IMC 2018 Q13
13. The diagram shows a rhombus and two sizes of regular hexagon.
What is the ratio of the area of the smaller hexagon to the area of the larger hexagon?
- A 1 : 2
- B 1 : 3
- C 1 : 4
- D 1 : 8
- E 1 : 9
Intermediate Mathematical Challenge 2018, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 1 : 4
Let the length of each edge of the rhombus be \(x\). Then, as the rhombus and the smaller regular hexagon have an edge in common, the length of each edge of this hexagon is also \(x\). So the length of each edge of the larger regular hexagon is \(2x\). The two regular hexagons are similar. Therefore the ratio of their areas is the square of the ratio of their edge-lengths, that is \(1^2 : 2^2 = 1 : 4\).
Solution © UK Mathematics Trust, from the IMC 2018 solutions. Longer versions with investigations: Solutions and investigations (PDF).