Puzzles

The square is split into 3 rectangles by drawing two lines on the rectangle. There are two cases: the lines can both go in the same direction; or the lines go in perpendicular directions, with one going all the way across the square and one going from an edge of the square to the other line.

For an $n$ by $n$ square, there are:

In total this makes $(n-1)(5n-6)$ ways to split the square. (10–1)×(5×10-6) is 396.

Let $a$ be the height of the rectangle. As the total of the blue lines is 50, the width of the rectangle is $50-a$. As the total of the red lines is 50, each red line segment is 25.

Using Pythagoras's theorem in one of the right-angled triangles, we see that:

$a$ is not zero, and so $a=20$. This means that the area of the rectangle is 20×30=600.

The only way this is possible is if the rectangle is this square:

The area of this square is 162.

Let $a$ be the height of the rectangle and $b$ be the width.

The length of the red line is $a+b$. The length of the blue line is $2\sqrt{a^2+\frac{b^2}{4}}$. These are equal so:

Therefore the ratio of the sides is 2:3.

The largest rectangle will be a square. 20 (double the radius) will be the length of its diagonal. By Pythagoras' Theorem, the sides of the square are $10\sqrt{2}$. Therefore the area of the square is 200.