Halving a square

Transcript

Halving a squareNow, let's go in reverse. We have a big square, and we want one that is half the size. The Method: Instead of building out on the diagonal, we build in by connecting the midpoints of the sides. Why is the smaller inside square (PQRS) half the area of the larger square? Look at the folded paper. When you fold the corners in to meet at the center, you are covering up the middle of the paper. The corners you folded in (blue arrows) exactly cover the inner square. This means the paper is now "double thick." If you unfolded it, you'd see that the inner square (PQRS) takes up exactly half the space, and the four corners take up the other half Triangle Count: The big square contains 8 small triangles. The inner square contains 4 small triangles. Since 4 is half of 8, the area is half! Why is PQRS a square? This is a great geometry proof question! Sides are equal: We connected the midpoints. By symmetry, the distance from P to Q is the same as Q to R, and so on. All four sides are equal. Angles are 90 degrees: When you fold the 45-degree angles of the corner triangles in, they meet to form straight lines and right angles at the vertices P, Q, R, and S. Since it has equal sides and 90-degree corners, PQRS is definitely a square.