Question

right triangle xyz has a right angle at vertex y and a hypotenuse that measures 24 cm. angle zxy measures 70°.
what is the length of line segment xy? round to the nearest tenth.
8.2 cm
8.7 cm
22.6 m
25.5 cm

Explanation:

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<reasoning>

Identify given values and trigonometric relationship

\[

$$\begin{aligned} &\text{Hypotenuse } XZ = 24\text{ cm} \\ &\text{Angle } \angle ZXY = 70^\circ \\ &\cos(\angle ZXY) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{XY}{XZ} \end{aligned}$$

\]

Solve for the length of segment XY

\[

$$\begin{aligned} &XY = XZ \cdot \cos(70^\circ) \\ &XY = 24 \cdot \cos(70^\circ) \end{aligned}$$

\]

Calculate the numerical value

\[

$$\begin{aligned} &\cos(70^\circ) \approx 0.34202 \\ &XY \approx 24 \cdot 0.34202 \approx 8.2085\text{ cm} \\ &XY \approx 8.2\text{ cm} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) 8.2 cm</mcq-correct>
<mcq-option>(B) 8.7 cm</mcq-option>
<mcq-option>(C) 22.6 m</mcq-option>
<mcq-option>(D) 25.5 cm</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Y",
"color": "#8C55F2",
"withLabel": true,
"labelPosition": ["left", "bottom"]
}
},
{
"type": "point",
"params": [[2.81, 0]],
"properties": {
"name": "X",
"color": "#8C55F2",
"withLabel": true,
"labelPosition": ["right", "bottom"]
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},
{
"type": "point",
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"name": "XY",
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},
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"name": "YZ",
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"properties": {
"strokeColor": "#F2557F",
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"name": "XZ",
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"params": [[1.5, 4.2], "24 cm"],
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Answer:

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<reasoning>

Identify given values and trigonometric relationship

\[

$$\begin{aligned} &\text{Hypotenuse } XZ = 24\text{ cm} \\ &\text{Angle } \angle ZXY = 70^\circ \\ &\cos(\angle ZXY) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{XY}{XZ} \end{aligned}$$

\]

Solve for the length of segment XY

\[

$$\begin{aligned} &XY = XZ \cdot \cos(70^\circ) \\ &XY = 24 \cdot \cos(70^\circ) \end{aligned}$$

\]

Calculate the numerical value

\[

$$\begin{aligned} &\cos(70^\circ) \approx 0.34202 \\ &XY \approx 24 \cdot 0.34202 \approx 8.2085\text{ cm} \\ &XY \approx 8.2\text{ cm} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) 8.2 cm</mcq-correct>
<mcq-option>(B) 8.7 cm</mcq-option>
<mcq-option>(C) 22.6 m</mcq-option>
<mcq-option>(D) 25.5 cm</mcq-option>
</answer>

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