Question

a horse corral is represented by triangle xyz. if ( mangle x = 65^circ ), ( mangle y = 78^circ ), and ( z = 32 ) ft, what is the length of ( x )?
52 ft
48 ft
41 ft
30 ft
question 2 (multiple choice worth 1 points)
(05.09 lc)
use the image below to answer the following question.
image of a right triangle with right angle at the bottom left, vertical leg 9, horizontal leg 12, hypotenuse 15, angle at top left is ( x^circ ), angle at bottom right is ( y^circ )
what relationship do the ratios of ( sin y^circ ) and ( cos x^circ ) share?
the ratios are both identical ( left( \frac{9}{15} \text{ and } \frac{9}{15}
ight) )
the ratios are opposites ( left( \frac{9}{15} \text{ and } \frac{9}{15}
ight) )
the ratios are reciprocals ( left( \frac{9}{15} \text{ and } \frac{15}{9}
ight) )
the ratios are both negative ( left( \frac{-15}{9} \text{ and } \frac{-9}{15}
ight) )

Explanation:

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First Question

Step1: Find $\angle Z$

The sum of angles in a triangle is $180^\circ$.
$\angle Z = 180^\circ - 65^\circ - 78^\circ = 37^\circ$

Step2: Apply Law of Sines

Relate sides and opposite angles: $\frac{x}{\sin Y} = \frac{z}{\sin Z}$
Substitute values: $\frac{x}{\sin 78^\circ} = \frac{32}{\sin 37^\circ}$

Step3: Solve for $x$

Rearrange to isolate $x$: $x = \frac{32 \times \sin 78^\circ}{\sin 37^\circ}$
Calculate: $\sin78^\circ\approx0.9781$, $\sin37^\circ\approx0.6018$
$x \approx \frac{32 \times 0.9781}{0.6018} \approx 52$

In a right triangle, the acute angles are complementary ($x^\circ + y^\circ = 90^\circ$). For complementary angles, $\sin(y^\circ) = \cos(90^\circ - y^\circ) = \cos(x^\circ)$. Calculate the ratios:
$\sin(y^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15}$
$\cos(x^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9}{15}$
The ratios are identical.

Answer:

52 ft

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Second Question