Geometry worksheet on 45-45-90 triangles with ten problems requiring calculation of missing side lengths.

Let's solve each of these 45-45-90 triangle problems step by step.

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🔷 Key Concept: 45-45-90 Triangle Properties

In a 45-45-90 right triangle:
- The two legs are equal.
- The hypotenuse is $ \text{leg} \times \sqrt{2} $.
- So, if the legs are $ x $, then the hypotenuse is $ x\sqrt{2} $.
- Conversely, if the hypotenuse is $ h $, then each leg is $ \frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2} $.

We’ll use this to find missing sides.

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1)

```
√2
/|
/ |
/ | y
/ |
/____|
x
```

- One leg = $ \sqrt{2} $
- Since it’s a 45-45-90 triangle, both legs are equal → $ y = \sqrt{2} $
- Hypotenuse $ x = \text{leg} \times \sqrt{2} = \sqrt{2} \cdot \sqrt{2} = 2 $

✔ Answer:
$ x = 2 $, $ y = \sqrt{2} $

---

2)

```
8
/|
/ |
/ | n
/ |
/____|
m
```

- Right angle at bottom-right, 45° at bottom-left → it’s a 45-45-90 triangle.
- Given side = 8 (this is one leg)
- Since both legs are equal → $ m = 8 $
- Hypotenuse $ n = 8\sqrt{2} $

✔ Answer:
$ m = 8 $, $ n = 8\sqrt{2} $

---

3)

```
b
/|
/ |
/ | 9√2
/ |
/____|
a
```

- One leg = $ 9\sqrt{2} $
- Since both legs are equal → $ b = 9\sqrt{2} $
- Hypotenuse $ a = \text{leg} \times \sqrt{2} = 9\sqrt{2} \cdot \sqrt{2} = 9 \cdot 2 = 18 $

✔ Answer:
$ a = 18 $, $ b = 9\sqrt{2} $

---

4)

```
x
/|
/ |
/ | 9
/ |
/____|
y
```

- Right angle at bottom-right, 45° at bottom-left → 45-45-90 triangle.
- Given side = 9 → this is a leg
- So $ y = 9 $ (other leg)
- Hypotenuse $ x = 9\sqrt{2} $

✔ Answer:
$ x = 9\sqrt{2} $, $ y = 9 $

---

5)

```
x
/|
/ |
/ | y
/ |
/____|
2
```

- Hypotenuse = 2
- In 45-45-90: each leg = $ \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{2}{\sqrt{2}} $
- Rationalize: $ \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2} $

So both legs $ x = y = \sqrt{2} $

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

---

6)

```
a
/|
/ |
/ | 5√2
/ |
/____|
b
```

- One leg = $ 5\sqrt{2} $
- Other leg $ b = 5\sqrt{2} $ (since legs are equal)
- Hypotenuse $ a = 5\sqrt{2} \cdot \sqrt{2} = 5 \cdot 2 = 10 $

✔ Answer:
$ a = 10 $, $ b = 5\sqrt{2} $

---

7)

```
u
/|
/ |
/ | v
/ |
/____|
10
```

- Hypotenuse = 10
- Legs: $ u = v = \frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2} $

✔ Answer:
$ u = 5\sqrt{2} $, $ v = 5\sqrt{2} $

---

8)

```
u
/|
/ |
/ | 9√2 / 2
/ |
/____|
v
```

- One leg = $ \frac{9\sqrt{2}}{2} $
- So other leg $ v = \frac{9\sqrt{2}}{2} $
- Hypotenuse $ u = \left(\frac{9\sqrt{2}}{2}\right) \cdot \sqrt{2} = \frac{9 \cdot 2}{2} = 9 $

✔ Answer:
$ u = 9 $, $ v = \frac{9\sqrt{2}}{2} $

---

9)

```
y
/|
/ |
/ | 3√5
/ |
/____|
x
```

- One leg = $ 3\sqrt{5} $
- So other leg $ y = 3\sqrt{5} $
- Hypotenuse $ x = 3\sqrt{5} \cdot \sqrt{2} = 3\sqrt{10} $

✔ Answer:
$ x = 3\sqrt{10} $, $ y = 3\sqrt{5} $

---

10)

```
y
/|
/ |
/ | 45°
/ |
/____|
5
```

- Right angle at bottom-left, 45° at top-right → 45-45-90 triangle
- One leg = 5
- So other leg $ y = 5 $
- Hypotenuse $ x = 5\sqrt{2} $

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

---

✔ Final Answers Summary:

| Problem | Missing Sides |
|--------|----------------|
| 1) | $ x = 2 $, $ y = \sqrt{2} $ |
| 2) | $ m = 8 $, $ n = 8\sqrt{2} $ |
| 3) | $ a = 18 $, $ b = 9\sqrt{2} $ |
| 4) | $ x = 9\sqrt{2} $, $ y = 9 $ |
| 5) | $ x = \sqrt{2} $, $ y = \sqrt{2} $ |
| 6) | $ a = 10 $, $ b = 5\sqrt{2} $ |
| 7) | $ u = 5\sqrt{2} $, $ v = 5\sqrt{2} $ |
| 8) | $ u = 9 $, $ v = \frac{9\sqrt{2}}{2} $ |
| 9) | $ x = 3\sqrt{10} $, $ y = 3\sqrt{5} $ |
| 10) | $ x = 5\sqrt{2} $, $ y = 5 $ |

All answers are in simplest radical form as requested. ✔