How big?
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
Diagram not to scale
The following solution came from Alistair Jacklin, class 9S, King Edward Camp Hill School for Boys. Kang Hong Joo of The Chinese High School, Singapore and Neil Wright of Madras College, St Andrew's and Jessica Zhang, Woodbridge School, Essex used the same method.
The first thing I did was to split the large triangle into 3 sections, a square and 2 smaller triangles. Next I then worked out the area of the large triangle using the formula:-
| Height * Width * 0.5 | = | area of the triangle |
| 3 * 4 * 0.5 | = | area of a triangle |
| 6 | = | area of a large triangle |
With this I used a formula to show the area of this triangle by adding the areas of the other parts:
x 2 + 0.5 x(3 - x) + 0.5 x(4 - x) = 6.The formula is then simplified in several easy steps to give:-
| x 2 + 1.5 x - 0.5 x 2 + 2 x - 0.5 x 2 | = | 6 |
| 3.5 x | = | 6 |
| x | = | 12/7 |
The area of the square is then x 2 = 144/49 exactly. Rounded to three decimal places this is 2.938.