IMC 2018 Q22
22. Three squares, with side-lengths 2, are placed together edge-to-edge to make an L-shape. The L-shape is placed inside a rectangle so that all five vertices of the L-shape lie on the rectangle, one of them at the midpoint of an edge, as shown.
What is the area of the rectangle?
- A 16
- B 18
- C 20
- D 22
- E 24
Intermediate Mathematical Challenge 2018, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 24
Let the area of each of the three squares be \(S\). Then the areas of triangles \(AJI\), \(AJB\), \(BKC\), \(CKD\) and \(GLF\) are all equal to \(\tfrac{1}{2}S\).
Also, the areas of triangles \(DEF\) and \(GHI\) are both equal to \(\tfrac{1}{4}S\).
So the total area inside the rectangle, but outside the three squares, is \(5 \times \tfrac{1}{2}S + 2 \times \tfrac{1}{4}S = 3S\). Therefore the area of the rectangle is \(6S = 6 \times 2 \times 2 = 24\).
(It is left as an exercise for the reader to show that the height and width of the rectangle are \(4\sqrt{2}\) and \(3\sqrt{2}\) respectively and therefore that its area is \(4\sqrt{2} \times 3\sqrt{2} = 24\).)
Solution © UK Mathematics Trust, from the IMC 2018 solutions. Longer versions with investigations: Solutions and investigations (PDF).