Problem 14 · 2023 Math Kangaroo
Hard
Geometry & Measurement
area-fractionarea-decomposition
The points M and N are the midpoints of two sides of the big rectangle (see diagram). Which part of the area of the big rectangle is shaded?
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Answer: C — \(\frac{1}{4}\)
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Hint 1 of 2
The line MN joins the midpoints of the two opposite sides, so it sits at half the height of the rectangle.
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Hint 2 of 2
Every shaded triangle has its base on MN and its tip on the top or bottom edge — so each has the same height.
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Approach: the shaded triangles all share one base line and a common height
- M and N are midpoints of the left and right sides, so the line MN runs across at half the rectangle's height h.
- Each shaded triangle has its base on MN and its tip on the top or bottom edge, so every one has height h/2.
- Their bases together cover the whole segment MN, which has length equal to the rectangle's width w.
- Total shaded area = ½ · w · (h/2) = wh/4, which is 1/4 of the rectangle.
Mark:
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