Today’s Question of the Day:

How do you find the side lengths of a right triangle?

Let’s start by reviewing some terms about a right triangle. A right triangle has two sides (or legs) and a hypotenuse (the side opposite the right angle). In the right triangle ΔABC, the legs are sides a and b and the hypotenuse is side c.

The Pythagorean Theorem is defined by the sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). Since the area of a square is defined by side2, the area of square on leg a is a2. Then, the Pythagorean Theorem is, a2 + b2 = c2.

We can now try to test our knowledge!

Question 1: Solve for c.

For this example, a = 3, b = 4, and we want to solve for variable c.
Using the Pythagorean Theorem, we plug in a, b, and c: 32 + 42 = c2

Now, we simplify the left side of the equation.

9 + 16 = c2 Simplify 32 and 42.
25 = c2 Add 9 and 16.
√25 = √c2 Square root both sides of the equation.
5 = c Solve for c.

So, for this right triangle, a = 3, b = 4, and c = 5.

Question 2: Solve for b.

For this example, a = 5, c = 13, and we want to solve for variable b.
Using the Pythagorean Theorem, we plug in a, b, and c: 52 + b2 = 132

Now, we simplify the left side of the equation.

25 + b2 = 169 Simplify 52 and 132.
b2 = 144 Subtract 25 from 169.
√b2 = √144 Square root both sides of the equation.
b = 12 Solve for b.

In this right triangle, a = 5, b = 12, and c = 13.

*Side Learning*
The right triangles from Question 1 (3-4-5) and Question 2 (5-12-13) are Common Pythagorean Triples. Some of the most well-known triples are listed below.

a b c
3 4 5
5 12 13
8 15 17
7 24 25
9 40 41

*Bonus Question*
We have already solved the first two Common Pythagorean triples. Show that the Pythagorean Theorem works for the other three triples.