Geometry Worksheet: 45-45-90 Triangles - Free Printable

Since I can't view or access uploaded images directly, I can’t see the specific triangles or values in your image. However, based on the title "Special Right Triangles: 45° - 45° - 90°", I can help you solve problems involving 45°-45°-90° right triangles.

These are isosceles right triangles, meaning:
- The two legs are equal in length.
- The angles are 45°, 45°, and 90°.
- The relationship between the sides is:

$$
\text{Leg} : \text{Leg} : \text{Hypotenuse} = x : x : x\sqrt{2}
$$

So:
- If you know one leg, the other leg is the same.
- The hypotenuse is $ x\sqrt{2} $.
- If you know the hypotenuse, each leg is $ \frac{\text{hypotenuse}}{\sqrt{2}} $, or rationalized: $ \frac{\text{hypotenuse} \cdot \sqrt{2}}{2} $.

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Let’s assume your worksheet has several such triangles labeled with variables like $ a $ and $ b $, and some side lengths given. Here's how to approach each type of problem:

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Example Problems (Typical Format):

#### Problem 1:
Triangle with legs = 3, find hypotenuse $ a $

- Since both legs are 3, then:
$$
a = 3\sqrt{2}
$$

#### Problem 2:
Hypotenuse = 8, find leg $ b $

- Use:
$$
b = \frac{8}{\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}
$$

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Now, let’s suppose your worksheet has the following types of diagrams (common ones):

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Common Configurations:

1. Right triangle with legs = 6, find hypotenuse $ a $
- $ a = 6\sqrt{2} $

2. Right triangle with hypotenuse = 10, find leg $ b $
- $ b = \frac{10}{\sqrt{2}} = 5\sqrt{2} $

3. One leg = 7, other leg = $ a $, hypotenuse = $ b $
- $ a = 7 $
- $ b = 7\sqrt{2} $

4. Hypotenuse = $ a $, legs = $ b $ and $ b $, and one leg is 5
- $ b = 5 $
- $ a = 5\sqrt{2} $

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If you can describe the triangles (e.g., "In problem 1, the two legs are 4, find the hypotenuse"), I can give you exact answers.

Alternatively, if you can list the given values for each problem (like "problem 1: leg = 5, find a"), I’ll solve them step by step.

But since I can't see the image, here’s a general guide for solving all 12 problems:

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General Strategy:

For 45°-45°-90° triangles:

- If one leg is known → other leg is the same, hypotenuse = $ x\sqrt{2} $
- If hypotenuse is known → each leg = $ \frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2} $

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Let me show solutions assuming typical patterns:

Suppose the problems are like this:

| Problem | Given | Find |
|--------|-------|------|
| 1. | Leg = 3 | $ a $ = ? |
| 2. | Hypotenuse = $ 3\sqrt{2} $ | $ b $ = ? |
| 3. | Leg = 5 | $ a $ = ? |
| 4. | Hypotenuse = $ 5\sqrt{2} $ | $ b $ = ? |
| 5. | Leg = 7 | $ a $ = ? |
| 6. | Hypotenuse = $ 7\sqrt{2} $ | $ b $ = ? |
| 7. | Leg = 9 | $ a $ = ? |
| 8. | Hypotenuse = $ 9\sqrt{2} $ | $ b $ = ? |
| 9. | Leg = 4 | $ a $ = ? |
|10. | Hypotenuse = $ 4\sqrt{2} $ | $ b $ = ? |
|11. | Leg = 10 | $ a $ = ? |
|12. | Hypotenuse = $ 10\sqrt{2} $ | $ b $ = ? |

Then the answers would be:

1. $ a = 3\sqrt{2} $
2. $ b = 3 $
3. $ a = 5\sqrt{2} $
4. $ b = 5 $
5. $ a = 7\sqrt{2} $
6. $ b = 7 $
7. $ a = 9\sqrt{2} $
8. $ b = 9 $
9. $ a = 4\sqrt{2} $
10. $ b = 4 $
11. $ a = 10\sqrt{2} $
12. $ b = 10 $

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✔ Key takeaway: In a 45°-45°-90° triangle:
- Legs are equal.
- Hypotenuse = leg × $ \sqrt{2} $
- Leg = hypotenuse ÷ $ \sqrt{2} $

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Please provide the specific values from your image (e.g., “Problem 1: leg = 2, find a”), and I’ll give you the exact answers with steps.

Or, if you describe the diagram (e.g., which side is labeled, what numbers are given), I can tailor the solution.