IMC 2015 Q19
19. The figure shows four smaller squares in the corners of a large square. The smaller squares have sides of length 1 cm, 2 cm, 3 cm and 4 cm (in anticlockwise order) and the sides of the large square have length 11 cm.
What is the area of the shaded quadrilateral?
- A 35 cm2
- B 36 cm2
- C 37 cm2
- D 38 cm2
- E 39 cm2
Intermediate Mathematical Challenge 2015, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 35 cm2
The small numbers in the figure show the lengths in cm of each line segment. The larger numbers inside the figure show the areas in cm2 of each square or trapezium. (The area of a trapezium is \(\tfrac{1}{2}(a + b)h\) where \(a\) and \(b\) are the lengths of the parallel sides and \(h\) is the perpendicular distance between them.)
So the area of the shaded portion in cm2 is
11 × 11 − (1 + 12 + 4 + 15 + 9 + 14 + 16 + 15) = 35.
(See the extended solutions for a beautifully elegant solution of this problem.)
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).