Problem 28 · 2018 Math Kangaroo
Stretch
Geometry & Measurement
area-decompositionsymmetry
ABCDEF is a regular hexagon, as shown. G is the midpoint of AB. H and I are the intersections of the line segments GD and GE respectively with the line segment FC. How big is the ratio of the areas of triangle GIF and trapezium IHDE?
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Answer: A — \(\tfrac{1}{2}\)
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Hint 1 of 2
Set coordinates for the regular hexagon and find H, I as intersections on FC.
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Hint 2 of 2
Compare the two areas directly.
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Approach: coordinate geometry on the regular hexagon
- Place the hexagon with F and C on a horizontal diagonal and G at the midpoint of the top side AB.
- Lines GD and GE cross FC at H and I, symmetric about the centre.
- Computing areas, triangle GIF is exactly half of trapezium IHDE.
- Ratio = 1/2.
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