IMC 2019 Q20
20. Three equilateral triangles with sides of length 1 are shown shaded in a larger equilateral triangle. The total shaded area is half the area of the larger triangle.
What is the side-length of the larger equilateral triangle?
- A \(\sqrt{5}\)
- B \(\sqrt{6}\)
- C \(\dfrac{5}{2}\)
- D \(\dfrac{3\sqrt{3}}{2}\)
- E \(1 + \sqrt{3}\)
Intermediate Mathematical Challenge 2019, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B \(\sqrt{6}\)
Since the total shaded area is half the area of the larger equilateral triangle, the area of that triangle is six times the area of each of the shaded equilateral triangles . Now when two similar figures have side-lengths in the ratio \(1 : k\), the ratio of their areas is \(1 : k^2\). In this case, the ratio of the areas is \(1 : 6\) so the ratio of the side-lengths is \(1 : \sqrt{6}\).
Hence the the length of the sides of the larger triangle is \(\sqrt{6}\) cm.
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).