Question

line segment xy is tangent to circle z at point u.
if the measure of \\(\widehat{uv}\\) is \\(84^\circ\\), what is the measure of \\(\angle yuv\\)?

\\(42^\circ\\)
\\(84^\circ\\)
\\(96^\circ\\)
\\(168^\circ\\)

Explanation:

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<reasoning>

Identify the given geometric properties

\[

$$\begin{aligned} &\text{Line } XY \text{ is tangent to circle } Z \text{ at point } U.\\ &V \text{ is a point on circle } Z.\\ &m\widehat{UV} = 84^\circ \end{aligned}$$

\]

Apply the Tangent-Chord Theorem

\[

$$\begin{aligned} &m\angle YUV = \frac{1}{2} \cdot m\widehat{UV} \end{aligned}$$

\]

Calculate the angle measure

\[

$$\begin{aligned} &m\angle YUV = \frac{1}{2} \cdot 84^\circ = 42^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(42^\circ\)</mcq-correct>
<mcq-option>(B) \(84^\circ\)</mcq-option>
<mcq-option>(C) \(96^\circ\)</mcq-option>
<mcq-option>(D) \(168^\circ\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
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"Tangent-Chord Theorem"
]
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Answer:

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<reasoning>

Identify the given geometric properties

\[

$$\begin{aligned} &\text{Line } XY \text{ is tangent to circle } Z \text{ at point } U.\\ &V \text{ is a point on circle } Z.\\ &m\widehat{UV} = 84^\circ \end{aligned}$$

\]

Apply the Tangent-Chord Theorem

\[

$$\begin{aligned} &m\angle YUV = \frac{1}{2} \cdot m\widehat{UV} \end{aligned}$$

\]

Calculate the angle measure

\[

$$\begin{aligned} &m\angle YUV = \frac{1}{2} \cdot 84^\circ = 42^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(42^\circ\)</mcq-correct>
<mcq-option>(B) \(84^\circ\)</mcq-option>
<mcq-option>(C) \(96^\circ\)</mcq-option>
<mcq-option>(D) \(168^\circ\)</mcq-option>
</answer>

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"Geometry",
"Tangent-Chord Theorem"
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