Problem 22 · 2025 Math Kangaroo
Hard
Geometry & Measurement
area-decompositionratio
The two small rectangles in the diagram are congruent and each has an area of 4 cm². What is the area of rectangle ABCD in cm²?
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Answer: D — 12
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Hint 1 of 3
The diagonal \(AC\) of the big rectangle passes through the meeting corner of the two small rectangles.
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Hint 2 of 3
The diagonal cuts ABCD into two equal halves; compare how many small-rectangle areas fit against that diagonal.
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Hint 3 of 3
Each small rectangle has its diagonal corner on \(AC\), so each is split into two equal triangles by the diagonal — use that to count areas.
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Approach: use the main diagonal to balance the areas
- Draw diagonal \(AC\); it runs through the common corner of the two congruent rectangles and splits ABCD into two equal triangles of area \(\tfrac12[ABCD]\) each.
- Below the diagonal the dotted rectangle and the leftover triangle make up one half; matching the congruent rectangles against the diagonal shows the big rectangle is built from three small-rectangle areas.
- So \([ABCD]=3\times 4=\) 12 cm², answer D.
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