Question

geometry
extra practice 45-45-90/30-60-90 right triangles
find the missing side lengths. leave your answers as radicals in simplest form.
1)
triangle with right angle, 45° angle, leg 20, hypotenuse x
2)
triangle with right angle, 30° angle, leg (6sqrt{3}), other leg b, hypotenuse a
3)
triangle with right angle, 45° angle, leg (7sqrt{2}), other leg y, hypotenuse x
4)
triangle with right angle, 60° angle, leg 17, other leg x, hypotenuse y
5)
triangle with right angle, 45° angle, hypotenuse (14sqrt{3}), legs x and y
6)
triangle with right angle, 30° angle, leg 19, other leg y, hypotenuse x
7)
triangle with right angle, 45° angle, hypotenuse (18sqrt{2}), legs m and n
8)
triangle with right angle, 60° angle, leg (sqrt{5}), other leg v, hypotenuse u
9)
triangle with right angle, 60° angle, leg (9sqrt{3}), other leg y, hypotenuse x
10)
triangle with right angle, 30° angle, leg (\frac{13}{2}), other leg y, hypotenuse x

Explanation:

1) 45-45-90 Triangle (Leg=20)

Step1: Identify equal legs

In 45-45-90 triangles, legs are equal.
$y = 20$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$
$x = 20\sqrt{2}$

2) 30-60-90 Triangle (Long Leg=$6\sqrt{3}$)

Step1: Find short leg

Short leg = $\frac{\text{Long Leg}}{\sqrt{3}}$
$b = \frac{6\sqrt{3}}{\sqrt{3}} = 6$

Step2: Calculate hypotenuse

Hypotenuse = short leg $\times 2$
$a = 6 \times 2 = 12$

3) 45-45-90 Triangle (Leg=$7\sqrt{2}$)

Step1: Identify equal legs

In 45-45-90 triangles, legs are equal.
$y = 7\sqrt{2}$

Step2: Calculate hypotenuse

Hypotenuse = leg $\times \sqrt{2}$
$x = 7\sqrt{2} \times \sqrt{2} = 14$

4) 30-60-90 Triangle (Short Leg=17)

Step1: Calculate long leg

Long leg = short leg $\times \sqrt{3}$
$y = 17\sqrt{3}$

Step2: Calculate hypotenuse

Hypotenuse = short leg $\times 2$
$x = 17 \times 2 = 34$

5) 45-45-90 Triangle (Hypotenuse=$14\sqrt{3}$)

Step1: Calculate leg length

Leg = $\frac{\text{Hypotenuse}}{\sqrt{2}}$
$x = y = \frac{14\sqrt{3}}{\sqrt{2}} = 7\sqrt{6}$

6) 30-60-90 Triangle (Short Leg=19)

Step1: Calculate hypotenuse

Hypotenuse = short leg $\times 2$
$x = 19 \times 2 = 38$

Step2: Calculate long leg

Long leg = short leg $\times \sqrt{3}$
$y = 19\sqrt{3}$

7) 45-45-90 Triangle (Hypotenuse=$18\sqrt{2}$)

Step1: Calculate leg length

Leg = $\frac{\text{Hypotenuse}}{\sqrt{2}}$
$m = n = \frac{18\sqrt{2}}{\sqrt{2}} = 18$

8) 30-60-90 Triangle (Short Leg=$\sqrt{5}$)

Step1: Calculate hypotenuse

Hypotenuse = short leg $\times 2$
$u = \sqrt{5} \times 2 = 2\sqrt{5}$

Step2: Calculate long leg

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

9) 30-60-90 Triangle (Long Leg=$8\sqrt{3}$)

Step1: Find short leg

Short leg = $\frac{\text{Long Leg}}{\sqrt{3}}$
$y = \frac{8\sqrt{3}}{\sqrt{3}} = 8$

Step2: Calculate hypotenuse

Hypotenuse = short leg $\times 2$
$x = 8 \times 2 = 16$

10) 30-60-90 Triangle (Long Leg=$\frac{13}{2}$)

Step1: Find short leg

Short leg = $\frac{\text{Long Leg}}{\sqrt{3}}$
$y = \frac{13}{2\sqrt{3}} = \frac{13\sqrt{3}}{6}$

Step2: Calculate hypotenuse

Hypotenuse = short leg $\times 2$
$x = \frac{13\sqrt{3}}{6} \times 2 = \frac{13\sqrt{3}}{3}$

Answer:

1. $y=20$, $x=20\sqrt{2}$
2. $b=6$, $a=12$
3. $y=7\sqrt{2}$, $x=14$
4. $y=17\sqrt{3}$, $x=34$
5. $x=y=7\sqrt{6}$
6. $x=38$, $y=19\sqrt{3}$
7. $m=n=18$
8. $u=2\sqrt{5}$, $v=\sqrt{15}$
9. $y=8$, $x=16$
10. $y=\frac{13\sqrt{3}}{6}$, $x=\frac{13\sqrt{3}}{3}$