Problem 24 · 2012 Math Kangaroo
Hard
Geometry & Measurement
areaarea-fraction
In a square ABCD, M is the midpoint of AB. MN is perpendicular to AC. Determine the ratio of the area of the grey triangle to the area of the square ABCD.
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Answer: D — 3 : 16
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Hint 1 of 2
Put the square on coordinates with side 2 and find N as the foot of the perpendicular from M to AC.
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Hint 2 of 2
Compute the grey triangle's area, then compare it to the square's area.
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Approach: coordinates and area
- Take A(0,0), B(2,0), C(2,2), D(0,2); M(1,0) is the midpoint of AB and AC is the line y = x.
- The foot of the perpendicular from M to AC is N(0.5, 0.5); the grey triangle has vertices M, N, C with area 3/4.
- The square's area is 4, so the ratio is (3/4) : 4 = 3 : 16.
Mark:
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