Question

question 18 (2 points)
19)
29
26°
34°
x
a) 8.6
b) 7.5
c) 11
d) 5.5

Explanation:

Step1: Find height of the triangle

Let $h$ be the height. Use $\sin(26^\circ)$ on left triangle:
$\sin(26^\circ) = \frac{h}{29}$
$h = 29 \times \sin(26^\circ) \approx 29 \times 0.4384 \approx 12.7136$

Step2: Find length $x$

Use $\tan(34^\circ)$ on right triangle:
$\tan(34^\circ) = \frac{h}{x}$
$x = \frac{h}{\tan(34^\circ)} \approx \frac{12.7136}{0.6745} \approx 18.85$
*Note: Re-evaluate using consistent triangle logic:
First, find the segment adjacent to $26^\circ$: $29\cos(26^\circ)\approx26.13$. The angle at the top is $180-26-34=120^\circ$. Use Law of Sines:
$\frac{x}{\sin(26^\circ)}=\frac{29}{\sin(34^\circ)}$
$x = 29 \times \frac{\sin(26^\circ)}{\sin(34^\circ)} \approx 29 \times \frac{0.4384}{0.5592} \approx 29 \times 0.784 \approx 22.74$
Correcting: The height $h=29\sin26\approx12.71$. For the right triangle, $x=h/\tan34\approx12.71/0.6745\approx18.85$, but matching options, recalculate precisely:
$\sin26^\circ\approx0.4383711467890774$, $29\times0.4383711467890774\approx12.712763256883245$
$\tan34^\circ\approx0.6745085168424265$, $12.712763256883245/0.6745085168424265\approx18.85$
*Adjustment: Recognize $x$ is only the right segment. The full base is $29\cos26 + x$, but question labels $x$ as the right base segment. Rounding to closest option, 18.85 is not listed, recalculate with Law of Sines for the triangle:
$\frac{x}{\sin(26^\circ)}=\frac{29}{\sin(34^\circ)}$
$x=29\times\frac{\sin26}{\sin34}\approx29\times0.784\approx22.74$
*Error correction: The vertical line splits the triangle into two right triangles. The angle at the top for the right triangle is $34^\circ$, so the top angle of the whole triangle is $180-26-90+34=98^\circ$? No, correct: The right triangle has angle $34^\circ$ at the bottom right, so the height $h$ is opposite $34^\circ$, so $h=x\sin34$. Also $h=29\sin26$. Set equal:
$x\sin34=29\sin26$
$x=29\times\frac{\sin26}{\sin34}\approx29\times\frac{0.4384}{0.5592}\approx29\times0.784\approx22.74$
*Note: The closest option is not present, but rechecking calculations with more precision:
$\sin26^\circ=0.4383711467890774$, $\sin34^\circ=0.5591929034707468$
$\frac{0.4383711467890774}{0.5591929034707468}\approx0.7839$
$29\times0.7839\approx22.73$
*Re-express: The question's $x$ is the adjacent side to $34^\circ$, so $h=x\tan34$, and $h=29\sin26$, so $x=\frac{29\sin26}{\tan34}=29\times\frac{\sin26\cos34}{\sin34}$
$\cos34^\circ\approx0.8290$
$x=29\times\frac{0.4384\times0.8290}{0.5592}\approx29\times\frac{0.3634}{0.5592}\approx29\times0.650\approx18.85$
*Conclusion: The closest option to 18.85 is not listed, but likely a misinterpretation. If $x$ is the full base:
Full base $=29\cos26 + x=29\times0.8988 + 18.85\approx26.07+18.85=44.92$, not matching.
*Correcting initial misread: The $34^\circ$ is at the top, so the right triangle has angle $34^\circ$ at the top, so $\sin34=\frac{x}{hypotenuse}$, no, height $h=29\sin26$, and $\cos34=\frac{h}{right side}$, so right side $=\frac{h}{\cos34}$, then $x=right side \sin34=h\tan34=29\sin26\tan34\approx29\times0.4384\times0.6745\approx29\times0.2957\approx8.57\approx8.6$

Step3: Match to option

$x\approx8.6$, which matches option A.

Answer:

A) 8.6