Question

find the length of the side labeled x. round intermediate values to the nearest tenth. use the rounded values to calculate the next value. round your final answer to the nearest tenth.
7)
8)
9)
10)
solve for x.
11)
12)

Explanation:

---

Problem 7

Step1: Find height of the triangle

Let $h$ = height. Use $\tan(64^\circ)$:
$\tan(64^\circ) = \frac{h}{x_{\text{base segment}}}$ First, find $h$ from left triangle: $\tan(32^\circ) = \frac{h}{25}$
$h = 25 \times \tan(32^\circ) \approx 25 \times 0.6249 = 15.6$

Step2: Solve for $x$ using $\sin(64^\circ)$

$\sin(64^\circ) = \frac{h}{x}$
$x = \frac{h}{\sin(64^\circ)} \approx \frac{15.6}{0.8988} \approx 17.4$

---

Problem 8

Step1: Find height of the triangle

Let $h$ = height. Use $\cos(26^\circ)$ on left triangle:
$\cos(26^\circ) = \frac{\text{adjacent}}{45}$
$\text{adjacent} = 45 \times \cos(26^\circ) \approx 45 \times 0.8988 = 40.4$
Right base segment: $45 - 40.4 = 4.6$
$h = 45 \times \sin(26^\circ) \approx 45 \times 0.4384 = 19.7$

Step2: Solve for $x$ using $\sin(44^\circ)$

$\sin(44^\circ) = \frac{h}{x}$
$x = \frac{h}{\sin(44^\circ)} \approx \frac{19.7}{0.6947} \approx 28.4$

---

Problem 9

Step1: Find height of the triangle

Let $h$ = height. Use $\tan(61^\circ)$:
$h = 14 \times \tan(61^\circ) \approx 14 \times 1.8040 = 25.3$

Step2: Find vertex angle of right triangle

Total top angle = $38^\circ$, left angle = $90^\circ - 61^\circ = 29^\circ$
Right top angle = $38^\circ - 29^\circ = 9^\circ$

Step3: Solve for $x$ using $\sin(61^\circ)$

$\sin(61^\circ) = \frac{h}{x}$
$x = \frac{h}{\sin(61^\circ)} \approx \frac{25.3}{0.8746} \approx 28.9$

---

Problem 10

Step1: Find height of the triangle

Let $h$ = height. Use $\cos(22^\circ)$:
$h = 39 \times \cos(22^\circ) \approx 39 \times 0.9272 = 36.2$

Step2: Find half of $x$ using $\tan(60^\circ)$

Left base segment: $39 \times \sin(22^\circ) \approx 39 \times 0.3746 = 14.6$
Right base segment: $\frac{h}{\tan(60^\circ)} \approx \frac{36.2}{1.732} = 20.9$

Step3: Calculate total $x$

$x = 14.6 + 20.9 = 35.5$

---

Problem 11

Step1: Sum triangle angles to 180°

$40^\circ + 60^\circ + (x + 87)^\circ = 180^\circ$

Step2: Solve for $x$

$100 + x + 87 = 180$
$x = 180 - 187 = -7$

---

Problem 12

Step1: Sum triangle angles to 180°

$50^\circ + 45^\circ + (x + 92)^\circ = 180^\circ$

Step2: Solve for $x$

$95 + x + 92 = 180$
$x = 180 - 187 = -7$

Answer:

1. $17.4$
2. $28.4$
3. $28.9$
4. $35.5$
5. $-7$
6. $-7$