Question

13)
given = 2√3
b= (2√3)/2 = √3
a=√3·√3 = 3
a=3, b=√3
14)
x=5·2=10
x=10, y=5
15)
v=5/√3 = 5√3/3
u=2·(5√3/3) = 10√3/3
u=10√3/3
v=5√3/3
16)
given = 3
n=3/√3 = √3
m=2·√3 = 2√3
m=2√3
n=√3
17)
leg y = 1/2
x= (1/2)·√2 = √2/2
x=√2/2
y=1/2
18)
x=18
y=9
19)
y=4/2 = 2
x=2√3
x=2√3
y=2
20)
√2·a = b = 7
a=7, b=7

Explanation:

Problem 13

Step1: Identify short leg $b$

The given side is the hypotenuse $2\sqrt{3}$. In a 30-60-90 triangle, the short leg is half the hypotenuse.
$b = \frac{2\sqrt{3}}{2} = \sqrt{3}$

Step2: Calculate long leg $a$

The long leg is $\sqrt{3}$ times the short leg.
$a = \sqrt{3} \times \sqrt{3} = 3$

Problem 14

Step1: Identify short leg $y$

The given side is the long leg $5\sqrt{3}$. In a 30-60-90 triangle, short leg $y = \frac{\text{long leg}}{\sqrt{3}}$.
$y = \frac{5\sqrt{3}}{\sqrt{3}} = 5$

Step2: Calculate hypotenuse $x$

The hypotenuse is twice the short leg.
$x = 2 \times 5 = 10$

Problem 15

Step1: Calculate short leg $v$

The given side is the long leg $5$. In a 30-60-90 triangle, short leg $v = \frac{\text{long leg}}{\sqrt{3}}$, rationalize the denominator.
$v = \frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

Step2: Calculate hypotenuse $u$

The hypotenuse is twice the short leg.
$u = 2 \times \frac{5\sqrt{3}}{3} = \frac{10\sqrt{3}}{3}$

Problem 16

Step1: Identify short leg $n$

The given side is the long leg $3$. In a 30-60-90 triangle, short leg $n = \frac{\text{long leg}}{\sqrt{3}}$, rationalize the denominator.
$n = \frac{3}{\sqrt{3}} = \sqrt{3}$

Step2: Calculate hypotenuse $m$

The hypotenuse is twice the short leg.
$m = 2 \times \sqrt{3} = 2\sqrt{3}$

Problem 17

Step1: Identify leg $y$

In a 45-45-90 triangle, legs are equal. The given leg is $\frac{1}{2}$.
$y = \frac{1}{2}$

Step2: Calculate hypotenuse $x$

The hypotenuse is $\sqrt{2}$ times a leg.
$x = \frac{1}{2} \times \sqrt{2} = \frac{\sqrt{2}}{2}$

Problem 18

Step1: Calculate short leg $y$

The given side is the long leg $9\sqrt{3}$. In a 30-60-90 triangle, short leg $y = \frac{\text{long leg}}{\sqrt{3}}$.
$y = \frac{9\sqrt{3}}{\sqrt{3}} = 9$

Step2: Calculate hypotenuse $x$

The hypotenuse is twice the short leg.
$x = 2 \times 9 = 18$

Problem 19

Step1: Identify short leg $y$

The given side is the hypotenuse $4$. In a 30-60-90 triangle, the short leg is half the hypotenuse.
$y = \frac{4}{2} = 2$

Step2: Calculate long leg $x$

The long leg is $\sqrt{3}$ times the short leg.
$x = 2 \times \sqrt{3} = 2\sqrt{3}$

Problem 20

Step1: Calculate leg $a$

In a 45-45-90 triangle, a leg is $\frac{\text{hypotenuse}}{\sqrt{2}}$. The hypotenuse is $7\sqrt{2}$.
$a = \frac{7\sqrt{2}}{\sqrt{2}} = 7$

Step2: Identify leg $b$

In a 45-45-90 triangle, legs are equal.
$b = 7$

Answer:

1. $a=3$, $b=\sqrt{3}$
2. $x=10$, $y=5$
3. $u=\frac{10\sqrt{3}}{3}$, $v=\frac{5\sqrt{3}}{3}$
4. $m=2\sqrt{3}$, $n=\sqrt{3}$
5. $x=\frac{\sqrt{2}}{2}$, $y=\frac{1}{2}$
6. $x=18$, $y=9$
7. $x=2\sqrt{3}$, $y=2$
8. $a=7$, $b=7$