45-45-90 Triangle Worksheet Answer Key Explained
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In the world of geometry, triangles hold a special place, especially the unique 45-45-90 triangles. Understanding these triangles is essential for both students and educators alike. In this article, we will delve into the intricacies of the 45-45-90 triangle, what it represents, and how to approach answering worksheets centered around this topic. We will also provide a detailed explanation of the answer key to a typical 45-45-90 triangle worksheet. Let's get started!
What is a 45-45-90 Triangle? 🔺
A 45-45-90 triangle is an isosceles right triangle. This means that two of its angles are 45 degrees, and the third angle is 90 degrees. The properties of this type of triangle make it unique:
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Equal Legs: The two legs (the sides opposite the 45-degree angles) are of equal length.
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Hypotenuse: The length of the hypotenuse (the side opposite the 90-degree angle) can be determined using the formula:
[ \text{Hypotenuse} = \text{Leg} \times \sqrt{2} ]
Properties of 45-45-90 Triangles
- If the length of each leg is ( x ), then the hypotenuse will always be ( x\sqrt{2} ).
- The triangle maintains a consistent relationship between its sides, making it easier to solve for unknown lengths.
Importance of Worksheets 📄
Worksheets serve as an essential tool for practicing the application of geometric principles. For students learning about 45-45-90 triangles, worksheets allow them to:
- Reinforce Knowledge: Applying the properties through problem-solving helps solidify understanding.
- Test Comprehension: Worksheets can reveal areas where students may need further clarification or practice.
- Prepare for Exams: Regular practice prepares students for tests that might include questions on triangles.
Overview of the Worksheet 📊
Typically, a 45-45-90 triangle worksheet might include various types of problems, such as:
- Finding the lengths of the legs given the hypotenuse.
- Finding the length of the hypotenuse given the legs.
- Word problems that incorporate real-world applications of 45-45-90 triangles.
Sample Problems
Here is an example of what a typical problem set might look like:
| Problem Type | Question Description | Expected Answer |
|---|---|---|
| Finding Legs | The hypotenuse is 10. What is the length of each leg? | 10/√2 or approximately 7.07 |
| Finding Hypotenuse | Each leg measures 5. What is the hypotenuse? | 5√2 or approximately 7.07 |
| Word Problem | A square has a diagonal measuring 8. What is the length of each side? | 8/√2 or approximately 5.66 |
Answer Key Explained 🗝️
Understanding the answer key is crucial for learning from mistakes. Let's break down how to solve each type of problem found in a 45-45-90 triangle worksheet.
1. Finding the Legs
Question: If the hypotenuse is 10, what is the length of each leg?
Solution:
Using the formula for the hypotenuse:
[ \text{Hypotenuse} = \text{Leg} \times \sqrt{2} ]
Rearranging gives:
[ \text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{10}{\sqrt{2}} \approx 7.07 ]
2. Finding the Hypotenuse
Question: If each leg is 5, what is the length of the hypotenuse?
Solution:
Using the hypotenuse formula:
[ \text{Hypotenuse} = \text{Leg} \times \sqrt{2} = 5\sqrt{2} \approx 7.07 ]
3. Word Problems
Question: A square has a diagonal measuring 8. What is the length of each side?
Solution:
Since the diagonal of a square forms two 45-45-90 triangles, we apply the hypotenuse formula:
[ \text{Side} = \frac{\text{Diagonal}}{\sqrt{2}} = \frac{8}{\sqrt{2}} \approx 5.66 ]
Common Mistakes to Avoid ⚠️
- Neglecting to Use the Correct Formula: Always check that you are using the 45-45-90 triangle relationships.
- Rounding Too Early: It's often better to keep numbers exact in calculations before rounding at the final answer.
- Forgetting to Check Work: Always review answers to catch potential mistakes.
Conclusion
Understanding the 45-45-90 triangle is essential for any student venturing into geometry. Worksheets focused on this type of triangle provide an excellent opportunity for reinforcement and mastery of the concept. By familiarizing yourself with the properties and practicing with worksheets, you can develop a solid foundation that will support further mathematical learning. The answer key serves as a crucial guide to not only verify answers but also to understand the reasoning behind each solution. Keep practicing, and you'll find that working with 45-45-90 triangles becomes second nature!