Problem 11 · 2017 Math Kangaroo
Hard
Geometry & Measurement
Algebra & Patterns
areasubstitution
ABCD is a trapezium with parallel sides AB and CD. Let AB = 50 and CD = 20. Point E lies on side AB in such a way that the straight line DE divides the trapezium into two shapes of equal area. How long is the straight line AE?
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Answer: C — 35
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Hint 1 of 2
The line DE makes triangle ADE on one side; its area is half base times height.
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Hint 2 of 2
Set that triangle equal to half the whole trapezium's area.
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Approach: express the half-area as a triangle and solve for the base AE
- The trapezium has area (50 + 20)/2 × h = 35h, so each half is 17.5h.
- Triangle ADE has base AE on AB and the same height h, area = ½ · AE · h.
- Setting ½ · AE · h = 17.5h gives AE = 35, choice C.
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