IMC 2020 Q19
19. The diagram shows two squares and four equal semicircles. The edges of the outer square have length 48 and the inner square joins the midpoints of the edges of the outer square. Each semicircle touches two edges of the outer square, and the diameter of each semicircle lies along an edge of the inner square.
What is the radius of each semicircle?
- A 10
- B 12
- C 14
- D 16
- E 18
Intermediate Mathematical Challenge 2020, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 12
The diagram shows the top left-hand corner of the original diagram. The centre of the semicircle shown is \(Q\). Also, \(U\) and \(S\) are the points where the edges of the bigger square touch the semicircle shown. Therefore both \(QU\) and \(QS\) are radii of the semicircle and \(\angle TSQ = \angle TUQ = 90°\). Also \(\angle UTS\) is a right angle as it is the corner of a square. Therefore \(UQST\) is a square. Hence \(QS = ST\).
Note that \(TR = TP\) because \(P\) and \(R\) are midpoints of the original large square. Therefore \(\angle PRT = 45°\). So \(QS = RS\).
Hence \(QS\) is half the length of \(TR\), which is itself half of the length of a side of the outer square, which is 48 cm.
So the radius of the semicircle is one quarter of 48 cm = 12 cm.
Solution © UK Mathematics Trust, from the IMC 2020 solutions. Longer versions with investigations: Solutions and investigations (PDF).