Problem 19 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
areapythagorean-triple
A circle is tangent to one side of a rectangle and passes through two of its vertices, as shown. A square of area 20 cm² has one side lying on a side of the rectangle and two vertices on the circle. What is the area of the rectangle?
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Answer: C — 50 cm²
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Hint 1 of 2
By symmetry put the circle's centre on the rectangle's top edge, on the vertical line of symmetry.
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Hint 2 of 2
The square's two upper corners sit a half-side left/right and a full side up from that centre — write that distance as the radius.
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Approach: put the circle's centre on the rectangle's top edge and use the square's corners to find the radius
- By symmetry the circle's centre lies on the rectangle's top edge; since the circle is tangent to the bottom edge and passes through the top vertices, the radius equals the rectangle's height and also its half-width.
- The square (side \(\sqrt{20}\)) stands on the top edge, so its upper corners are \(\tfrac{\sqrt{20}}{2}\) sideways and \(\sqrt{20}\) up from the centre: \(R^2 = \tfrac{20}{4} + 20 = 25\), so \(R = 5\).
- Thus the rectangle is \(10 \times 5\), with area \(50\) cm², option C.
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