IMC 2018 Q18
18. The diagram shows a regular pentagon and an equilateral triangle placed inside a square.
What is the value of \(x\)?
- A 24
- B 26
- C 28
- D 30
- E 32
Intermediate Mathematical Challenge 2018, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 24
In the diagram, line segment \(AB\) is parallel to the base of the square.
First note that each exterior angle of a regular pentagon equals 360° ÷ 5 = 72°.
So each interior angle of a regular pentagon equals 180° − 72° = 108°.
Therefore obtuse angle \(CDA = (60 + 108)° = 168°\) and reflex angle \(CDA = (360 - 168)° = 192°\). The sum of the interior angles of a quadrilateral is 360°. Therefore, in quadrilateral \(ABCD\), \(\angle BAD = (360 - 192 - 72 - 90)° = 6°\). Angle \(BAE\) is a right angle. So \((6 + 60 + x)° = 90°\). Therefore \(x = 90 - 66 = 24\).
Solution © UK Mathematics Trust, from the IMC 2018 solutions. Longer versions with investigations: Solutions and investigations (PDF).