Problem 7 · 2022 Math Kangaroo
Medium
Geometry & Measurement
area-fractionarea-decomposition
The midpoints of both longer sides of a rectangle are connected with the vertices (see diagram). Which fraction of the rectangle is shaded?
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Answer: B — \(\tfrac{1}{4}\)
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Hint 1 of 2
Set up coordinates with the rectangle 2 wide and 2 tall for easy midpoints.
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Hint 2 of 2
The shaded rhombus is built from the triangles formed by the crossing lines.
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Approach: coordinate area of the central rhombus
- Let the rectangle be 2 wide and 2 tall; the long-side midpoints are the top and bottom centres.
- The slanted lines from those midpoints to the vertices bound a central rhombus.
- Computing its area against the 2x2 rectangle gives the shaded part as 1/4 of the whole.
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