8.2 Special Right Triangles Worksheet Answers Explained
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In the world of geometry, special right triangles play a crucial role in understanding the relationships between angles and side lengths. These unique triangles often simplify complex problems, making them an essential topic for students. In this post, we will explore the answers and explanations to problems commonly found in 8.2 Special Right Triangles worksheets, including the properties of 30-60-90 triangles and 45-45-90 triangles. So grab your notebook, and let’s dive in! 📐✨
Understanding Special Right Triangles
Special right triangles are two specific types of triangles that have unique angle and side relationships.
30-60-90 Triangles
A 30-60-90 triangle has angles measuring 30°, 60°, and 90°. The side lengths follow a specific ratio:
- The side opposite the 30° angle is ( x )
- The side opposite the 60° angle is ( x\sqrt{3} )
- The side opposite the 90° angle (the hypotenuse) is ( 2x )
45-45-90 Triangles
A 45-45-90 triangle has angles measuring 45°, 45°, and 90°. The side lengths also follow a special ratio:
- Each leg of the triangle is ( x )
- The hypotenuse is ( x\sqrt{2} )
This ratio makes it easy to quickly solve problems involving these triangles.
Common Problems and Their Answers
Let’s look at some typical problems you might encounter in an 8.2 Special Right Triangles worksheet and provide explanations for each.
Problem 1: Finding the Hypotenuse of a 30-60-90 Triangle
Given: One leg is 4 cm (the side opposite the 30° angle).
Question: What is the length of the hypotenuse?
Solution: Using the properties of 30-60-90 triangles:
[ \text{Hypotenuse} = 2 \times x = 2 \times 4 = 8 \text{ cm} ]
Answer: The hypotenuse is 8 cm. ✅
Problem 2: Finding the Longer Leg of a 30-60-90 Triangle
Given: The hypotenuse is 10 cm.
Question: What is the length of the longer leg?
Solution: First, find ( x ):
[ x = \frac{\text{Hypotenuse}}{2} = \frac{10}{2} = 5 ]
Now calculate the longer leg:
[ \text{Longer Leg} = x\sqrt{3} = 5\sqrt{3} \approx 8.66 \text{ cm} ]
Answer: The longer leg is approximately 8.66 cm. 📏
Problem 3: Finding the Legs of a 45-45-90 Triangle
Given: The hypotenuse is 14 cm.
Question: What are the lengths of the legs?
Solution: Using the ratio for 45-45-90 triangles:
[ \text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{14}{\sqrt{2}} = 14 \times \frac{\sqrt{2}}{2} = 7\sqrt{2} \approx 9.90 \text{ cm} ]
Answer: Each leg is approximately 9.90 cm. 🎉
Key Formulas Summary
Below is a quick reference table summarizing the key formulas for special right triangles:
<table> <tr> <th>Triangle Type</th> <th>Angles</th> <th>Side Ratios</th> </tr> <tr> <td>30-60-90</td> <td>30°, 60°, 90°</td> <td>1 : √3 : 2</td> </tr> <tr> <td>45-45-90</td> <td>45°, 45°, 90°</td> <td>1 : 1 : √2</td> </tr> </table>
Important Note:
"It’s essential to always double-check your calculations when working with special right triangles. Mistakes can easily happen with square roots and simplifications."
Real-Life Applications
Understanding special right triangles is not just an academic exercise; it has real-world applications in various fields:
- Architecture: When designing buildings, architects must calculate angles and distances accurately.
- Engineering: Engineers often use these principles to design structures, bridges, and even mechanical parts.
- Art: Artists use geometry to create perspective and balance in their artwork.
Conclusion
Mastering the properties and applications of special right triangles is crucial for success in geometry and many real-world situations. Whether you're working through an 8.2 Special Right Triangles worksheet or applying these concepts in everyday life, understanding how to calculate the sides and angles will empower you to solve a variety of problems with ease.
Remember, practice makes perfect! 💪 Keep working on different problems, and soon you’ll find yourself confident in your understanding of special right triangles.