## Introduction

As a website operator, it is important to provide valuable resources to students, educators, and other individuals seeking to improve their knowledge and skills in various subjects. In this article, we will be addressing one critical topic in mathematics – trigonometry. Specifically, we will delve into the Pythagorean Theorem, one of the fundamental concepts in trigonometry. We will provide answers to homework questions that deal with the application of the Pythagorean Theorem.

## What is the Pythagorean Theorem?

The Pythagorean Theorem is named after the ancient Greek mathematician, Pythagoras. It is a mathematical concept that relates to the relationship between the sides of a right-angled triangle. The theorem states that the sum of the squares of the two shorter sides (the legs) of a right-angled triangle is equal to the square of the longest side (the hypotenuse). In other words, the Pythagorean Theorem can be written in the equation form as follows:

a² + b² = c²

Where ‘a’ and ‘b’ are the lengths of the two shorter sides (legs), and ‘c’ is the length of the longest side (hypotenuse).

## How to Apply the Pythagorean Theorem?

To apply the Pythagorean Theorem, you need to follow these steps:

Step 1: Identify the right-angled triangle and label the sides as ‘a,’ ‘b,’ and ‘c’.

Step 2: Determine which of the sides is the hypotenuse (the longest side). Label it ‘c’.

Step 3: Determine the lengths of the two legs (sides ‘a’ and ‘b’).

Step 4: Substitute the lengths of the two legs into the Pythagorean Theorem equation.

Step 5: Simplify the equation and solve for the missing side.

## Pythagorean Theorem Homework Answers

1. Find the length of the hypotenuse of a right-angled triangle with sides of 5 units and 12 units.

By applying the Pythagorean theorem, we get:

a² + b² = c²

5² + 12² = c²

25 + 144 = c²

c² = 169

Taking the square root of both sides, we get:

c = 13 units

Therefore, the length of the hypotenuse is 13 units.

2. Find the length of the shorter side of a right-angled triangle with a hypotenuse of 17 units and the longer leg of 12 units.

By applying the Pythagorean theorem, we get:

a² + b² = c²

a² + 12² = 17²

a² + 144 = 289

a² = 145

Taking the square root of both sides, we get:

a = 12.04 (rounded to 2 decimal places)

Therefore, the length of the shorter side is approximately 12.04 units.

3. Find the length of the missing side of a right-angled triangle with a hypotenuse of 5 units and one leg of 3 units.

By applying the Pythagorean theorem, we get:

a² + b² = c²

3² + b² = 5²

9 + b² = 25

b² = 16

Taking the square root of both sides, we get:

b = 4 units

Therefore, the length of the missing side is 4 units.

## Conclusion

The Pythagorean Theorem is a critical concept in trigonometry that helps in the calculation of the sides of a right-angled triangle. By following the required steps, students can easily apply the theorem to get accurate results. The three homework questions discussed in this article provide an insight into how the theorem can be applied in real-life situations. It is essential that website operators provide comprehensive resources, including solved problems, to enable students to excel in their studies.