Question

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.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$
$z = \underline{\quad\quad}$

1. find the perimeter of the triangle:
2. find the perimeter of the square
3. 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.
4. 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.

Explanation:

---

Problem 15

Step1: Solve for shared hypotenuse

In top right 45-45-90 triangle, hypotenuse $h = 14\sqrt{2} \times \sqrt{2} = 28$

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

$x = 14\sqrt{2}$ (equal leg of 45-45-90 triangle)

Step3: Find $z$ (30-60-90 short leg)

$z = \frac{h}{2} = \frac{28}{2} = 14$

Step4: Find $y$ (30-60-90 long leg)

$y = z \times \sqrt{3} = 14\sqrt{3}$

Problem 16

Step1: Solve for shared hypotenuse

In top right 30-60-90 triangle, hypotenuse $h = 16\sqrt{3} \times \frac{2}{\sqrt{3}} = 32$

Step2: Find $x$ (30-60-90 short leg)

$x = \frac{h}{2} = \frac{32}{2} = 16$

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

$y = \frac{h}{\sqrt{2}} = \frac{32}{\sqrt{2}} = 16\sqrt{2}$

Step4: Find $z$ (45-45-90 leg)

$z = y = 16\sqrt{2}$

Problem 17

Step1: Find height of right triangles

In right 45-45-90 triangle, height $h = 39$ (equal leg)

Step2: Find $y$ (60-30-90 short leg)

$y = \frac{h}{\sqrt{3}} = \frac{39}{\sqrt{3}} = 13\sqrt{3}$

Step3: Find $x$ (60-30-90 hypotenuse)

$x = 2y = 2 \times 13\sqrt{3} = 26\sqrt{3}$

Step4: Find $z$ (45-45-90 hypotenuse)

$z = h\sqrt{2} = 39\sqrt{2}$

Problem 18

Step1: Find $y$ (45-45-90 leg)

$y = \frac{20}{\sqrt{2}} = 10\sqrt{2}$

Step2: Find $x$ (30-60-90 short leg)

$x = \frac{y}{\sqrt{3}} = \frac{10\sqrt{2}}{\sqrt{3}} = \frac{10\sqrt{6}}{3}$

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

$z = 2x = 2 \times \frac{10\sqrt{6}}{3} = \frac{20\sqrt{6}}{3}$

Problem 19

Step1: Find $x$ (30-60-90 short leg)

$x = \frac{6\sqrt{3}}{\sqrt{3}} = 6$

Step2: Find shared hypotenuse

Hypotenuse $h = 2x = 2 \times 6 = 12$

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

$y = \frac{h}{\sqrt{2}} = \frac{12}{\sqrt{2}} = 6\sqrt{2}$

Step4: Find $z$ (45-45-90 hypotenuse)

$z = h\sqrt{2} = 12\sqrt{2}$

Problem 20

Step1: Find height of right triangles

In right 45-45-90 triangle, height $h = 10\sqrt{6} \times \frac{1}{\sqrt{2}} = 10\sqrt{3}$

Step2: Find $z$ (45-45-90 leg)

$z = h = 10\sqrt{3}$

Step3: Find $y$ (30-60-90 short leg)

$y = \frac{h}{\sqrt{3}} = \frac{10\sqrt{3}}{\sqrt{3}} = 10$

Step4: Find $x$ (30-60-90 hypotenuse)

$x = 2y = 2 \times 10 = 20$

Problem 21

Step1: Find side length of triangle

For 60-60-60 triangle, side $s = \frac{4\sqrt{15}}{\sin60^\circ} = \frac{4\sqrt{15}}{\frac{\sqrt{3}}{2}} = 8\sqrt{5}$

Step2: Calculate perimeter

Perimeter $P = 3s = 3 \times 8\sqrt{5} = 24\sqrt{5}$

Problem 22

Step1: Find side length of square

Square side $s = \frac{28}{\sqrt{2}} = 14\sqrt{2}$

Step2: Calculate perimeter

Perimeter $P = 4s = 4 \times 14\sqrt{2} = 56\sqrt{2}$

Problem 23

Step1: Convert height to feet

Height $h = \frac{37.5}{12} = 3.125$ feet

Step2: Calculate ramp length

Ramp length $L = \frac{h}{\sin30^\circ} = \frac{3.125}{0.5} = 6.25$ feet

Problem 24

Step1: Find distance between bases

Distance $d = 90\sqrt{2} \approx 127.279$ feet

Step2: Calculate speed

Speed $v = \frac{d}{1.2} = \frac{127.279}{1.2} \approx 106.1$ ft/s
---

Answer:

1. $x=14\sqrt{2}$, $y=14\sqrt{3}$, $z=14$
2. $x=16$, $y=16\sqrt{2}$, $z=16\sqrt{2}$
3. $x=26\sqrt{3}$, $y=13\sqrt{3}$, $z=39\sqrt{2}$
4. $x=\frac{10\sqrt{6}}{3}$, $y=10\sqrt{2}$, $z=\frac{20\sqrt{6}}{3}$
5. $x=6$, $y=6\sqrt{2}$, $z=12\sqrt{2}$
6. $x=20$, $y=10$, $z=10\sqrt{3}$
7. $24\sqrt{5}$
8. $56\sqrt{2}$
9. $6.25$ feet
10. $106.1$ feet per second