Question

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use trigonometry and similar triangles to find the missing segment.
a. 90.19 units
b. 91 units
c. 86.67 units
d. 83.34 units

Explanation:

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<reasoning>

Identify the triangles

We have two right triangles sharing a vertical side of length \(115\).
The smaller right triangle has an angle of \(56^\circ\).
The larger right triangle has an angle of \(35^\circ\).

Calculate the smaller base

Let \(y\) be the base of the smaller right triangle.
Using the tangent ratio:
\[
\tan(56^\circ) = \frac{115}{y}
\]
Solving for \(y\):
\[
y = \frac{115}{\tan(56^\circ)} \approx \frac{115}{1.48256} \approx 77.57 \text{ units}
\]

Calculate the larger base

Let the total base of the larger right triangle be \(y + x\).
Using the tangent ratio:
\[
\tan(35^\circ) = \frac{115}{y + x}
\]
Solving for \(y + x\):
\[
y + x = \frac{115}{\tan(35^\circ)} \approx \frac{115}{0.70021} \approx 164.24 \text{ units}
\]

Solve for the missing segment \(x\)

Subtract the smaller base from the total base:
\[
x = (y + x) - y \approx 164.24 - 77.57 = 86.67 \text{ units}
\]
This matches option C.
</reasoning>

<answer>
<mcq-option>(A) 90.19 units</mcq-option>
<mcq-option>(B) 91 units</mcq-option>
<mcq-correct>(C) 86.67 units</mcq-correct>
<mcq-option>(D) 83.34 units</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the triangles

We have two right triangles sharing a vertical side of length \(115\).
The smaller right triangle has an angle of \(56^\circ\).
The larger right triangle has an angle of \(35^\circ\).

Calculate the smaller base

Let \(y\) be the base of the smaller right triangle.
Using the tangent ratio:
\[
\tan(56^\circ) = \frac{115}{y}
\]
Solving for \(y\):
\[
y = \frac{115}{\tan(56^\circ)} \approx \frac{115}{1.48256} \approx 77.57 \text{ units}
\]

Calculate the larger base

Let the total base of the larger right triangle be \(y + x\).
Using the tangent ratio:
\[
\tan(35^\circ) = \frac{115}{y + x}
\]
Solving for \(y + x\):
\[
y + x = \frac{115}{\tan(35^\circ)} \approx \frac{115}{0.70021} \approx 164.24 \text{ units}
\]

Solve for the missing segment \(x\)

Subtract the smaller base from the total base:
\[
x = (y + x) - y \approx 164.24 - 77.57 = 86.67 \text{ units}
\]
This matches option C.
</reasoning>

<answer>
<mcq-option>(A) 90.19 units</mcq-option>
<mcq-option>(B) 91 units</mcq-option>
<mcq-correct>(C) 86.67 units</mcq-correct>
<mcq-option>(D) 83.34 units</mcq-option>
</answer>

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