Given ABC is a square of area 100 sq. units and M and N the midpoints of the sides on which they lie, to find the areas 1, 2, 3 and 4.
Solution
The small triangle (3) can be got from triangle (4) by a rotation of 90° followed by a scaling. The hypotenuse of triangle (4) is equal to the side length of the square, and that of triangle (3) is half the side length of the square, so the dimensions of (3) are half those of (4) and its area one-quarter the area of (4).
But (3) and (4) between them make up one-quarter of the square, which represents an area of 25 sq. units and given their areas are in the ratio 1:4 the area of (3) must be 5 sq. units and that of (4) 20 sq. units.
The combined areas of regions (2) and (3) is also one-quarter of the square, hence the area of region (2) must also be 20 sq. units.
Finally the areas of these three regions add up to (20 + 5 + 20) sq. units, which leaves 55 sq. units for region 1.
Solution to puzzle 21/10/2025
This is the solution to the puzzle at https://learn1.open.ac.uk/mod/oublog/viewpost.php?post=312323
Given ABC is a square of area 100 sq. units and M and N the midpoints of the sides on which they lie, to find the areas 1, 2, 3 and 4.
Solution
The small triangle (3) can be got from triangle (4) by a rotation of 90° followed by a scaling. The hypotenuse of triangle (4) is equal to the side length of the square, and that of triangle (3) is half the side length of the square, so the dimensions of (3) are half those of (4) and its area one-quarter the area of (4).
But (3) and (4) between them make up one-quarter of the square, which represents an area of 25 sq. units and given their areas are in the ratio 1:4 the area of (3) must be 5 sq. units and that of (4) 20 sq. units.
The combined areas of regions (2) and (3) is also one-quarter of the square, hence the area of region (2) must also be 20 sq. units.
Finally the areas of these three regions add up to (20 + 5 + 20) sq. units, which leaves 55 sq. units for region 1.
So finally we have
Region 1: 55 sq. units
Region 2: 20 sq. units
Region 3: 5 sq. units
Region 4: 20 sq. units