Pythagorean Theorem

The Pythagorean Theorem is one of the fundamental principles of Euclidean geometry and defines the relationship between the three sides of a right triangle. It was first proven by an ancient Greek mathematician and religious leader named Pythagoras of Samos who believed that everything in the universe was comprised of and could be described by numbers. Modern science sure makes use of this idea!

The Pythagorean Theorem (sometimes called Pythagoras' Theorem) states that in a right triangle with legs of lengths a and b and a hypotenuse of length c, the area of the square whose side is the hypotenuse is equal to the sum of the areas on the squares on the other two sides. If that doesn't make much sense to you, consider the following diagram:

All the Pythagorean Theorem is saying is that the area of square c is equal to the combined areas of squares a and b. Or, the green square has the same area as the blue and red squares added together. That's not so bad, right?

Expressing the Pythagorean theorem mathematically

Saying the Pythagorean Theorem in words can be difficult and you won't always have access to a diagram like the one above. Fortunately, it can be expressed as a simple equation:

$a^{2} + b^{2} = c^{2}$ , where c represents the length of the hypotenuse and a and b represent the lengths of the other two sides of the triangle.

The validity of this equation (sometimes called the Pythagorean Equation) has been proven countless times through various mechanisms, some of which date back thousands of years. Any three whole numbers satisfying this equation are called Pythagorean Triples. As long as you memorize this equation, you'll be prepared to apply the Pythagorean Theorem whenever it comes up.

Remember, however, that the Pythagorean Theorem only applies to right triangles. If you're working with a triangle that doesn't have an angle measuring exactly 90 degrees, you cannot apply the Pythagorean Theorem.

Applying the Pythagorean theorem

As you know from your study of algebraic equations, you can manipulate both sides of an equation to isolate a specific variable and determine more information than you were originally provided with. The same thing applies to the Pythagorean Equation.

If you know the length of both legs of a right triangle but not the hypotenuse, you can use the following equation to solve for c:

$\sqrt{a^{2}} + \sqrt{b^{2}} = \sqrt{c^{2}}$

Likewise, if you know the length of the hypotenuse and one of the legs but not the other, you can use the following equations to get the length of the remaining leg:

$a = \sqrt{c^{2} - b^{2}}$ OR $b = \sqrt{c^{2} - a^{2}}$

Since all of these formulas were derived from the Pythagorean Equation, they only apply to right triangles. Eventually, you'll study the law of cosines and discover how to solve for any side of any triangle with the lengths of two sides and the angle between them. When you do, you might notice that the equation can be reduced to the Pythagorean Equation when you're working with a right triangle. Isn't it great when mathematics works out so perfectly?

The converse of the Pythagorean theorem

While the most obvious application of the Pythagorean Theorem is to determine the length of the sides of a right triangle, the converse is also true. If a triangle has sides a, b, and c, and if $a^{2} + b^{2} = c^{2}$ , then it must be a right triangle. Similarly, any triangle with three sides that do not satisfy the $a^{2} + b^{2} = c^{2}$ equation cannot be a right triangle. This is a great way to identify right triangles when they aren't directly labeled for you.

Practical applications of the Pythagorean theorem

Many students struggle with math concepts that feel abstract, often asking, "How am I ever going to use this?" The Pythagorean Theorem is frequently among the biggest offenders since the entire concept revolves around working with very specific triangles you wouldn't expect to find outside of your mathematics textbook.

The Pythagorean Theorem is important for two reasons. First, professionals in STEM fields often use it to describe distances or theoretical objects that haven't been created yet. For example, surveyors will use it to ensure their maps accurately depict the slope and height of geographic features they're describing while engineers utilize it to accurately predict how much material they'll need to design prototypes. There are also practical applications in activities such as flying a kite, though obviously, you can do that without knowing what a hypotenuse is.

Second, the Pythagorean Theorem represents a breakthrough in humanity's creative critical thinking skills. Pythagoras didn't have a modern high school math book to work with and still came up with a theory that can be applied to an infinite number of triangles. Later mathematicians were able to build on his work and discover hundreds of independent proofs and applications, firmly identifying mathematics as an outlet for creative thinking.

If you're considering a career in a STEM field, those abstract triangles might not remain abstract for too much longer. If you aren't, you can still look at the Pythagorean Theorem as an example of what can be achieved if you put your mind to it.

Pythagorean theorem practice problems

a. If the two non-hypotenuse legs of a right triangle measure 3 and 4 meters, respectively, what is the perimeter of the triangle? (Remember that you find the perimeter by adding all three sides of the triangle together)

$3^{2} + 4^{2} = c^{2}$

$9 + 16 = c^{2}$

$25 = c^{2}$

$c = \sqrt{25}$

$c = 5 meters$

Now we can find the perimeter by adding all three sides together:

$Perimeter = 3 + 4 + 5 = 12 meters$

b. If a right triangle's hypotenuse measures 50 inches and one of its legs measures 14 inches, what is the length of the third leg?

$14^{2} + b^{2} = 50^{2}$

$196 + b^{2} = 2500$

$b^{2} = 2304$

$b = \sqrt{2304}$

$b = 48 inches$

c. What is the area of a right triangle with a hypotenuse measuring 70 centimeters and one leg measuring 42 centimeters? (The formula for the area of a triangle is ½ x base x height)

$42^{2} + b^{2} = 70^{2}$

$1764 + b^{2} = 4900$

$b^{2} = 3136$

$b = \sqrt{3136}$

$b = 56 centimeters$

Now we can find the area using the formula for the area of a triangle ( $A = \frac{1}{2} \times base \times height$ ):

$Area = \frac{1}{2} \times 42 \times 56 = 1176 {cm}^{2}$

d. The three sides of a triangle measure 10 feet, 20 feet, and 40 feet. Could this be a right triangle? Why or why not?

No, the numbers don't satisfy the equation $a^{2} + b^{2} = c^{2}$

e. The three sides of a triangle measure 30 inches, 40 inches, and 50 inches. Could this be a right triangle? Why or why not?

$30^{2} + 40^{2} = 50^{2}$

$900 + 1600 = 2500$

$2500 = 2500$

Since the equation holds true, this can be a right triangle.

Topics related to the Pythagorean Theorem

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Converse of Pythagorean Theorem

Congruent Angles

Flashcards covering the Pythagorean Theorem

8th Grade Math Flashcards

Basic Geometry Flashcards

Practice tests covering the Pythagorean Theorem

Common Core: High School - Geometry Diagnostic Tests

Intermediate Geometry Diagnostic Tests

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