Problem 2 · 2011 Math Kangaroo
Medium
Geometry & Measurement
areaarea-fraction
The area of the grey rectangle shown is 13 cm². X and Y are the midpoints of the slanted sides of the trapezium. How big is the area of the trapezium?
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Answer: C — 26 cm²
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Hint 1 of 2
X and Y are midpoints, so the rectangle's top edge is the trapezium's midsegment and its height is half the trapezium's height.
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Hint 2 of 2
Express the rectangle's area in terms of the trapezium's, then invert.
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Approach: relate the midpoint rectangle to the trapezium area
- The midsegment XY has length equal to the average of the two parallel sides, (b₁+b₂)/2.
- The grey rectangle's height is half the trapezium's height, h/2.
- Rectangle area = (b₁+b₂)/2 × h/2 = ½ × [(b₁+b₂)/2 × h] = half the trapezium's area.
- So the trapezium is 2 × 13 = 26 cm², choice (C).
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