The Pythagorean Theorem is one of those “cornerstones” of math. It is the basis for trigonometry and is used throughout Algebra and Geometry courses.
The Pythagorean Theorem states that if given a right triangle, the sum of the legs squared will equal the hypotenuse squared. This theorem is named after the famous Greek mathematician Pythagoras.
We represent “the sum of the legs squared is equal to the hypotenuse squared” by the infamous formula a2 + b2 = c2, where a and b are the legs and c is the hypotenuse of a right triangle.
Let’s look at a right triangle and note what it means to square a length. Squaring is multiplying a number by itself as we learned in the last module. Squaring, then, is the same as finding the area of a square since a square has an equal length and width.
In this picture, we can see how the sides squared of a right triangle look.
Now, look what happens with c2. It is exactly the same size as a2 + b2 areas!
This is the Pythagorean Theorem!! Let’s look at the formula again now with greater clarity!
a2 + b2 = c2
Seeing is believing! I want you to try.
Get some graph paper and draw out a square of 3 units for the length and 3 units for the width. Now draw a second square with length and width of 4 units each. Set those aside. Now, since the theorem says that the sum of those two should equal the hypotenuse, we should be able to add the units from both of our squares together and cut a square of that size. The square with side lengths of 3 will have an area of 9. The square with side lengths 4 will have an area of 16. When we add that together, we get 25. In order to create a square that has an area of 25, we would need the side lengths to be 5. So now, cut out a 5×5 square. You should now have 3 squares: one with side lengths 3, one with side lengths 4, and a third with side lengths 5. On your graph paper, draw a triangle that has these same side lengths. You’ll see that your squares can be placed just like the ones in the pictures above this text. Here’s the cool part….place the 4×4 square on the 5×5 square. Now, take your 3×3 square and cut it so that it fits in the leftover units of the 5×5 square! So cool, right!!! This is proof for your eyes that the sum of the sides squared is equal to the hypotenuse squared when you have a right triangle.
The converse of the Pythagorean Theorem is also true. In other words, if you know all the side lengths of a triangle, but you aren’t sure if it is a right triangle, you can test it by substituting into the formula a2 + b2 = c2. If both sides are equal, then, yes, it is a right triangle.
Let’s take a look at a few examples.
To help you see an example of the Pythagorean Theorem in a real-world problem, take a look at these two examples:
Let’s take a closer look at why this happens in this visual proof.
Let’s now examine a few problems looking at the converse of the Pythagorean Theorem that we discussed earlier.
Remember, the Pythagorean Theorem works both ways….
IF you have a right triangle, THEN the sum of the legs squared is equal to the hypotenuse squared.
IF the sum of the legs squared is equal to the hypotenuse squared, THEN you have a right triangle.
Based on the converse of the Pythagorean Theorem, determine if the following are Pythagorean Triples. For example 3, 4, 5 is a Pythagorean Triple because those are the side lengths of a triangle that satisfies a2 + b2 = c2.
Is the set 9, 12, 15 a Pythagorean Triple?
To see, we’ll test 9, 12, and 15 in our formula defined by the Pythagorean Theorem. If it is true, then yes, we have a right triangle. If it is not true, then we do not have a right triangle.
Let’s test and see!
The set 9, 12, 15 represents a Pythagorean Triple and is therefore a right triangle.
Let’s consider one more area where you use the Pythagorean Theorem. The problems above were all in two dimensions. Let’s see what happens when we have a three dimensional space.
We can use the Pythagorean Theorem to help us find the distance between two points. In fact, the well-known “distance formula” is derived from the Pythagorean Theorem.