Problem 17 · 2024 Math Kangaroo
Stretch
Geometry & Measurement
area-fractionarea
The two large squares have the same area. Parts of them are coloured grey (see picture). In the left square, the dots divide the sides into two equal pieces. In the right square, the dots divide the sides into three equal pieces. The four grey parts in the left square have a combined area of 9 cm². What is the area of the four grey parts in the right square?
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Answer: B — 8 cm²
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Hint 1 of 2
Use the left square first: the midpoint construction tells you what fraction of the square the grey corners cover, which fixes the area of the whole square.
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Hint 2 of 2
Then express the right square's grey pieces as a fraction of that same total area.
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Approach: find the common square area from the left figure, then take the right figure's fraction
- In the left square the midpoints make the inner tilted square exactly half the big square, so the four grey corner triangles are the other half: 9 cm² is half, giving a square of 18 cm².
- In the right square the dots cut each side into thirds, and the four grey pieces there make up 4/9 of the square.
- So the grey area on the right is 4/9 × 18 = 8 cm².
- The four grey parts on the right have area 8 cm².
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