Question

what is the measure of arc qr?

\\(26^\circ\\)
\\(52^\circ\\)
\\(104^\circ\\)
\\(128^\circ\\)

Explanation:

<pre_analysis>
{
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"skills_matched": [
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"multiple_choice_formatter",
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"needs_drawing": false,
"concepts_used": [
"Inscribed Angle Theorem"
],
"new_concepts": [],
"current_concepts": [
"Inscribed Angle Theorem",
"Circle Geometry"
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</pre_analysis>

<reasoning>

Identify the inscribed angle and its intercepted arc

\[

$$\begin{aligned} &\angle QSR = 52^\circ\\ &\text{Intercepted Arc} = \text{arc } QR \end{aligned}$$

\]

Apply the Inscribed Angle Theorem

\[

$$\begin{aligned} &m\angle QSR = \frac{1}{2} \cdot m(\text{arc } QR)\\ &52^\circ = \frac{1}{2} \cdot m(\text{arc } QR) \end{aligned}$$

\]

Solve for the measure of arc QR

\[

$$\begin{aligned} &m(\text{arc } QR) = 2 \cdot 52^\circ = 104^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(26^\circ\)</mcq-option>
<mcq-option>(B) \(52^\circ\)</mcq-option>
<mcq-correct>(C) \(104^\circ\)</mcq-correct>
<mcq-option>(D) \(128^\circ\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Inscribed Angle Theorem"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Inscribed Angle Theorem"
],
"new_concepts": [],
"current_concepts": [
"Inscribed Angle Theorem",
"Circle Geometry"
]
}
</pre_analysis>

<reasoning>

Identify the inscribed angle and its intercepted arc

\[

$$\begin{aligned} &\angle QSR = 52^\circ\\ &\text{Intercepted Arc} = \text{arc } QR \end{aligned}$$

\]

Apply the Inscribed Angle Theorem

\[

$$\begin{aligned} &m\angle QSR = \frac{1}{2} \cdot m(\text{arc } QR)\\ &52^\circ = \frac{1}{2} \cdot m(\text{arc } QR) \end{aligned}$$

\]

Solve for the measure of arc QR

\[

$$\begin{aligned} &m(\text{arc } QR) = 2 \cdot 52^\circ = 104^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(26^\circ\)</mcq-option>
<mcq-option>(B) \(52^\circ\)</mcq-option>
<mcq-correct>(C) \(104^\circ\)</mcq-correct>
<mcq-option>(D) \(128^\circ\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Inscribed Angle Theorem"
]
}
</post_analysis>