Problem 26 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplesquare-area
The big square shown is split into four small squares. The circle touches the right side of the square in its midpoint. How big is the side length of the big square? (Hint: the diagram is not drawn to scale.)
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Answer: A — 18 cm
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Hint 1 of 3
Tangency at the midpoint of the right side puts the circle's centre on the horizontal midline, a radius in from that side.
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Hint 2 of 3
Read the 8 cm and 6 cm marks as the gaps from the square's edges to where the circle crosses the two midlines.
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Hint 3 of 3
Set side \(s\) and radius \(r\); the two marks give two equations to solve together.
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Approach: place coordinates and turn the two marks into equations in side and radius
- Put the origin at the square's centre; tangency at the right side's midpoint puts the circle's centre at \((\tfrac{s}{2}-r,\,0)\) with radius \(r\).
- The 8 cm mark is the stretch of the horizontal midline from the left edge to the circle, so \(s - 2r = 8\); the 6 cm mark is the drop on the vertical midline from the top edge to the circle, so \(\tfrac{s}{2} - \sqrt{r^2-(\tfrac{s}{2}-r)^2} = 6\).
- Solving the pair gives \(r = 5\) and \(s = \mathbf{18}\) cm.
Mark:
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