Solving Right Triangles Using Trigonometry & the Pythagorean Theorem

Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

The Pythagorean Theorem is one of the most well-known and useful math equations. Its simple form and abundant uses make it one of the most helpful processes to learn in math. In this lesson, learn how to solve right triangles using the Pythagorean Theorem.

Practical Application of Pythagorean Theorem

You're playing a dogleg left in golf (a hole that bends about 90 degrees to the left). You could play it safe and hit it short onto the fairway. Or, you could try to hit it over the trees onto the green. The problem is, you don't know how far it is.

A Hole Illustrating the Use of Pythagorean Theorem
A Hole Illustrating the Use of Pythagorean Theorem

Too bad you didn't pay more attention in math class. If you did, you'd know how to use the Pythagorean Theorem to help you make this choice. But first, a review of some basics.

Right Triangles

Right angles hold a hallowed place in math. However, before getting into the most well-used right triangle theorem, it is first necessary to understand some basic right triangle terminology.

Illustration of Pythagorean Theorem
Illustration of Pythagorean Theorem

The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. The letter 'c' usually represents its length, as we see in the diagram. The other two sides are called legs, and their lengths are usually represented by the letters 'a' and 'b'. The interesting (and useful) thing about right triangles is that if you square the hypotenuse length (c), it will be equal to the square of leg a and the square of leg b added together. This is represented by the famous equation, a 2 + b 2 = c 2 , otherwise known as the Pythagorean Theorem.

Since we know lengths of the two legs, we can use that information to find the hypotenuse (i.e., the distance to carry over the trees in our golf problem). Before getting there, though, let's do a quick review of solving for squared variables.

Solving for Squared Variables

In solving equations, we always do the opposite operation to 'get rid of' anything on the variable's side. To solve x - 16 = 1, we add 16 to each side, giving us 17. On the left-hand side of that equation, adding 16 gets rid of the 16 (since -16 + 16 = 0). On the right-hand side, 1 + 16 = 17. Similarly, to solve x / 4 = 2, we multiply each side by 4, giving us a solution of 8. To solve problems with squared variables, the opposite operation of squared is square root. Let's see that in action.

Common sense tells us that for the equation x 2 = 9, the solution for x is 3 (it is actually +/- 3, but we're not going to worry about the -3 in this lesson since we are dealing with lengths, which are always positive). How to find this solution? Simple: just perform the opposite operation, square root to each side. The square root of x 2 is x, and the square root of 9 is 3. This skill is necessary in Pythagorean Theorem problems.

Finding a Hypotenuse

Consider a right triangle with legs of length 3 and 4. What is the length of the hypotenuse?

Right triangle with legs 3 and 4
Right triangle with legs 3 and 4

The legs are sides a and b. Plugging into the equation, we see:

a 2 + b 2 = c 2

3 2 + 4 2 = c 2

9 + 16 = c 2

c 2 = 25

d

c = 5

The length of the hypotenuse is 5.

Finding a Leg Length

Consider a right triangle with a leg of 10 and a hypotenuse of 26. What is the other leg's length?

Right Triangle with Leg of 10 and Hypotenuse of 26
Right Triangle

Well, first we should probably switch the equation around to isolate one of the leg lengths. It doesn't matter which letter we use for the short leg or long leg. For now, we'll say that we're trying to find a. If a 2 + b 2 = c 2 , then a 2 = c 2 - b 2 . Plugging in the numbers:

a 2 = c 2 - b 2

a 2 = 26 2 - 10 2

a 2 = 676 - 100

a 2 = 576

d

a = 24

The missing leg length is 24.

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