IMC 2013 Q18
18. The triangle \(T\) has sides of length 6, 5, 5. The triangle \(U\) has sides of length 8, 5, 5.
What is the ratio area \(T\) : area \(U\)?
- A 9 : 16
- B 3 : 4
- C 1 : 1
- D 4 : 3
- E 16 : 9
Intermediate Mathematical Challenge 2013, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 1 : 1
The diagrams show isosceles triangles \(T\) and \(U\).
The perpendicular from the top vertex to the base divides an isosceles triangle into two congruent right-angled triangles as shown in both \(T\) and \(U\).
Evidently, by Pythagoras' Theorem, \(h_1 = 4\) and \(h_2 = 3\). So both triangles \(T\) and \(U\) consist of two ‘3, 4, 5’ triangles and therefore have equal areas.
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).