Mastering 45-45-90 Triangles: Essential Worksheet Answers

Unlocking the Secrets of 45-45-90 Triangles: A Comprehensive Guide

When it comes to geometry, certain types of triangles hold a special place in the world of mathematics. Among these, the 45-45-90 triangle stands out for its unique properties and wide range of applications. This special right triangle, characterized by its two equal 45-degree angles and one 90-degree angle, is a fundamental concept in various fields, including architecture, engineering, and design. In this article, we will delve into the world of 45-45-90 triangles, exploring their properties, theorems, and worksheet answers to help you master this essential topic.

Properties of 45-45-90 Triangles

One of the defining features of a 45-45-90 triangle is the relationship between its sides. In a 45-45-90 triangle, the two legs (the sides opposite the 45-degree angles) are congruent, meaning they have the same length. The hypotenuse (the side opposite the 90-degree angle) is equal to √2 times the length of either leg. This unique ratio makes 45-45-90 triangles easily recognizable and invaluable in various mathematical and real-world applications.

The Pythagorean Theorem and 45-45-90 Triangles

The Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, is a fundamental concept that can be applied to 45-45-90 triangles. Given the unique ratio of sides in a 45-45-90 triangle, we can use the Pythagorean theorem to derive the following equation:

a^2 + a^2 = (√2a)^2

where a represents the length of one leg. Simplifying this equation, we get:

2a^2 = 2a^2

This equation demonstrates the intrinsic relationship between the sides of a 45-45-90 triangle, making it an essential tool for solving problems involving these special triangles.

Worksheet Answers and Examples

To help you master 45-45-90 triangles, we’ve compiled a set of worksheet answers and examples to guide you through the process.

Example 1: In a 45-45-90 triangle, the length of one leg is 5 inches. Find the length of the hypotenuse.

Answer: Using the ratio of sides in a 45-45-90 triangle, we can calculate the length of the hypotenuse as follows:

hypotenuse = √2 × leg = √2 × 5 = 7.07 inches

Example 2: In a 45-45-90 triangle, the length of the hypotenuse is 10 inches. Find the length of one leg.

Answer: Using the ratio of sides in a 45-45-90 triangle, we can calculate the length of one leg as follows:

leg = hypotenuse / √2 = 10 / √2 = 7.07 inches

Example 3: In a 45-45-90 triangle, the length of one leg is 3x inches. Find the length of the hypotenuse in terms of x.

Answer: Using the ratio of sides in a 45-45-90 triangle, we can calculate the length of the hypotenuse as follows:

hypotenuse = √2 × leg = √2 × 3x = 3√2x inches

📝 Note: These examples demonstrate the use of the 45-45-90 triangle ratio to solve problems involving these special triangles. Practice working with different values to become more proficient in applying this concept.

Real-World Applications of 45-45-90 Triangles

45-45-90 triangles have a wide range of applications in various fields, including:

• Architecture: 45-45-90 triangles are used in the design of buildings, bridges, and other structures to create stable and balanced frameworks.
• Engineering: 45-45-90 triangles are used in the design of mechanical systems, such as gears and pulleys, to transmit power and motion efficiently.
• Design: 45-45-90 triangles are used in graphic design, typography, and visual arts to create balanced and harmonious compositions.

Conclusion

Mastering 45-45-90 triangles is an essential skill for anyone studying geometry and mathematics. By understanding the unique properties and theorems associated with these special triangles, you’ll be able to tackle a wide range of problems and applications with confidence. Remember to practice working with different values and scenarios to become more proficient in applying this concept.