Problem 24 · 2019 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplesquare-area
Two vertices of a square lie on a semi-circle, as shown, while the other two lie on its diameter. The radius of the circle is 1 cm. How big is the area of the square?
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Answer: A — 45 cm2
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Hint 1 of 2
Put the centre of the diameter at the origin; the square sits symmetrically on it.
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Hint 2 of 2
If the side is s, the top corners (±s/2, s) lie on the radius-1 circle.
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Approach: put a corner on the circle and solve
- By symmetry the base runs from (−s/2, 0) to (s/2, 0) with top corners (±s/2, s).
- These lie on the circle: (s/2)² + s² = 1, so (5/4)s² = 1.
- Thus s² = 4/5, and the area is 4/5 cm².
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