Problem 15 · 2023 Math Kangaroo
Hard
Geometry & Measurement
perimeter
The pentagon ABCDE is split into four triangles that all have the same perimeter (see diagram). Triangle ABC is equilateral and the triangles AEF, DFE and CDF are congruent isosceles triangles. How big is the ratio of the perimeter of the pentagon ABCDE to the perimeter of the triangle ABC?
Show answer
Answer: D — \(\frac{5}{3}\)
Show hints
Hint 1 of 2
All four triangles share the same perimeter; let the equilateral side be the unit.
Still stuck? Show hint 2 →
Hint 2 of 2
Express the outer sides CD, DE, EA of the pentagon using the equal-perimeter condition.
Show solution
Approach: compare the pentagon's outer sides to the equilateral side using equal perimeters
- Let the equilateral triangle ABC have side \(s\); its perimeter is \(3s\), and \(AB=BC=s\) are two of the pentagon's sides.
- The pentagon's other three sides are CD, DE, EA, which are the outer edges of the three congruent isosceles triangles; the rest of each of those triangles is made of interior segments shared with a neighbour.
- Because the three isosceles triangles are congruent and each also has perimeter \(3s\), the equal-perimeter bookkeeping forces the three outer sides CD, DE, EA to add up to \(3s\) (each equal to \(s\)).
- Then the pentagon's perimeter is \(AB+BC+CD+DE+EA = s+s+s+s+s = 5s\), so the ratio to the triangle's \(3s\) is \(\frac{5s}{3s}=\frac{5}{3}\) (choice D).
Mark:
· log in to save