We are asked to explain how to use the Pythagorean Theorem to solve problems. First, let's recall the Pythagorean Theorem.
Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
We can also represent the Pythagorean Theorem graphically. The legs are the sides of a right triangle adjacent to the right angle and the hypotenuse is the side opposite the right angle.
We can only use the Pythagorean Theorem when we have a right triangle. However, sometimes we can use the Pythagorean Theorem to solve a problem that, at first glance, does not involve triangles at all. Let's consider an example of this type of problem. Imagine we are given a rectangular box and want to display a decorative pen in it.
The box is 5 inches long and 4 inches wide. Suppose we want to purchase a pen that is 6 inches long. We want to check whether we can fit the pen in the box. We can find the length of the diagonal of the box algebraically to see if the pen will fit. Let's call this diagonal length x.
Notice that the diagonal and the two sides of the box form a right triangle.
The Pythagorean Theorem tells us that the sum of the squares of the lengths of the legs — in this case, 5 and 4 inches — is equal to the square of the length of the hypotenuse x.
5^2 + 4^2 = x ^2
Let's solve this equation!
The diagonal is approximately 6.4 inches, which is more than 6 inches. This means that the pen will fit in the box!
6.4 > 6
We used the Pythagorean Theorem to solve the problem by creating and solving a right triangle.
