Problem 21 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
arearatio
The window shown is a square of area 1 m² and is made up of four triangles whose areas, indicated in the figure, satisfy the ratios \(3A = 4B\) and \(2C = 3D\). A fly is placed exactly at the point where these four triangles touch each other. The fly flies straight to the nearest side of the window. How far does it fly?
Show answer
Answer: A — 40 cm
Show hints
Hint 1 of 2
Each triangle has a full side of the 1 m square as its base; its area fixes the height to that side.
Still stuck? Show hint 2 →
Hint 2 of 2
Opposite heights add to 1 m; use the ratios to find all four heights, then take the smallest.
Show solution
Approach: turn the area ratios into heights and pick the shortest
- Each triangle’s area equals ½·(100 cm)·(height to its side), and opposite heights sum to 100 cm.
- From 3A = 4B the left/right heights are 400/7 and 300/7 cm.
- From 2C = 3D the top/bottom heights are 60 and 40 cm.
- The closest side is the bottom, 40 cm away, so the fly flies 40 cm.
Mark:
· log in to save