Question

in the diagram of circle c, \\(m\angle vwx\\) is \\(43^\circ\\). what is \\(m\widehat{ux}\\)?

\\(39^\circ\\)
\\(41^\circ\\)
\\(78^\circ\\)
\\(82^\circ\\)

Explanation:

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"Intersecting Secants Theorem"
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<reasoning>

Identify the given values and the geometric relationship

\[

$$\begin{aligned} &m\angle VWX = 43^\circ \\ &m\widehat{UY} = 128^\circ \\ &m\angle VWX = \frac{1}{2}(m\widehat{UY} - m\widehat{VX}) \end{aligned}$$

\]

Solve for the measure of arc VX

\[

$$\begin{aligned} &43^\circ = \frac{1}{2}(128^\circ - m\widehat{VX}) \\ &86^\circ = 128^\circ - m\widehat{VX} \\ &m\widehat{VX} = 128^\circ - 86^\circ \\ &m\widehat{VX} = 42^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 39°</mcq-option>
<mcq-option>(B) 41°</mcq-option>
<mcq-option>(C) 78°</mcq-option>
<mcq-option>(D) 82°</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circle Geometry"
]
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Answer:

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<reasoning>

Identify the given values and the geometric relationship

\[

$$\begin{aligned} &m\angle VWX = 43^\circ \\ &m\widehat{UY} = 128^\circ \\ &m\angle VWX = \frac{1}{2}(m\widehat{UY} - m\widehat{VX}) \end{aligned}$$

\]

Solve for the measure of arc VX

\[

$$\begin{aligned} &43^\circ = \frac{1}{2}(128^\circ - m\widehat{VX}) \\ &86^\circ = 128^\circ - m\widehat{VX} \\ &m\widehat{VX} = 128^\circ - 86^\circ \\ &m\widehat{VX} = 42^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 39°</mcq-option>
<mcq-option>(B) 41°</mcq-option>
<mcq-option>(C) 78°</mcq-option>
<mcq-option>(D) 82°</mcq-option>
</answer>

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