Emily says she can prove the Pythagorean Theorem using the following diagram. She explains that she can divide the squares on th
e two shorter sides into grids with equal-sized grid squares. She says she can then rearrange the grid squares to cover the area of the square on the hypotenuse, which proves that the sum of the squares on the two shorter sides equals the square on the hypotenuse. A 3 by 3 square, a 4 by 4 square, and a 5 by 5 square, put together to form a right triangle with legs 3 and 4 and hypotenuse 5. Is this a valid geometrical proof that a2 + b2 = c2 for all right triangles? Yes, because there are 9 grid squares on one side, 16 grid squares on the other side, and 25 grid squares on the hypotenuse, and 9 + 16 = 25. Yes, because it shows that you can rearrange the squares on the shorter sides to fill up the area of the square on the larger side. No, because a proof needs to have equations in order to be valid. No, because this proof only works when you can draw grids on the squares. It does not work for all right triangles.
<h2><em>y = ± 5⁄2 x</em></h2><h2><em>y = ± 5⁄2 xThe equation is the left half of the hyperbola. The domain is ( – ∞, – 1⁄5 ]. The range is ( – ∞, ∞ ). The vertical line test indicates that this is not the graph of a function.</em></h2>
