## Introduction

The Pythagorean Theorem is one of the fundamental principles in mathematics that relates to right triangles. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem has numerous applications in real-life situations, allowing us to solve various problems involving distance, measurements, and more. In this article, we will explore some word problems that involve the Pythagorean Theorem and provide answers to help you understand the concept better.

## Problem 1: The Fence

John wants to build a fence around his rectangular backyard. The length of the backyard is 15 feet, and the width is 10 feet. How long should the diagonal of the fence be?

To solve this problem, we can use the Pythagorean Theorem. Let’s assume the length of the diagonal is ‘d’. According to the theorem, we can write the equation as:

d^2 = 15^2 + 10^2

d^2 = 225 + 100

d^2 = 325

d = sqrt(325)

d ≈ 18.03 feet

Therefore, the length of the diagonal of the fence should be approximately 18.03 feet.

## Problem 2: The Ladder

Sarah wants to reach the top of a wall that is 12 feet high. She has a ladder that is 15 feet long. How far should she place the ladder from the wall?

Again, we can use the Pythagorean Theorem to solve this problem. Let’s assume the distance between the wall and the ladder’s base is ‘x’. According to the theorem, we can write the equation as:

x^2 + 12^2 = 15^2

x^2 + 144 = 225

x^2 = 225 – 144

x^2 = 81

x = sqrt(81)

x = 9 feet

Therefore, Sarah should place the ladder 9 feet away from the wall.

## Problem 3: The TV

Tom wants to buy a new TV for his living room. He has a TV stand against the wall, and the distance between the stand and the opposite corner of the room is 8 feet. If he wants to buy a TV with a diagonal length of 50 inches, how wide should the stand be?

Let’s assume the width of the TV stand is ‘w’. According to the Pythagorean Theorem, we can write the equation as:

w^2 + 8^2 = 50^2

w^2 + 64 = 2500

w^2 = 2500 – 64

w^2 = 2436

w = sqrt(2436)

w ≈ 49.35 inches

Therefore, the TV stand should be approximately 49.35 inches wide.

## Problem 4: The Roof

Emily’s house has a triangular roof. The height of the roof is 8 feet, and the base is 10 feet. What is the length of the diagonal of the roof?

Let’s assume the length of the diagonal is ‘d’. According to the Pythagorean Theorem, we can write the equation as:

d^2 = 8^2 + 10^2

d^2 = 64 + 100

d^2 = 164

d = sqrt(164)

d ≈ 12.81 feet

Therefore, the length of the diagonal of the roof is approximately 12.81 feet.

## Frequently Asked Questions (FAQ)

### Q1: What is the Pythagorean Theorem?

The Pythagorean Theorem is a mathematical principle that relates to right triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. It can be represented as a^2 + b^2 = c^2, where ‘a’ and ‘b’ are the lengths of the legs of the triangle, and ‘c’ is the length of the hypotenuse.

### Q2: How can I use the Pythagorean Theorem to solve word problems?

To solve word problems using the Pythagorean Theorem, you first need to identify the right triangle in the problem. Then, assign variables to the lengths of the sides, and use the theorem to set up an equation. Solve the equation to find the missing side or distance.

### Q3: What are some common applications of the Pythagorean Theorem?

The Pythagorean Theorem has various real-life applications, including measuring distances, calculating areas, determining heights or lengths, solving navigation problems, and more. It is widely used in fields such as architecture, engineering, construction, and surveying.

### Q4: Can the Pythagorean Theorem be used for non-right triangles?

No, the Pythagorean Theorem only applies to right triangles. For non-right triangles, you would need to use other trigonometric principles such as the Law of Sines or the Law of Cosines.

### Q5: How can I remember the Pythagorean Theorem?

One popular mnemonic to remember the Pythagorean Theorem is “A squared plus B squared equals C squared”. You can also create your own mnemonic or practice solving different types of problems to reinforce the concept.

## Tags

Pythagorean Theorem, Word Problems, Geometry, Right Triangle, Distance, Measurements, Applications, Hypotenuse, Legs, Equation, Triangular Roof