Pythagorean Theorem Worksheets - Free Printable

You've uploaded an answer key for a Pythagorean Theorem word problems worksheet (Math Worksheet 17). Since you asked to “solve the problem and explain the solution,” I’ll go through each of the four problems step-by-step, explaining how the Pythagorean Theorem is applied.

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## 🔷 Problem 1: Television Diagonal

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

✔ Solution:

We treat the TV as a rectangle. The diagonal forms the hypotenuse of a right triangle with legs 50 in and 30 in.

Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Where:
- \(a = 50\)
- \(b = 30\)
- \(c\) = diagonal (hypotenuse)

Step-by-step:
\[
50^2 + 30^2 = c^2 \\
2500 + 900 = c^2 \\
3400 = c^2 \\
c = \sqrt{3400} \approx 58.31
\]

✔ Answer: 58.31 inches

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## 🔷 Problem 2: Heather’s Walk 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:
- One leg = 3 miles (North)
- Other leg = 4 miles (East)
- Shortest path = straight line (hypotenuse) back to home

Pythagorean Theorem:
\[
3^2 + 4^2 = c^2 \\
9 + 16 = c^2 \\
25 = c^2 \\
c = \sqrt{25} = 5
\]

✔ Answer: 5 miles

*(Note: This is a classic 3-4-5 right triangle — very common!)*

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## 🔷 Problem 3: Mr. Cooper’s 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:

We know:
- Hypotenuse \(c = 200\)
- One leg \(b = 172\)
- Need to find the other leg \(a\)

Use rearranged Pythagorean Theorem:
\[
a^2 = c^2 - b^2
\]

Step-by-step:
\[
a^2 = 200^2 - 172^2 \\
a^2 = 40000 - 29584 \\
a^2 = 10416 \\
a = \sqrt{10416} \approx 102.06
\]

✔ Answer: 102.06 mm

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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 is the hypotenuse (18 ft), the wall is one leg (15 ft), and we need to find the distance from the base (other leg).

Use:
\[
a^2 = c^2 - b^2
\]

Where:
- \(c = 18\)
- \(b = 15\)
- \(a\) = distance from base

Step-by-step:
\[
a^2 = 18^2 - 15^2 \\
a^2 = 324 - 225 \\
a^2 = 99 \\
a = \sqrt{99} \approx 9.95
\]

✔ Answer: 9.95 feet

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## 🧠 Summary of Key Concepts:

- The Pythagorean Theorem applies only to right triangles:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the hypotenuse (longest side, opposite the right angle).

- When solving for a leg, rearrange:
\[
a = \sqrt{c^2 - b^2}
\]

- Always round to the nearest tenth unless otherwise specified.

- Real-world applications include:
- Diagonals of rectangles (TVs, screens)
- Shortest paths (walking, flying, driving)
- Ladders, ramps, construction

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✔ All answers in the worksheet are correct and properly rounded.

Let me know if you’d like to practice similar problems or see visual diagrams!