Problem 29 · 2019 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
In square ABCD, the points P, Q and R are the midpoints of the edges DA, BC and CD. Which fraction of the square ABCD is shaded in the diagram?
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Answer: E — 38
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Hint 1 of 2
Place the square on coordinates with side 1 and find the corner points and midpoints.
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Hint 2 of 2
The shaded region is the big triangle from R down to the base, minus the little triangle below the crossing.
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Approach: coordinates and area subtraction
- Take A(0,0), B(1,0), C(1,1), D(0,1), so R(1/2,1), P(0,1/2), Q(1,1/2); lines A→Q and B→P cross at (1/2, 1/4).
- Triangle R-A-B has area 1/2; the small triangle A-B-crossing has area 1/2 × 1 × 1/4 = 1/8.
- Shaded = 1/2 − 1/8 = 3/8 of the square.
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