## Introduction

Understanding and applying the Pythagorean theorem and special right triangles is a fundamental skill in mathematics. These concepts not only help in solving geometric problems but also have numerous real-world applications. In this article, we will explore the answer key to the Pythagorean theorem and special right triangles, providing step-by-step explanations and examples to enhance your understanding.

## Pythagorean Theorem Answer Key

### What is the Pythagorean Theorem?

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

### Formula

The Pythagorean theorem can be represented by the formula:

a² + b² = c²

where 'a' and 'b' are the lengths of the two legs of the right triangle, and 'c' is the length of the hypotenuse.

### Solving for the Hypotenuse (c)

To find the length of the hypotenuse (c) using the Pythagorean theorem, follow these steps:

- Identify the lengths of the two legs (a and b) of the right triangle.
- Square each leg: a² and b².
- Add the squared values: a² + b².
- Take the square root of the sum to find the length of the hypotenuse (c).

### Solving for a Leg (a or b)

To find the length of a leg (a or b) using the Pythagorean theorem, follow these steps:

- Identify the length of the hypotenuse (c) and the length of the other leg (a or b).
- Square the length of the hypotenuse: c².
- Subtract the squared value of the known leg from the squared value of the hypotenuse: c² - a² or c² - b².
- Take the square root of the difference to find the length of the unknown leg (a or b).

### Example 1: Finding the Hypotenuse

Let's consider a right triangle with leg lengths of 3 units and 4 units. To find the length of the hypotenuse:

a = 3, b = 4

a² + b² = c²

3² + 4² = c²

9 + 16 = c²

25 = c²

c = √25

c = 5

Therefore, the length of the hypotenuse is 5 units.

### Example 2: Finding a Leg

Consider a right triangle with leg length of 5 units and hypotenuse length of 13 units. To find the length of the other leg:

a = 5, c = 13

c² - a² = b²

13² - 5² = b²

169 - 25 = b²

144 = b²

b = √144

b = 12

Therefore, the length of the other leg is 12 units.

## Special Right Triangles Answer Key

### What are Special Right Triangles?

Special right triangles are triangles that have specific angles and side length ratios. These triangles are often encountered in geometry and have unique properties that make them easier to solve.

### 45-45-90 Triangle

A 45-45-90 triangle is a special right triangle where the two acute angles are equal, measuring 45 degrees each. The ratio of the side lengths in a 45-45-90 triangle is:

Leg length: Leg length: Hypotenuse length = 1:1:√2

### Example: 45-45-90 Triangle

Consider a 45-45-90 triangle with a leg length of 3 units. To find the length of the hypotenuse:

a = 3

a : a : c = 1 : 1 : √2

3 : 3 : c = 1 : 1 : √2

3 : 3 : c = 1√2 : 1√2 : √2 * √2

3 : 3 : c = √2 : √2 : 2

3 : 3 : c = √2 : √2 : 2

c = 2

Therefore, the length of the hypotenuse is 2 units.

### 30-60-90 Triangle

A 30-60-90 triangle is a special right triangle where the angles measure 30 degrees, 60 degrees, and 90 degrees. The ratio of the side lengths in a 30-60-90 triangle is:

Short leg: Long leg: Hypotenuse length = 1:√3:2

### Example: 30-60-90 Triangle

Consider a 30-60-90 triangle with a short leg length of 4 units. To find the length of the other leg:

a = 4

a : b : c = 1 : √3 : 2

4 : b : c = 1 : √3 : 2

4 : b : c = 1√3 : √3√3 : 2√3

4 : b : c = √3 : 3 : 2√3

b = 3√3

Therefore, the length of the other leg is 3√3 units.

## Conclusion

The Pythagorean theorem and special right triangles are essential concepts in geometry. By understanding the answer key to these concepts, you can confidently solve problems involving right triangles and apply this knowledge to real-world scenarios. Remember to practice these concepts regularly to enhance your mathematical skills and problem-solving abilities.