Question

directions: 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 30-60-90 triangle rules

In a 30-60-90 triangle, sides are in ratio $1:\sqrt{3}:2$, where the side opposite 30° is the shortest. Here, hypotenuse $=10$, so side $y$ (opposite 30°) is half the hypotenuse.
$y = \frac{1}{2} \times 10 = 5$

1) Step2: Calculate $x$ (opposite 60°)

$x = y \times \sqrt{3} = 5\sqrt{3}$

2) Step1: Find hypotenuse $x$ (30-60-90)

Side opposite 60° is $\sqrt{3}$, so hypotenuse $x = \frac{\sqrt{3}}{\sqrt{3}} \times 2 = 2$

2) Step2: Calculate $y$ (opposite 30°)

$y = \frac{1}{2}x = 1$

3) Step1: Find hypotenuse $x$ (30-60-90)

Side opposite 60° is 5, so $x = \frac{5}{\sqrt{3}} \times 2 = \frac{10\sqrt{3}}{3}$

3) Step2: Calculate $y$ (opposite 30°)

$y = \frac{1}{2}x = \frac{5\sqrt{3}}{3}$

4) Step1: Find $v$ (opposite 30°)

Side opposite 60° is $\frac{10\sqrt{3}}{3}$, so $v = \frac{\frac{10\sqrt{3}}{3}}{\sqrt{3}} = \frac{10}{3}$

4) Step2: Calculate $u$ (hypotenuse)

$u = 2v = \frac{20}{3}$

5) Step1: Find $b$ (opposite 30°)

Side opposite 60° is $\frac{4\sqrt{3}}{3}$, so $b = \frac{\frac{4\sqrt{3}}{3}}{\sqrt{3}} = \frac{4}{3}$

5) Step2: Calculate $a$ (hypotenuse)

$a = 2b = \frac{8}{3}$

6) Step1: Find $y$ (opposite 30°)

Side opposite 60° is $\sqrt{3}$, so $y = \frac{\sqrt{3}}{\sqrt{3}} = 1$

6) Step2: Calculate $x$ (hypotenuse)

$x = 2y = 2$

7) Step1: Find $y$ (hypotenuse, 30-60-90)

Side opposite 60° is $\frac{1}{3}$, so $y = \frac{\frac{1}{3}}{\sqrt{3}} \times 2 = \frac{2\sqrt{3}}{9}$

7) Step2: Calculate $x$ (opposite 30°)

$x = \frac{1}{2}y = \frac{\sqrt{3}}{9}$

8) Step1: Find $n$ (opposite 60°)

Side opposite 30° is 1, so $n = 1 \times \sqrt{3} = \sqrt{3}$

8) Step2: Calculate $m$ (hypotenuse)

$m = 2 \times 1 = 2$

9) Step1: Find $y$ (opposite 30°)

Side opposite 60° is 4, so $y = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3}$

9) Step2: Calculate $x$ (hypotenuse)

$x = 2y = \frac{8\sqrt{3}}{3}$

10) Step1: Find $y$ (opposite 30°)

Side opposite 60° is $2\sqrt{3}$, so $y = \frac{2\sqrt{3}}{\sqrt{3}} = 2$

10) Step2: Calculate $x$ (hypotenuse)

$x = 2y = 4$

Answer:

1. $x=5\sqrt{3}$, $y=5$
2. $x=2$, $y=1$
3. $x=\frac{10\sqrt{3}}{3}$, $y=\frac{5\sqrt{3}}{3}$
4. $u=\frac{20}{3}$, $v=\frac{10}{3}$
5. $a=\frac{8}{3}$, $b=\frac{4}{3}$
6. $x=2$, $y=1$
7. $x=\frac{\sqrt{3}}{9}$, $y=\frac{2\sqrt{3}}{9}$
8. $m=2$, $n=\sqrt{3}$
9. $x=\frac{8\sqrt{3}}{3}$, $y=\frac{4\sqrt{3}}{3}$
10. $x=4$, $y=2$