Problem 13 · 2017 Math Kangaroo
Hard
Geometry & Measurement
area-fractionsymmetry
In an equilateral triangle with area 1, we draw the six perpendicular lines from the midpoints of each side to the other two sides as seen in the diagram. How big is the area of the grey hexagon that has been created this way?
Show answer
Answer: D — 12
Show hints
Hint 1 of 2
The figure is highly symmetric — the hexagon sits at the centre.
Still stuck? Show hint 2 →
Hint 2 of 2
Compare the grey region with the whole triangle using that symmetry.
Show solution
Approach: use the threefold symmetry to pair the grey hexagon against the cut-off corner regions
- The construction has the triangle's full threefold rotational symmetry, so the central hexagon and the regions cut off around it all repeat in three identical copies.
- Tracking those repeating pieces (or checking with coordinates) shows the cut-off regions outside the hexagon add up to exactly the same area as the hexagon itself.
- So the hexagon is exactly half of the triangle; with the triangle's area equal to 1 that is 1/2, choice D.
Mark:
· log in to save