45-45-90 Triangle Worksheet Answer Key: Simplified Guide
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A 45-45-90 triangle is a special type of right triangle in which the two legs are equal in length, and the angles are 45 degrees, 45 degrees, and 90 degrees. This triangle is commonly encountered in geometry and trigonometry, making it essential for students to understand how to work with it effectively. In this article, we will provide a simplified guide to solving problems involving 45-45-90 triangles, including an answer key for a worksheet designed to reinforce understanding.
Characteristics of 45-45-90 Triangles
Basic Properties
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Angles: As mentioned, the angles in a 45-45-90 triangle are 45°, 45°, and 90°.
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Sides: The lengths of the legs (the two sides opposite the 45° angles) are equal. If we denote the length of one leg as ( x ), then the length of the hypotenuse (the side opposite the 90° angle) can be determined using the formula:
[ \text{Hypotenuse} = x\sqrt{2} ]
Visual Representation
To illustrate a 45-45-90 triangle, consider the diagram below:
|\
| \
x | \ x√2
| \
|____\
x
In this diagram, ( x ) represents the lengths of the legs, while ( x\sqrt{2} ) represents the hypotenuse.
Important Notes
Note: The unique ratios of the side lengths in 45-45-90 triangles make them easy to work with. Whenever you encounter a problem involving these triangles, you can always rely on the relationships between the angles and sides.
Solving Problems Involving 45-45-90 Triangles
Step-by-Step Approach
- Identify the Known Values: Begin by identifying any known values, such as the length of the legs or the hypotenuse.
- Use the Ratios: Apply the ratios derived from the properties of 45-45-90 triangles:
- If you know the hypotenuse, use ( x = \frac{\text{Hypotenuse}}{\sqrt{2}} ).
- If you know the length of one leg, the hypotenuse can be found using ( \text{Hypotenuse} = x\sqrt{2} ).
- Perform Calculations: Carry out the necessary calculations to find the unknown values.
Example Problems
Below are some examples to practice:
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Given Leg Length: If one leg of a 45-45-90 triangle is 5 cm, what is the length of the hypotenuse?
- Solution: Hypotenuse = ( 5\sqrt{2} \approx 7.07 ) cm.
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Given Hypotenuse: If the hypotenuse is 10 cm, what is the length of each leg?
- Solution: Leg = ( \frac{10}{\sqrt{2}} = 5\sqrt{2} \approx 7.07 ) cm.
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Word Problem: A square has a diagonal length of 12 cm. What is the length of each side?
- Solution: Since the square can be divided into two 45-45-90 triangles, the side length is ( \frac{12}{\sqrt{2}} = 6\sqrt{2} \approx 8.49 ) cm.
45-45-90 Triangle Worksheet
To practice solving problems involving 45-45-90 triangles, a worksheet can be an effective tool. Below is an example of what a simplified worksheet might look like, followed by an answer key.
Example Worksheet Problems
| Problem Number | Problem Description |
|---|---|
| 1 | Find the hypotenuse if the leg is 8 cm. |
| 2 | If the hypotenuse is 14 cm, what is the leg? |
| 3 | A triangle has legs measuring 6 cm. Find the hypotenuse. |
| 4 | A square has a diagonal of 10√2 cm. What is the side length? |
Answer Key
<table> <tr> <th>Problem Number</th> <th>Answer</th> </tr> <tr> <td>1</td> <td>8√2 ≈ 11.31 cm</td> </tr> <tr> <td>2</td> <td>7 cm</td> </tr> <tr> <td>3</td> <td>6√2 ≈ 8.49 cm</td> </tr> <tr> <td>4</td> <td>10 cm</td> </tr> </table>
Additional Tips
- Practice Regularly: The more problems you solve, the more comfortable you’ll become with these triangles.
- Visualize: Drawing triangles can help you better understand the relationships between their sides.
- Use Tools: Don’t hesitate to use calculators to simplify square roots for precise measurements.
By familiarizing yourself with the properties of 45-45-90 triangles and practicing with worksheets, you'll gain confidence in handling geometry problems involving these unique shapes. Happy studying! 📐📏