Question

find the value of the variables. show your work and leave answers in simplified radical form.
19.
right triangle with legs x, x and hypotenuse 4√2, angles 45°, 45°
x =
20.
right triangle with leg x, leg 2√3, angle n° and angle 2n°
n =
x =
y =
21.
right triangle with leg x, leg x + 4, angle 60°
x =
22.
isosceles right triangle with legs x, x and hypotenuse x + 4
x =
23.
quadrilateral divided into two right triangles, one with leg 1, angle 60°, the other with leg x, right angle
x =
24.
figure with triangles, one with side 1, angle 45°, another with angle 30°, side x
x =
options: 1/2, √2, √3, 2 + 2√3, 3, 4, 4 + 4√2, 30

Explanation:

Problem 19

Step1: Use 45-45-90 triangle rules

In a 45-45-90 triangle, hypotenuse = $x\sqrt{2}$, where $x$ is leg length.
$4\sqrt{2} = x\sqrt{2}$

Step2: Solve for $x$

Divide both sides by $\sqrt{2}$:
$x = \frac{4\sqrt{2}}{\sqrt{2}} = 4$

Problem 20

Step1: Solve for $n$

Sum of triangle angles = $180^\circ$.
$n + 2n + 90 = 180$
$3n = 90$
$n = 30$

Step2: Identify triangle type

Angles are $30^\circ, 60^\circ, 90^\circ$. Shorter leg $x$ is opposite $30^\circ$, hypotenuse $y$ is twice the shorter leg, longer leg is $x\sqrt{3}$.

Step3: Solve for $x$

Longer leg = $2\sqrt{3} = x\sqrt{3}$
$x = \frac{2\sqrt{3}}{\sqrt{3}} = 2$

Step4: Solve for $y$

$y = 2x = 2\times2 = 4$

Problem 21

Step1: Use 30-60-90 triangle rules

In 30-60-90 triangle, longer leg = $x\sqrt{3}$, where $x$ is shorter leg.
$x+4 = x\sqrt{3}$

Step2: Rearrange and solve for $x$

$x\sqrt{3} - x = 4$
$x(\sqrt{3}-1) = 4$
$x = \frac{4}{\sqrt{3}-1}$
Rationalize denominator:
$x = \frac{4(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = \frac{4(\sqrt{3}+1)}{2} = 2+2\sqrt{3}$

Problem 22

Step1: Use 45-45-90 triangle rules

Hypotenuse = $x\sqrt{2}$, here hypotenuse is $x+4$.
$x+4 = x\sqrt{2}$

Step2: Rearrange and solve for $x$

$x\sqrt{2} - x = 4$
$x(\sqrt{2}-1) = 4$
$x = \frac{4}{\sqrt{2}-1}$
Rationalize denominator:
$x = \frac{4(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1)} = \frac{4(\sqrt{2}+1)}{1} = 4+4\sqrt{2}$

Problem 23

Step1: Find segment in 30-60-90 triangle

Left triangle: shorter leg = 1, longer leg (shared segment) = $1\times\sqrt{3} = \sqrt{3}$

Step2: Use 45-45-90 triangle rules

Right triangle is 45-45-90, so legs are equal: $x = \sqrt{3}$

Problem 24

Step1: Find segment in 45-45-90 triangle

Top triangle: leg = 1, hypotenuse (shared segment) = $1\times\sqrt{2} = \sqrt{2}$

Step2: Use 30-60-90 triangle rules

Bottom triangle: side opposite $30^\circ$ is $x$, hypotenuse is $\sqrt{2}$.
$x = \frac{\sqrt{2}}{2}$
Rationalize:
$x = \frac{\sqrt{2}}{2}$

Answer:

1. $x=4$
2. $n=30$, $x=2$, $y=4$
3. $x=2+2\sqrt{3}$
4. $x=4+4\sqrt{2}$
5. $x=\sqrt{3}$
6. $x=\frac{\sqrt{2}}{2}$