This solution is from David Lowe, age 16, from Trinity School, Carlisle, Cumbria.
A square is folded so that the corner E coincides with the midpoint of an opposite edge as shown in the diagram. The length of the edge of the square is 1 unit and the problem is to find the lengths and areas of the three triangles.
David uses trigonometry but alternatively, once you have found the dimensions of triangle EGF you can find all the other lengths and areas using the fact that the triangles EGF, CED and CAB are similar 3:4:5 triangles.
The table shows the solutions:
| Triangle | Lengths of sides 3 : 4 : 5 |
Ratio of lengths in 3 triangles |
Area | Ratio of areas of 3 triangles |
| EGF | 3/8 : 1/2 : 5/8 | 3 | 3/32 | 9 |
| CED | 1/2 : 2/3 : 5/6 | 4 | 1/6 | 16 |
| CAB | 1/8 : 1/6 : 5/24 | 1 | 1/96 | 1 |
This is David's method: