Special Right Triangles Worksheets - Math Monks - Free Printable

Let's solve each problem on the 30-60-90 and 45-45-90 Triangle Worksheet step by step.

---

🔷 Key Properties:

#### 45°-45°-90° Triangle:
- Legs are equal: $ a = a $
- Hypotenuse: $ a\sqrt{2} $
- So if leg = $ x $, then hypotenuse = $ x\sqrt{2} $

#### 30°-60°-90° Triangle:
- Side opposite 30°: $ x $
- Side opposite 60°: $ x\sqrt{3} $
- Hypotenuse (opposite 90°): $ 2x $

---

Now let’s go through each problem.

---

1.

```
Right triangle with angles 45°, 45°, 90°
One leg = 2√2
Find x (hypotenuse), y (other leg)
```

This is a 45-45-90 triangle.

Since both legs are equal:
- $ y = 2\sqrt{2} $

Hypotenuse $ x = \text{leg} \times \sqrt{2} = 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4 $

✔ Answer:
$ x = 4 $, $ y = 2\sqrt{2} $

---

2.

```
Right triangle: 30°, 60°, 90°
Hypotenuse = 20
Find x (side opposite 30°), y (side opposite 60°)
```

In a 30-60-90 triangle:
- Hypotenuse = $ 2x $ → $ 2x = 20 $ → $ x = 10 $
- Side opposite 60° = $ x\sqrt{3} = 10\sqrt{3} $

So:
- $ x = 10 $ (opposite 30°)
- $ y = 10\sqrt{3} $ (opposite 60°)

✔ Answer:
$ x = 10 $, $ y = 10\sqrt{3} $

---

3.

```
Right triangle: 30°, 60°, 90°
Side opposite 60° = 3√12
Find x (hypotenuse), y (side opposite 30°)
```

First simplify $ 3\sqrt{12} $:
- $ \sqrt{12} = 2\sqrt{3} $
- So $ 3\sqrt{12} = 3 \times 2\sqrt{3} = 6\sqrt{3} $

This side is opposite 60°, so it is $ x\sqrt{3} $ in the ratio.

So:
- $ x\sqrt{3} = 6\sqrt{3} $ → $ x = 6 $

Then:
- Hypotenuse $ x = 6 $? Wait — confusion in notation.

Wait! Let’s clarify:

In 30-60-90:
- Opposite 30°: $ x $
- Opposite 60°: $ x\sqrt{3} $
- Hypotenuse: $ 2x $

Here, side opposite 60° = $ 6\sqrt{3} $

So:
- $ x\sqrt{3} = 6\sqrt{3} $ → $ x = 6 $

Then:
- Side opposite 30° = $ x = 6 $
- Hypotenuse = $ 2x = 12 $

But in diagram:
- $ x $ is labeled on the hypotenuse?
- $ y $ is labeled on the shorter leg?

Looking at the diagram:
- Right angle at bottom right.
- 30° at top, 60° at bottom left.
- So:
- Side opposite 30° = $ y $
- Side opposite 60° = $ 3\sqrt{12} = 6\sqrt{3} $
- Hypotenuse = $ x $

So:
- Opposite 60° = $ x_{\text{ratio}} \cdot \sqrt{3} = 6\sqrt{3} $ → $ x_{\text{ratio}} = 6 $
- Then:
- $ y = x_{\text{ratio}} = 6 $
- $ x = 2 \cdot x_{\text{ratio}} = 12 $

✔ Answer:
$ x = 12 $, $ y = 6 $

---

4.

```
Right triangle: 30°, 60°, 90°
Side opposite 30° = 11√3
Find x (hypotenuse), y (side opposite 60°)
```

Side opposite 30° = $ x_{\text{ratio}} = 11\sqrt{3} $

Then:
- Hypotenuse $ x = 2 \cdot x_{\text{ratio}} = 2 \cdot 11\sqrt{3} = 22\sqrt{3} $
- Side opposite 60° $ y = x_{\text{ratio}} \cdot \sqrt{3} = 11\sqrt{3} \cdot \sqrt{3} = 11 \cdot 3 = 33 $

✔ Answer:
$ x = 22\sqrt{3} $, $ y = 33 $

---

5.

```
Right triangle: 45°, 45°, 90°
Hypotenuse = 10
Find x (one leg), y (other leg)
```

In 45-45-90:
- Hypotenuse = $ x\sqrt{2} = 10 $
- So $ x = \frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2} $

Both legs are equal.

✔ Answer:
$ x = 5\sqrt{2} $, $ y = 5\sqrt{2} $

---

6.

```
Right triangle: 45°, 45°, 90°
One leg = 6√2
Find x (hypotenuse), y (other leg)
```

Legs are equal → $ y = 6\sqrt{2} $

Hypotenuse $ x = 6\sqrt{2} \cdot \sqrt{2} = 6 \cdot 2 = 12 $

✔ Answer:
$ x = 12 $, $ y = 6\sqrt{2} $

---

7.

```
Isosceles right triangle (45-45-90)
Hypotenuse = 8√3
Find x and y (legs)
```

Hypotenuse = $ x\sqrt{2} = 8\sqrt{3} $

Solve for $ x $:
- $ x = \frac{8\sqrt{3}}{\sqrt{2}} = 8 \cdot \frac{\sqrt{3}}{\sqrt{2}} = 8 \cdot \frac{\sqrt{6}}{2} = 4\sqrt{6} $

So both legs $ x = y = 4\sqrt{6} $

✔ Answer:
$ x = 4\sqrt{6} $, $ y = 4\sqrt{6} $

---

8.

```
Right triangle: 30°, 60°, 90°
Hypotenuse = 8√5
Find x (side opposite 60°), y (side opposite 30°)
```

Hypotenuse = $ 2x_{\text{ratio}} = 8\sqrt{5} $ → $ x_{\text{ratio}} = 4\sqrt{5} $

Then:
- Side opposite 30° = $ x_{\text{ratio}} = 4\sqrt{5} $
- Side opposite 60° = $ x_{\text{ratio}} \cdot \sqrt{3} = 4\sqrt{5} \cdot \sqrt{3} = 4\sqrt{15} $

From diagram:
- $ x $ is opposite 60° → $ x = 4\sqrt{15} $
- $ y $ is opposite 30° → $ y = 4\sqrt{5} $

✔ Answer:
$ x = 4\sqrt{15} $, $ y = 4\sqrt{5} $

---

✔ Final Answers Summary:

| Problem | $ x $ | $ y $ |
|--------|-------------------|---------------------|
| 1 | 4 | $ 2\sqrt{2} $ |
| 2 | 10 | $ 10\sqrt{3} $ |
| 3 | 12 | 6 |
| 4 | $ 22\sqrt{3} $ | 33 |
| 5 | $ 5\sqrt{2} $ | $ 5\sqrt{2} $ |
| 6 | 12 | $ 6\sqrt{2} $ |
| 7 | $ 4\sqrt{6} $ | $ 4\sqrt{6} $ |
| 8 | $ 4\sqrt{15} $ | $ 4\sqrt{5} $ |

---

Let me know if you'd like this formatted as a printable answer key!