since \(VWXYZ\) is a pentagon, the sum of its interior angles is 540°. Now \(\angle ZVW\) is an interior angle of a regular pentagon and so is 108°. Both \(\angle VWX\) and \(\angle XYZ\) are 90°; and the reflex angle \(\angle YZV\) = 90° + 108°.
Therefore 540° = 108° + 90° + 90° + 90° + 108° + \(x\)°.
Hence \(x = 540 - 486 = 54\).
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).