Problem 7 · 2012 Math Kangaroo
Medium
Geometry & Measurement
foldingarea
A rectangular piece of paper ABCD with the measurements 4 cm × 16 cm is folded along the line MN so that point C coincides with point A as shown. How big is the area of the quadrilateral ANMD′?
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Answer: C — 32 cm²
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Hint 1 of 2
The fold sends C onto A, so the crease MN reflects one part of the paper exactly onto the other.
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Hint 2 of 2
The quadrilateral ANMD' is the mirror image of region MNCD, so it has the same area.
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Approach: folding preserves area; find the reflected region
- Folding along MN reflects corner C onto A, mapping region MNCD onto quadrilateral ANMD'.
- Placing B=(0,0), C=(16,0), the equal-distance conditions give N=(7.5,0) on BC and M=(8.5,4) on AD.
- Region MNCD is a trapezoid with parallel sides 7.5 and 8.5 and height 4, area (7.5+8.5)/2·4 = 32.
- So ANMD' also has area 32 cm².
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