Question

45-45-90 triangles
date
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find the missing side lengths. leave your answers as radicals in simplest form.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)

Explanation:

1)

Step1: Identify equal legs

In 45-45-90 triangles, legs are equal: $y = \sqrt{2}$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$: $h = \sqrt{2} \times \sqrt{2} = 2$

2)

Step1: Identify equal legs

Legs are equal: $n = 8$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$: $h = 8 \times \sqrt{2} = 8\sqrt{2}$

3)

Step1: Find leg length

Leg = $\frac{\text{hypotenuse}}{\sqrt{2}}$: $a = b = \frac{9\sqrt{2}}{\sqrt{2}} = 9$

4)

Step1: Identify equal legs

Legs are equal: $y = 9$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$: $h = 9 \times \sqrt{2} = 9\sqrt{2}$

5)

Step1: Find leg length

Leg = $\frac{\text{hypotenuse}}{\sqrt{2}}$: $x = y = \frac{2}{\sqrt{2}} = \sqrt{2}$

6)

Step1: Identify equal legs

Legs are equal: $b = 5\sqrt{2}$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$: $a = 5\sqrt{2} \times \sqrt{2} = 10$

7)

Step1: Find leg length

Leg = $\frac{\text{hypotenuse}}{\sqrt{2}}$: $u = v = \frac{10}{\sqrt{2}} = 5\sqrt{2}$

8)

Step1: Identify equal legs

Legs are equal: $v = \frac{9\sqrt{2}}{2}$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$: $u = \frac{9\sqrt{2}}{2} \times \sqrt{2} = 9$

9)

Step1: Identify equal legs

Legs are equal: $x = y$

Step2: Calculate leg length

Leg = $\frac{\text{hypotenuse}}{\sqrt{2}}$: $x = y = \frac{3\sqrt{5}}{\sqrt{2}} = \frac{3\sqrt{10}}{2}$

10)

Step1: Identify equal legs

Legs are equal: $y = 5$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$: $x = 5 \times \sqrt{2} = 5\sqrt{2}$

Answer:

1. $y=\sqrt{2}$, $h=2$
2. $n=8$, $h=8\sqrt{2}$
3. $a=9$, $b=9$
4. $y=9$, $h=9\sqrt{2}$
5. $x=\sqrt{2}$, $y=\sqrt{2}$
6. $b=5\sqrt{2}$, $a=10$
7. $u=5\sqrt{2}$, $v=5\sqrt{2}$
8. $v=\frac{9\sqrt{2}}{2}$, $u=9$
9. $x=\frac{3\sqrt{10}}{2}$, $y=\frac{3\sqrt{10}}{2}$
10. $y=5$, $x=5\sqrt{2}$