Problem 14 · 2022 Math Kangaroo
Hard
Geometry & Measurement
area-fraction
The square pictured is split into two squares and two rectangles. The vertices of the shaded quadrilateral with area 3 are the midpoints of the sides of the smaller squares. What is the area of the non-shaded part of the big square?
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Answer: D — 21
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Hint 1 of 2
The shaded kite's corners are the side-midpoints of the two small squares around the centre.
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Hint 2 of 2
Compute the kite's area as a fraction of the whole square — it doesn't depend on how the square is split.
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Approach: kite is one-eighth of the big square
- Place the big square as side a+b with the smaller squares of sides a and b meeting at the centre cross.
- The kite's four vertices are the relevant side-midpoints; its area works out to exactly 1/8 of the big square (independent of a and b).
- So the big square has area 8 × 3 = 24.
- Non-shaded part = 24 − 3 = 21.
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