Pythagorean Theorem Worksheets - Free Printable

The image you've uploaded is a Pythagorean Theorem Word Problems Worksheet (Math Worksheet 17) with an answer key provided. It contains four word problems, each solved using the Pythagorean Theorem:
$$
a^2 + b^2 = c^2
$$
where $c$ is the hypotenuse (the longest side of a right triangle), and $a$ and $b$ are the other two sides.

Let me walk through each problem, explain the solution, and verify the answers:

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Problem 1: Diagonal of a Television

> Gavin bought a new television for her apartment. It is 50 inches wide and 30 inches tall. Find the diagonal length of the television.

Solution:

- The width (50 in) and height (30 in) form the two legs of a right triangle.
- The diagonal is the hypotenuse ($c$).

Using the Pythagorean Theorem:
$$
a^2 + b^2 = c^2 \\
50^2 + 30^2 = c^2 \\
2500 + 900 = c^2 \\
3400 = c^2 \\
c = \sqrt{3400} \approx 58.31
$$

✔ Answer: 58.3 inches (rounded to the nearest tenth)

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Problem 2: Shortest Distance Home

> Heather is on her way home from school. She walks 3 miles North and then 4 miles East. What is the shortest distance she can cover to reach home the fastest?

Solution:

- This forms a right triangle: 3 miles north (vertical leg), 4 miles east (horizontal leg).
- The straight-line path (hypotenuse) is the shortest distance.

$$
a^2 + b^2 = c^2 \\
3^2 + 4^2 = c^2 \\
9 + 16 = c^2 \\
25 = c^2 \\
c = \sqrt{25} = 5
$$

✔ Answer: 5 miles

This is a classic 3-4-5 right triangle.

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Problem 3: Finding the Missing Leg of a Right Triangle

> Mr. Cooper tells you that a right triangle has a hypotenuse of 200 mm and a leg of 172 mm. Find the length of the other leg.

Solution:

- Let $a = ?$, $b = 172$, $c = 200$

$$
a^2 + b^2 = c^2 \\
a^2 + 172^2 = 200^2 \\
a^2 + 29584 = 40000 \\
a^2 = 40000 - 29584 = 10416 \\
a = \sqrt{10416} \approx 102.06
$$

✔ Answer: 102.06 mm (rounded to the nearest hundredth)

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Problem 4: Ladder Against a Wall

> How far apart from the base of the house should an 18-foot ladder be placed so that it exactly reaches the top of a 15-foot wall?

Solution:

- The ladder acts as the hypotenuse (18 ft).
- The wall is one leg (15 ft).
- We need to find the distance from the base of the wall to the foot of the ladder (let’s call it $a$).

$$
a^2 + b^2 = c^2 \\
a^2 + 15^2 = 18^2 \\
a^2 + 225 = 324 \\
a^2 = 324 - 225 = 99 \\
a = \sqrt{99} \approx 9.95
$$

✔ Answer: 9.95 feet (rounded to the nearest hundredth)

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✔ Summary of Answers:

| Problem | Question | Answer |
|--------|---------|--------|
| 1 | TV diagonal | 58.3 inches |
| 2 | Shortest path home | 5 miles |
| 3 | Missing leg | 102.06 mm |
| 4 | Ladder placement | 9.95 feet |

All solutions correctly apply the Pythagorean Theorem, and the answers are properly rounded as instructed.

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🔍 Key Concept:

The Pythagorean Theorem applies only to right triangles. When given two sides, you can always find the third using:
$$
a^2 + b^2 = c^2
$$
Just identify which side is the hypotenuse (always opposite the right angle), and solve accordingly.

Let me know if you'd like this worksheet converted into a printable format or explained further!