Exercise 2 Page 384

We are asked whether any three squares could form the sides of a right triangle. Let's try to form a right triangle using the sides of three identical squares with side length $a.$

Notice that the squares form an equilateral triangle. The measure of each angle in an equilateral triangle is $60^{\circ },$ so this triangle is not a right triangle.

This means that it is not true that any three squares will form the sides of a right triangle. In general, given three squares, we can use the Pythagorean Theorem to justify whether they will form a right triangle. Let's consider three example squares where $a,$ $b,$ and $c$ are the side lengths of the squares.

If the sides of the squares form the sides of a right triangle, then their lengths must satisfy the Pythagorean Theorem. This means that the sum of squares of the side lengths of the two smaller squares $a$ and $b$ must be equal to the square of the side length of the biggest square $c.$ If that is not the case, then the triangle formed by the sides of the squares cannot be a right triangle.