Problem 23 · 2013 Math Kangaroo
Stretch
Geometry & Measurement
areacasework
We are looking at rectangles where one side has length 5.0 cm. Among them are some that can be cut into a square and a rectangle, one of which has an area of 4.0 cm². How many such rectangles are there?
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Answer: D — 4
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Hint 1 of 2
Cutting a 5-by-L rectangle once gives a square (side = the shorter dimension) plus a leftover rectangle.
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Hint 2 of 2
Set either the square's area or the leftover's area to 4 and solve in each case.
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Approach: case-split on which piece has area 4
- If the side of length 5 is the short one: square is 5 × 5, and leftover 5 × (L − 5) = 4 gives L = 5.8.
- If the side of length 5 is the long one: square L × L gives L² = 4 so L = 2; or leftover L · (5 − L) = 4 gives L = 1 and L = 4.
- That is 4 distinct rectangles.
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