Lesson 85: Solving Problems Using the Pythagorean Theorem

New Concept

If a triangle is a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$, then

What’s next

Next, you'll apply this theorem to find unknown side lengths and verify if triangles are right triangles.

Property

If a triangle is a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$, then

Explanation

This famous theorem is your superpower for right triangles. It reveals a hidden connection: the area of the squares on the two shorter legs perfectly adds up to the area of the square on the hypotenuse. You can use this balanced equation to find any missing side length, as long as you know the other two sides.

Examples

• Find the hypotenuse $c$ for a right triangle with legs $a=5$ and $b=12$: $5^2 + 12^2 = c^2 \implies 25 + 144 = c^2 \implies 169 = c^2 \implies c = 13$.
• Find the leg $b$ for a right triangle with leg $a=8$ and hypotenuse $c=10$: $8^2 + b^2 = 10^2 \implies 64 + b^2 = 100 \implies b^2 = 36 \implies b = 6$.
• Find the hypotenuse $m$ for legs of 4 and 6: $4^2 + 6^2 = m^2 \implies 16 + 36 = m^2 \implies m = \sqrt{52} = 2\sqrt{13}$.

Let's see how the Pythagorean theorem helps us find a missing leg, not just the hypotenuse. This example applies the first key idea: using the theorem to solve for an unknown side.

Example Problem

Find the missing side length, $x$, to the nearest tenth.

(A right triangle is shown with one leg of length 5, the other leg labeled $x$, and the hypotenuse of length 9.)

Property

If a triangle has side lengths $a$, $b$, and $c$ that satisfy the equation $a^2 + b^2 = c^2$, then the triangle is a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$.

Explanation

Think of this as the theorem in reverse! Instead of starting with a right triangle, you start with three side lengths and play detective. You can test them with the $a^2 + b^2 = c^2$ formula. If the equation balances perfectly, congratulations, you have proven it is a right triangle! If not, it is just some other impostor triangle.

Examples

• Sides 8, 15, 17: Check if $8^2 + 15^2 = 17^2 \implies 64 + 225 = 289$. Yes, $289 = 289$. This is a right triangle.
• Sides 7, 9, 11: Check if $7^2 + 9^2 = 11^2 \implies 49 + 81 = 121$. No, $130 \ne 121$. This is not a right triangle.
• Sides 6, 8, 10: Check if $6^2 + 8^2 = 10^2 \implies 36 + 64 = 100$. Yes, $100 = 100$. This is a right triangle.

Property

A Pythagorean triple is a group of three nonzero whole numbers $a, b,$ and $c$ that represent the lengths of the sides of a right triangle.

Explanation

These are the VIPs of the right triangle world! A Pythagorean triple is a special team of three positive whole numbers that fit the Pythagorean theorem perfectly, like the famous trio 3, 4, and 5. There are no messy decimals or radicals involved, making your calculations clean and easy. They are the perfect whole-number side lengths for a right triangle.

Examples

• The numbers 3, 4, 5 form a triple because they are whole numbers and $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.
• The numbers 5, 12, 13 form a triple because they are whole numbers and $5^2 + 12^2 = 25 + 144 = 169 = 13^2$.
• The set 7, 11, $\sqrt{170}$ is not a triple because $\sqrt{170}$ is not a whole number, even though the sides form a right triangle.