# Square in a Square

In the larger square, there is a smaller square formed by lines connecting each corner to the midpoint of one of the sides.

Question: What fraction of the larger square is the smaller filled square?

#### Solution

If you rotate the small triangles next to the trapeziums, they appear to form squares identical to the one in the center. Since there are 5 squares, it would appear the center square is 1/5 the area of the larger square.

However, we need to prove that this method actually forms squares identical to the one in the center.

First, the small triangle and the large triangle share an angle, and they are both right triangles, so they are similar triangles.

Since we already know the length of the hypotenuse of the smaller triangle is half the length of the hypotenuse of the larger triangle, the other sides of the smaller triangle must also be half the length of the corresponding sides of the larger triangle.

This means if we rotate the smaller triangle so it fits against the trapezium from the lower portion of the larger triangle, it forms a square (all the angles are right angles, and all the sides are the same length).

And we know the side length of this square is the same as the side length of the center square. Therefore we have shown the area of the larger square can be manipulated into five identical smaller squares, so the square in the center must be 1/5 of the area of the larger square.