The Pythagorean Theorem Examples

Find the value of x.

Like we always do, we start with Mr. Pythagoras's theorem.

a² + b² = c²

Substitute in the side lengths we know.

(24)² + x² = (30)²

Achin' for some simplification?

576 + x² = 900

Subtracting 576 from both sides, we get:

x² = 324

Just square root both sides, and presto.

x = 18

For brownie points, would this triangle be considered a Pythagorean triple? The side lengths are 18, 24, and 30, which are all whole numbers. Yes, it would be admitted into the Pythagorean Triple Club. Actually, it's probably already soaking in the Jacuzzi.

Find the value of x.

We have two triangles here, but since we're looking for x, we'll ignore the one on the right. Once again, start with our good old Pythagorean Theorem.

a² + b² = c²

In this case, x is our hypotenuse and the other two values are legs. Substituting them into the equation, we get:

(29)² + (44)² = x²

Simplify and add.

841 + 1936 = x²
2777 = x²

Take the square root and we're done.

x ≈ 52.7

Is this a right triangle?

All right triangles must follow the Pythagorean Theorem, and all triangles that satisfy the Pythagorean Theorem are right triangles. Need we say more?

a² + b² = c²

Substitute in all the values.

8² + 9² ≟ 10²

Solve and see if the right side equals the left side.

64 + 81 ≟ 100

145 ≠ 100

Nope, 145 definitely doesn't equal 100. This triangle might not be wrong, but it's certainly not right.

A triangle has three points at A (-10, -3), B (-5, -5), and C (-7, -10). Is ∆ABC a right triangle?

In order to find out whether these points make a right triangle or not, we need to find the distances between them. The distance between two points is given by the formula , where d is the distance and x and y are the coordinates of points 1 and 2. We'll need to do this three times, so let's get to it.

We'll start with the distance between A and B.

Substituting in the coordinates, we get:

Simplifying the numbers gives us:

Square the numbers and then add.

Since plugging this into our calculater will only give us an approximate, we'll leave the answer as-is for now. Next up, the distance from B to C:

Go through the same thing again, making sure to substitute the numbers in the right places.

Simplify, square, and add.

Hmmm. The same as d_AB. We'll do this one last time for A and C.

Plug it in, plug it in. Then simplify, of course.

Whew. We figured out the distances between the points: , , and . Those are the lengths of the triangle's sides. They don't tell us much alone, but if these numbers satisfy the Pythagorean Theorem, that means the triangle is right.

Wait. What's the Pythagorean Theorem, again?

a² + b² = c²

Oh, yeah. Now plug in the distances. The hypotenuse, by default, is always the longest side so that will be c. And those square roots will get canceled out by the exponents. How much simpler can simplifying get?

29 + 29 ≟ 58

Wait for it. Wait for it…

58 = 58

Yes! That means ∆ABC is indeed a right triangle. Success has never smelled sweeter.