45-45-90 Triangle Worksheet Answers For Students
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In the realm of geometry, the 45-45-90 triangle holds a special place due to its unique properties and relationships. Students often encounter this triangle when learning about right triangles and trigonometric ratios. Understanding the answers to a 45-45-90 triangle worksheet can significantly enhance students' comprehension and mastery of these concepts. In this article, we will explore the characteristics of the 45-45-90 triangle, provide example problems, and present the answers to common worksheet questions. 📝
Understanding the 45-45-90 Triangle
A 45-45-90 triangle is an isosceles right triangle, meaning that its angles are 45 degrees, 45 degrees, and 90 degrees. The sides opposite the 45-degree angles are equal, and the side opposite the 90-degree angle is the hypotenuse.
Side Length Ratios
In a 45-45-90 triangle, the ratio of the lengths of the sides can be expressed as follows:
- The legs (the sides opposite the 45-degree angles) are of equal length, denoted as ( x ).
- The hypotenuse (the side opposite the 90-degree angle) is ( x\sqrt{2} ).
This relationship is a crucial part of solving problems related to 45-45-90 triangles. Here’s a quick look at how the relationships work in a table format:
<table> <tr> <th>Component</th> <th>Expression</th> </tr> <tr> <td>Legs</td> <td>x</td> </tr> <tr> <td>Hypotenuse</td> <td>x√2</td> </tr> </table>
Example Problems
To demonstrate the application of these ratios, let's take a look at a few example problems that students might find on a 45-45-90 triangle worksheet.
Problem 1: Finding the Hypotenuse
Given: The length of each leg of the triangle is 5 units.
Find: The length of the hypotenuse.
Solution:
Using the ratio, the hypotenuse is calculated as follows:
[ \text{Hypotenuse} = x\sqrt{2} = 5\sqrt{2} \approx 7.07 \text{ units} ]
Problem 2: Finding the Length of a Leg
Given: The hypotenuse of the triangle is 10 units.
Find: The length of each leg.
Solution:
To find the leg length, rearrange the formula:
[ x = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{10}{\sqrt{2}} \approx 7.07 \text{ units} ]
Problem 3: Perimeter of the Triangle
Given: The length of each leg is 6 units.
Find: The perimeter of the triangle.
Solution:
The perimeter ( P ) can be calculated as follows:
[ P = \text{Leg 1} + \text{Leg 2} + \text{Hypotenuse} = 6 + 6 + 6\sqrt{2} \approx 6 + 6 + 8.49 = 20.49 \text{ units} ]
Common Worksheet Questions and Answers
Let’s compile a list of common questions and their answers that students may face when completing a worksheet focused on 45-45-90 triangles.
Question 1: If one leg measures 8 units, what is the hypotenuse?
Answer:
The hypotenuse is calculated as follows:
[ \text{Hypotenuse} = 8\sqrt{2} \approx 11.31 \text{ units} ]
Question 2: The hypotenuse is 14 units. What is the length of each leg?
Answer:
Each leg is:
[ x = \frac{14}{\sqrt{2}} \approx 9.90 \text{ units} ]
Question 3: A triangle has legs measuring 10 units. Calculate its area.
Answer:
The area ( A ) of a triangle is given by:
[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 10 = 50 \text{ square units} ]
Question 4: What is the perimeter of a 45-45-90 triangle with legs of length 12 units?
Answer:
The perimeter can be calculated as follows:
[ P = 12 + 12 + 12\sqrt{2} \approx 12 + 12 + 16.97 = 40.97 \text{ units} ]
Tips for Students
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Memorize the Ratios: Knowing the side length ratios is critical. Remember, in a 45-45-90 triangle, the hypotenuse is always ( x\sqrt{2} ).
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Practice with Different Values: The more you practice, the more comfortable you’ll become with various problems related to 45-45-90 triangles.
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Visualize the Triangle: Draw diagrams to help visualize the relationships between the sides and angles. This can improve your understanding of the problem.
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Check Your Work: After solving a problem, revisit your calculations to ensure accuracy, especially when dealing with square roots.
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Understand Applications: 45-45-90 triangles frequently appear in real-world applications, such as architecture and design. Understanding their properties can help in practical situations.
Conclusion
The 45-45-90 triangle is a fundamental concept in geometry, and worksheets focused on this topic provide an excellent opportunity for students to reinforce their understanding. By practicing with a variety of problems, utilizing the ratios, and following structured approaches, students can successfully navigate through their geometry challenges. With these insights and answers at hand, tackling any 45-45-90 triangle worksheet will be a breeze! 🌟