IMC 2017 Q11
11. The diagram shows four equilateral triangles with sides of lengths 1, 2, 3 and 4. The area of the shaded region is equal to \(n\) times the area of the unshaded triangle of side-length 1.
What is the value of \(n\)?
- A 8
- B 11
- C 18
- D 23
- E 26
Intermediate Mathematical Challenge 2017, Question 11. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 26
Let the area of the equilateral triangle of side 1 be \(A\). If similar figures have sides which are in the ratio \(k : 1\), then the ratio of their areas is \(k^2 : 1\). So the areas of the triangles with sides 2, 3, 4 are \(4A\), \(9A\), \(16A\) respectively.
So, as shown in the diagram, the total shaded area is \(3A + 8A + 15A = 26A\). Therefore \(n = 26\).
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).