Question

9)
10)
8√3
60°
y
x
11)
x
5√3
60°
y
12)
10
x
60°
y
13)
u
45°
8√2
14)
y
12
30°
x
15)
a
b
60°
3
16)
11√3
30°
a
b
17)
2√2
a
60°
b
18)
7
m
45°
n

Explanation:

Problem 9

Step1: Identify triangle type

This is a 30-60-90 right triangle (given 60° and right angle). The side length opposite 30° is $8$.

Step2: Find hypotenuse $u$

In 30-60-90 triangles, hypotenuse = 2 × shorter leg.
$u = 2 \times 8 = 16$

Step3: Find longer leg $v$

Longer leg = shorter leg × $\sqrt{3}$.
$v = 8 \times \sqrt{3} = 8\sqrt{3}$

Problem 10

Step1: Identify triangle type

30-60-90 right triangle (given 60° and right angle). Hypotenuse = $8\sqrt{5}$.

Step2: Find shorter leg $y$

Shorter leg = $\frac{1}{2}$ × hypotenuse.
$y = \frac{1}{2} \times 8\sqrt{5} = 4\sqrt{5}$

Step3: Find longer leg $x$

Longer leg = shorter leg × $\sqrt{3}$.
$x = 4\sqrt{5} \times \sqrt{3} = 4\sqrt{15}$

Problem 11

Step1: Identify triangle type

30-60-90 right triangle (given 60° and right angle). Longer leg = $5\sqrt{3}$.

Step2: Find shorter leg $y$

Shorter leg = $\frac{\text{longer leg}}{\sqrt{3}}$.
$y = \frac{5\sqrt{3}}{\sqrt{3}} = 5$

Step3: Find hypotenuse $x$

Hypotenuse = 2 × shorter leg.
$x = 2 \times 5 = 10$

Problem 12

Step1: Identify triangle type

30-60-90 right triangle (given 60° and right angle). Hypotenuse = $10$.

Step2: Find shorter leg $x$

Shorter leg = $\frac{1}{2}$ × hypotenuse.
$x = \frac{1}{2} \times 10 = 5$

Step3: Find longer leg $y$

Longer leg = shorter leg × $\sqrt{3}$.
$y = 5 \times \sqrt{3} = 5\sqrt{3}$

Problem 13

Step1: Identify triangle type

45-45-90 right triangle (given 45° and right angle). Hypotenuse = $8\sqrt{2}$.

Step2: Find leg length $u=v$

Leg length = $\frac{\text{hypotenuse}}{\sqrt{2}}$.
$u = v = \frac{8\sqrt{2}}{\sqrt{2}} = 8$

Problem 14

Step1: Identify triangle type

30-60-90 right triangle (given 30° and right angle). Longer leg = $12$.

Step2: Find shorter leg $y$

Shorter leg = $\frac{\text{longer leg}}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3}$

Step3: Find hypotenuse $x$

Hypotenuse = 2 × shorter leg.
$x = 2 \times 4\sqrt{3} = 8\sqrt{3}$

Problem 15

Step1: Identify triangle type

30-60-90 right triangle (given 60° and right angle). Hypotenuse = $3$.

Step2: Find shorter leg $b$

Shorter leg = $\frac{1}{2}$ × hypotenuse.
$b = \frac{1}{2} \times 3 = 1.5 = \frac{3}{2}$

Step3: Find longer leg $a$

Longer leg = shorter leg × $\sqrt{3}$.
$a = \frac{3}{2} \times \sqrt{3} = \frac{3\sqrt{3}}{2}$

Problem 16

Step1: Identify triangle type

30-60-90 right triangle (given 30° and right angle). Longer leg = $11\sqrt{3}$.

Step2: Find shorter leg $b$

Shorter leg = $\frac{\text{longer leg}}{\sqrt{3}}$.
$b = \frac{11\sqrt{3}}{\sqrt{3}} = 11$

Step3: Find hypotenuse $a$

Hypotenuse = 2 × shorter leg.
$a = 2 \times 11 = 22$

Problem 17

Step1: Identify triangle type

30-60-90 right triangle (given 60° and right angle). Hypotenuse = $2\sqrt{2}$.

Step2: Find shorter leg $a$

Shorter leg = $\frac{1}{2}$ × hypotenuse.
$a = \frac{1}{2} \times 2\sqrt{2} = \sqrt{2}$

Step3: Find longer leg $b$

Longer leg = shorter leg × $\sqrt{3}$.
$b = \sqrt{2} \times \sqrt{3} = \sqrt{6}$

Problem 18

Step1: Identify triangle type

45-45-90 right triangle (given 45° and right angle). Hypotenuse = $7$.

Step2: Find leg lengths $m=n$

Leg length = $\frac{\text{hypotenuse}}{\sqrt{2}} = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2}$

Answer:

1. $u=16$, $v=8\sqrt{3}$
2. $y=4\sqrt{5}$, $x=4\sqrt{15}$
3. $y=5$, $x=10$
4. $x=5$, $y=5\sqrt{3}$
5. $u=8$, $v=8$
6. $y=4\sqrt{3}$, $x=8\sqrt{3}$
7. $b=\frac{3}{2}$, $a=\frac{3\sqrt{3}}{2}$
8. $b=11$, $a=22$
9. $a=\sqrt{2}$, $b=\sqrt{6}$
10. $m=\frac{7\sqrt{2}}{2}$, $n=\frac{7\sqrt{2}}{2}$