IMC 2006 Q24
24. The diagram shows a square of area \(x\) square units inscribed inside a semicircle and a larger square of area \(y\) square units inscribed inside a circle.
What is the ratio \(x : y\)?
- A \(1 : \sqrt{2}\)
- B \(1 : 2\)
- C \(2 : 5\)
- D \(1 : 3\)
- E \(\sqrt{3} : 4\)
Intermediate Mathematical Challenge 2006, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C \(2 : 5\)
Let \(r\) be the radius of the circle. Then, in the smaller square:
\(r^2 = (\sqrt{x})^2 + (\tfrac{1}{2}\sqrt{x})^2 = x + \tfrac{x}{4} = \tfrac{5x}{4}\)
and in the larger square:
\(r^2 = (\tfrac{1}{2}\sqrt{y})^2 + (\tfrac{1}{2}\sqrt{y})^2 = \tfrac{y}{4} + \tfrac{y}{4} = \tfrac{y}{2}\)
So \(\tfrac{5x}{4} = \tfrac{y}{2}\) and we deduce that \(x : y = 2 : 5\).
Solution © UK Mathematics Trust, from the IMC 2006 solutions.