IMC 2005 Q14
14. Ten stones, of identical shape and size, are used to make an arch, as shown in the diagram. Each stone has a cross-section in the shape of a trapezium with three equal sides. What is the size of the smallest angles of the trapezium?
- A 72°
- B 75°
- C 81°
- D 83°
- E 85°
Intermediate Mathematical Challenge 2005, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 81°
Let \(ABCD\) be the cross-section of one of the stones, as shown. As \(AD = BC\), \(ABCD\) is an isosceles trapezium with \(\angle ADC = \angle BCD\) (the proof of which is left to the reader). If \(\angle ADC = \angle BCD = \theta\), then \(2\theta\) is the interior angle of a regular 20-sided polygon, namely (180 − 360/20)°, which equals 162°. So \(\theta\) is 81°.
Solution © UK Mathematics Trust, from the IMC 2005 solutions.