In our next triangle, the measure of one leg is 12 inches and the measure of the hypotenuse is 20 inches. For this triangle, we have the value of the hypotenuse, but must find the length of the missing leg. Therefore, we will let a = 12 and c = 20.
By substituting this information into the formula, we have 12^2 + b^2 = 20^2. When we simplify, we get 144 + b^2 = 400. At this point, we should subtract 144 from both sides to get b^2 = 256. Once we take the square root of both sides, we see that b, the missing leg in our triangle, has a value of 16 inches.
Did anything about the numbers in that last example stand out to you? Notice that all of the side lengths are whole numbers, not fractions or decimals. This means that the side lengths for this triangle are a Pythagorean triple, which is defined as a set of nonzero whole numbers that satisfy the Pythagorean theorem.
A few more examples of Pythagorean triples are 3, 4, 5, and 28, 45, 53. If you plug these numbers into the Pythagorean theorem, you will see that 3^2 + 4^2 = 5^2, and 28^2 + 45^2 = 53^2.
Pythagorean Theorem in Everyday Life
Now that we have an understanding of the Pythagorean theorem, let's apply it to a few everyday examples, beginning with Jessica's garden from the beginning of the lesson.
Let's review the problem again. Jessica has a square garden. To make watering her plants easier, she would like to create a diagonal pathway that connects opposite corners. If one side of her garden is 13 feet long, what will be the length of her diagonal pathway?
For this situation, we should definitely use the Pythagorean theorem. We know that in a square, all four angles measure 90 degrees and all four sides are congruent. So if one side has a length of 13 feet, then all the other sides have a length of 13 feet. By taking half of this square, the diagonal becomes the hypotenuse and we have a right triangle with both legs equal to 13 feet.
We are now ready to plug in our information to find the length of the hypotenuse. When we do, we will see that 13^2 + 13^2 = c^2. By simplifying and combining like terms, we end up with 338 = c^2, and c = 18.38 as our final answer, which will be the length of Jessica's diagonal pathway.
Let's do another. Elizabeth needs to repair her roof. She props a 21-foot ladder against the side of her 20-foot home. If the top of the ladder reaches the top of her house, how far away from her home is the base of the ladder?
In this problem, we are still working with a right triangle. The ladder is the hypotenuse, while the height of the home and the ground between the home and the ladder are the legs of this right triangle.
Using the given information, our beginning equation is 20^2 + b^2 = 21^2. Simplifying brings us to 400 + b^2 = 441. From here, we subtract 400 from both sides to get b^2 = 41, from which we see that b = 6.40. Therefore, we can conclude that the base of the ladder is 6.40 feet away from the home.
Lesson Summary
In summary, the Pythagorean theorem, which can only be used with right triangles, is a very important theorem that allows us to find the length of a missing side in a triangle. The theorem is interpreted as the formula a^2 + b^2 = c^2. The legs are represented by a and b and can be labeled interchangeably, but the hypotenuse must be labeled c. In order for this theorem to be used, two sides of the right triangle must be known.