The Pythagorean Theorem allows you to find the lengths of the sides of a right triangle, which is a triangle with one angle (known as the right angle). An example of a right triangle is depicted below.
A right triangle is composed of three sides: two legs, which are labeled in the diagram as and , and a hypotenuse, which is the side opposite to the right angle. The hypotenuse is always the longest of the three sides. Typically, we denote the right angle with a small square, as shown above, but this is not required.
The Pythagorean Theorem states that the length of the hypotenuse squared equals the sum of the squares of the two legs. This is written mathematically as:
To verify this statement, first explicitly expressed by Pythagoreans so many years ago, let’s look at an example.
Consider the right triangle below. Does the Pythagorean Theorem hold for this triangle?
As labeled, this right triangle has sides with lengths 3, 4, and 5. The side with length 5, the longest side, is the hypotenuse because it is opposite to the right angle. Let’s say the side of length 4 is and the side of length 3 is .
Recall that the Pythagorean Theorem states:
Although it is clear that the theorem holds for this specific triangle, we have not yet proved that the theorem will hold for all right triangles. A simple proof, however, will demonstrate that the Pythagorean Theorem is universally valid.
Proof Based on Similar Triangles
The diagram below depicts a large right triangle (triangle ) with an altitude (labeled ) drawn from one of its vertices. An altitude is a line drawn from a vertex to the side opposite it, intersecting the side perpendicularly and forming a angle.
In this example, the altitude hits side at point and creates two smaller right triangles within the larger right triangle. In this case, triangle is similar to triangles and . When a triangle is similar to another triangle, corresponding sides are proportional in lengths and corresponding angles are equal. In other words, in a set of similar triangles, one triangle is simply an enlarged version of the other.
Similar triangles are often used in proving the Pythagorean Theorem, as they will be in this proof. In this proof, we will first compare similar triangles and , then triangles and .
Comparing Triangles and
In the diagram above, side corresponds to side . Similarly, side corresponds to side , and side corresponds to side . It is possible to tell which side corresponds to the appropriate side on a similar triangle by using angles; for example, corresponding sides and are both opposite a right angle.
Because corresponding sides are proportional and have the same ratio, we can set the ratios of their lengths equal to one another. For example, the ratio of side to side in triangle is equal to the ratio of side to corresponding side in triangle :
Comparing Triangles and
Triangle is also similar to triangle . Side corresponds to side , side corresponds to side , and side corresponds to side .
Using this set of similar triangles, we can say that:
Earlier, we found that . If we replace with , we obtain . This is just another way to express the Pythagorean Theorem. In the triangle , side is the hypotenuse, while sides and are the two legs of the triangle.