15.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$
$z = \underline{\quad\quad}$
16.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$
$z = \underline{\quad\quad}$
17.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$
$z = \underline{\quad\quad}$
18.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$
$z = \underline{\quad\quad}$
19.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$
$z = \underline{\quad\quad}$
20.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$
$z = \underline{\quad\quad}$
21. find the perimeter of the triangle:
22. find the perimeter of the square:
23. steel loading ramps are used to load a lawn mower onto a truckbed 37.5 inches above ground. if the ramps make a $30^\circ$ angle with the ground, find the length of the ramps in feet.
24. the infield of a baseball field is a square with sides measuring 90 feet. a ball thrown from third to first base is caught in 1.2 seconds. find the speed of the ball in feet per second. round to the nearest tenth.

Question

Accepted Answer

### Problem 15
# Explanation:
## Step1: Solve for shared side
The top triangle is a 45-45-90 triangle. For such triangles, legs are equal, hypotenuse = leg$	imes\sqrt{2}$. Let shared side = $s$.
$14\sqrt{2} = s\sqrt{2} \implies s=14$
## Step2: Find $x$ (30-60-90 triangle)
In 30-60-90 triangle, shorter leg = $\frac{1}{2}$ hypotenuse. $s$ is the longer leg, so shorter leg $x = \frac{s}{\sqrt{3}} = \frac{14}{\sqrt{3}} = \frac{14\sqrt{3}}{3}$
## Step3: Find $z$ (30-60-90 triangle)
Hypotenuse $z = 2x = \frac{28\sqrt{3}}{3}$
# Answer:
$x=\frac{14\sqrt{3}}{3}$, $y=14$, $z=\frac{28\sqrt{3}}{3}$

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### Problem 16
# Explanation:
## Step1: Solve for shared side
The right triangle with 30° angle: shared side $s = \frac{16\sqrt{3}}{2}=8\sqrt{3}$ (30-60-90: shorter leg = $\frac{1}{2}$ hypotenuse)
## Step2: Find $x$ (45-45-90 triangle)
45-45-90 triangle: leg = $s=8\sqrt{3}$, hypotenuse $x = 8\sqrt{3}	imes\sqrt{2}=8\sqrt{6}$
## Step3: Find $y$ (45-45-90 triangle)
Legs are equal, so $y=8\sqrt{3}$
## Step4: Find $z$ (30-60-90 triangle)
Longer leg $z = s\sqrt{3}=8\sqrt{3}	imes\sqrt{3}=24$
# Answer:
$x=8\sqrt{6}$, $y=8\sqrt{3}$, $z=24$

---

### Problem 17
# Explanation:
## Step1: Find height $h$ (45-45-90 triangle)
45-45-90 triangle: leg = 30, so height $h=30$
## Step2: Find $x$ (60-30-90 triangle)
$	an60^\circ=\frac{h}{y} \implies y=\frac{h}{	an60^\circ}=\frac{30}{\sqrt{3}}=10\sqrt{3}$
Hypotenuse $x=2y=20\sqrt{3}$
## Step3: Find $z$ (45-45-90 triangle)
Hypotenuse $z=30\sqrt{2}$
# Answer:
$x=20\sqrt{3}$, $y=10\sqrt{3}$, $z=30\sqrt{2}$

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### Problem 18
# Explanation:
## Step1: Find $y$ (45-45-90 triangle)
45-45-90 triangle: leg $y = \frac{20}{\sqrt{2}}=10\sqrt{2}$
## Step2: Find $x$ (30-60-90 triangle)
30-60-90 triangle: hypotenuse $x=2y=20\sqrt{2}$
## Step3: Find $z$ (30-60-90 triangle)
Longer leg $z = y\sqrt{3}=10\sqrt{6}$
# Answer:
$x=20\sqrt{2}$, $y=10\sqrt{2}$, $z=10\sqrt{6}$

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### Problem 19
# Explanation:
## Step1: Find shared side $s$ (30-60-90 triangle)
30-60-90 triangle: longer leg $6\sqrt{3}=s\sqrt{3} \implies s=6$
## Step2: Find $x$ (30-60-90 triangle)
Hypotenuse $x=2s=12$
## Step3: Find $y$ (45-45-90 triangle)
45-45-90 triangle: leg $y=s=6$
## Step4: Find $z$ (45-45-90 triangle)
Hypotenuse $z=6\sqrt{2}$
# Answer:
$x=12$, $y=6$, $z=6\sqrt{2}$

---

### Problem 20
# Explanation:
## Step1: Find shared height $h$ (45-45-90 triangle)
45-45-90 triangle: $10\sqrt{6}=h\sqrt{2} \implies h=10\sqrt{3}$
## Step2: Find $z$ (45-45-90 triangle)
Legs equal, so $z=h=10\sqrt{3}$
## Step3: Find $x$ (30-60-90 triangle)
Hypotenuse $x=2h=20\sqrt{3}$
## Step4: Find $y$ (30-60-90 triangle)
Longer leg $y=h\sqrt{3}=10\sqrt{3}	imes\sqrt{3}=30$
# Answer:
$x=20\sqrt{3}$, $y=30$, $z=10\sqrt{3}$

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### Problem 21
# Explanation:
## Step1: Identify triangle type
Two 60° angles mean it's equilateral. Height $h=4\sqrt{3}$.
## Step2: Calculate side length $s$
For equilateral triangle, $h=\frac{\sqrt{3}}{2}s \implies s=\frac{2h}{\sqrt{3}}=\frac{2	imes4\sqrt{3}}{\sqrt{3}}=8$
## Step3: Calculate perimeter
Perimeter = $3s=3	imes8=24$
# Answer:
24

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### Problem 22
# Explanation:
## Step1: Find square side length $s$
Square diagonal $d=20$, $d=s\sqrt{2} \implies s=\frac{20}{\sqrt{2}}=10\sqrt{2}$
## Step2: Calculate perimeter
Perimeter = $4s=4	imes10\sqrt{2}=40\sqrt{2} \approx 56.57$
# Answer:
$40\sqrt{2}$ (or approximately 56.6)

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### Problem 23
# Explanation:
## Step1: Convert height to feet
$37.5$ inches $=\frac{37.5}{12}=3.125$ feet
## Step2: Use sine to find ramp length
$\sin30^\circ=\frac{	ext{height}}{	ext{ramp length}} \implies 	ext{ramp length}=\frac{3.125}{\sin30^\circ}=\frac{3.125}{0.5}=6.25$
# Answer:
6.25 feet

---

### Problem 24
# Explanation:
## Step1: Find distance between bases
Square side = 90 ft, diagonal $d=90\sqrt{2}\approx127.279$ feet
## Step2: Calculate speed
$	ext{speed}=\frac{	ext{distance}}{	ext{time}}=\frac{127.279}{1.2}\approx106.1$
# Answer:
106.1 feet per second

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QUESTION IMAGE

Question

Explanation:

Problem 15

Step1: Solve for shared side

Step2: Find $x$ (30-60-90 triangle)

Step3: Find $z$ (30-60-90 triangle)

Step1: Solve for shared side

Step2: Find $x$ (45-45-90 triangle)

Step3: Find $y$ (45-45-90 triangle)

Step4: Find $z$ (30-60-90 triangle)

Step1: Find height $h$ (45-45-90 triangle)

Step2: Find $x$ (60-30-90 triangle)

Step3: Find $z$ (45-45-90 triangle)

Answer:

Problem 16