# Pythagorean Theorem Worksheet

January 25, 2023

The Pythagorean Theorem describes the relationship between the lengths of the legs and the hypotenuse of a right triangle.

## Pythagorean Theorem Formula

\( a^2 + b^2 = c^2 \) where a and b are the lengths of the two shorter sides and c is the length of the hypotenuse.

The theorem is used in many areas of mathematics, including geometry, trigonometry, and algebra. It is also used to calculate the length of a side of a triangle when the other two sides and the angle between them are known. The Pythagorean Theorem can also be used to calculate the area of a right triangle. It is one of the most well-known and useful theorems in mathematics.

## Pythagorean Theorem Worksheet

**Question: Write whether the following lengths form a right triangle by applying the Pythagorean theorem.**

**Step1: **

**Step 2:**

**Determine if the given side lengths a, b, and c form a right triangle using the Pythagorean theorem.**

**Step 3:** a = 12 in , b = 5 in, and c = 13 in

**Step 4:** a = 4 ft, b = 3.5 ft, and c = 6 ft

## Pythagorean Theorem Worksheet Answers with Solutions

**Example 1: Prove if it is a right triangle or not.**

a = 12 in

b = 4 in

c = 10 in

Putting in the formula, we get

\( a^2 + b^2 = c^2 \)

\( 12^2 + 4^2 = 10^2 \)

\( 144 + 16 = 100 \)

160 **≠** 100

Since RHS is not equal to LHS, this is not a right triangle.

**Example 2:** Prove if it is a right triangle or not.

a = 8 ft

b = 15 ft

c = 17 ft

Putting in the formula, we get

\( a^2 + b^2 = c^2 \)

\( 8^2 + 15^2 = 17^2 \)

\( 64 + 225 = 289 \)

\( 289 = 289 \)

Since RHS is equal to LHS, this is a right triangle.

** Example 3: Prove if it is a right triangle or not. We have a = 12 in, b = 5 in, and c = 13 in**.

Putting in the formula, we get

\( a^2 + b^2 = c^2 \)

\( 12^2 + 5^2 = 13^2 \)

\( 144 + 25 = 169 \)

\( 169 = 169 \)

Since RHS is equal to LHS, this is a right triangle.

** Example 4: Prove if it is a right triangle or not. We have A = 4 ft, b = 3.5 ft, and c = 6 ft**.

Putting in the formula, we get

\( a^2 + b^2 = c^2 \)

\( 4^2 + 3.5^2 = 6^2 \)

\( 16 + 12.25 = 36 \)

28.25 **≠** 36

Since RHS is not equal to LHS, this is not a right triangle.

## Pythagorean Theorem Word Problems with Solutions

**Example 1: Stefan is on his way home from work. He drives 30 miles due North and then 39 miles due West. Find the quickest distance he can cover to reach home early. **

From the above diagram, we have:

a = 39 miles

b = 30 miles

c = ?

We put values in the Pythagorean formula.

\( a^2 + b^2 = c^2 \)

\( 39^2 + 30^2 = c^2 \)

\( 2421 = c^2 \)

c = 49.20 miles is the quickest distance he can use to reach home early.

**Example 2: Martina bought a rug for her room. The rug is 12 feet long and 8 feet wide. Find the length of the diagonal of the rug.**

From the above picture of the rug, we have:

a = 12 ft

b = 8 ft

c = ?

We put values in the Pythagorean formula.

\( a^2 + b^2 = c^2 \)

\( 12^2 + 8^2 = c^2 \)

C = 14.42 is the length of the diagonal of the rug.

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