Question

special right triangles (30-60-90)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 triangle sides

Hypotenuse $c=16$, angle $60^\circ$.
Short leg (opposite $30^\circ$): $\frac{1}{2}c$
$\frac{1}{2} \times 16 = 8$

Step2: Find long leg

Long leg = short leg $\times \sqrt{3}$
$8 \times \sqrt{3} = 8\sqrt{3}$

2)

Step1: Identify triangle sides

Short leg (opposite $30^\circ$) = 2.
Hypotenuse = $2 \times$ short leg
$2 \times 2 = 4$

Step2: Find long leg

Long leg = short leg $\times \sqrt{3}$
$2 \times \sqrt{3} = 2\sqrt{3}$

3)

Step1: Identify triangle sides

Short leg (adjacent to $60^\circ$) = 8.
Hypotenuse = $2 \times$ short leg
$2 \times 8 = 16$

Step2: Find long leg

Long leg = short leg $\times \sqrt{3}$
$8 \times \sqrt{3} = 8\sqrt{3}$

4)

Step1: Identify triangle sides

Hypotenuse $c=8\sqrt{5}$, angle $60^\circ$.
Short leg (opposite $30^\circ$): $\frac{1}{2}c$
$\frac{1}{2} \times 8\sqrt{5} = 4\sqrt{5}$

Step2: Find long leg

Long leg = short leg $\times \sqrt{3}$
$4\sqrt{5} \times \sqrt{3} = 4\sqrt{15}$

5)

Step1: Identify triangle sides

Long leg (opposite $60^\circ$) = $5\sqrt{3}$.
Short leg = $\frac{\text{long leg}}{\sqrt{3}}$
$\frac{5\sqrt{3}}{\sqrt{3}} = 5$

Step2: Find hypotenuse

Hypotenuse = $2 \times$ short leg
$2 \times 5 = 10$

6)

Step1: Identify triangle sides

Hypotenuse $c=10$, angle $60^\circ$.
Short leg (adjacent to $60^\circ$): $\frac{1}{2}c$
$\frac{1}{2} \times 10 = 5$

Step2: Find long leg

Long leg = short leg $\times \sqrt{3}$
$5 \times \sqrt{3} = 5\sqrt{3}$

7)

Step1: Identify triangle sides

Hypotenuse $c=2\sqrt{2}$, angle $60^\circ$.
Short leg (adjacent to $60^\circ$): $\frac{1}{2}c$
$\frac{1}{2} \times 2\sqrt{2} = \sqrt{2}$

Step2: Find long leg

Long leg = short leg $\times \sqrt{3}$
$\sqrt{2} \times \sqrt{3} = \sqrt{6}$

8)

Step1: Identify triangle sides

Long leg (adjacent to $30^\circ$) = 12.
Short leg = $\frac{\text{long leg}}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3}$

Step2: Find hypotenuse

Hypotenuse = $2 \times$ short leg
$2 \times 4\sqrt{3} = 8\sqrt{3}$

9)

Step1: Identify triangle sides

Hypotenuse $c=3$, angle $60^\circ$.
Short leg (opposite $30^\circ$): $\frac{1}{2}c$
$\frac{1}{2} \times 3 = \frac{3}{2}$

Step2: Find long leg

Long leg = short leg $\times \sqrt{3}$
$\frac{3}{2} \times \sqrt{3} = \frac{3\sqrt{3}}{2}$

10)

Step1: Identify triangle sides

Long leg (adjacent to $30^\circ$) = $11\sqrt{3}$.
Short leg = $\frac{\text{long leg}}{\sqrt{3}}$
$\frac{11\sqrt{3}}{\sqrt{3}} = 11$

Step2: Find hypotenuse

Hypotenuse = $2 \times$ short leg
$2 \times 11 = 22$

Answer:

1. Short leg: $8$, Long leg: $8\sqrt{3}$
2. Hypotenuse: $4$, Long leg: $2\sqrt{3}$
3. Hypotenuse: $16$, Long leg: $8\sqrt{3}$
4. Short leg: $4\sqrt{5}$, Long leg: $4\sqrt{15}$
5. Short leg: $5$, Hypotenuse: $10$
6. Short leg: $5$, Long leg: $5\sqrt{3}$
7. Short leg: $\sqrt{2}$, Long leg: $\sqrt{6}$
8. Short leg: $4\sqrt{3}$, Hypotenuse: $8\sqrt{3}$
9. Short leg: $\frac{3}{2}$, Long leg: $\frac{3\sqrt{3}}{2}$
10. Short leg: $11$, Hypotenuse: $22$