Question

point f is on circle c.
what is the length of line segment gf?
12.5 units
15.0 units
17.5 units
20.0 units

Explanation:

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"concepts_used": [],
"new_concepts": [
"Tangent Line to Circle",
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"Circle Radius"
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"Pythagorean Theorem",
"Circle Radius"
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<reasoning>

Identify the right triangle and given lengths

\[

$$\begin{aligned} &\angle GEC = 90^\circ \quad (\text{tangent } GE \perp \text{radius } EC) \\ &GE = 10 \\ &EC = 7.5 \quad (\text{radius } r) \end{aligned}$$

\]

Calculate the hypotenuse GC

\[

$$\begin{aligned} &GC = \sqrt{GE^2 + EC^2} \\ &GC = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25} = 12.5 \end{aligned}$$

\]

Calculate the length of segment GF

\[

$$\begin{aligned} &CF = r = 7.5 \\ &GF = GC + CF = 12.5 + 7.5 = 20.0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>12.5 units</mcq-option>
<mcq-option>15.0 units</mcq-option>
<mcq-option>17.5 units</mcq-option>
<mcq-correct>20.0 units</mcq-correct>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Tangent Line to Circle"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
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"Tangent Line to Circle",
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],
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"Tangent Line to Circle",
"Pythagorean Theorem",
"Circle Radius"
]
}
</pre_analysis>

<reasoning>

Identify the right triangle and given lengths

\[

$$\begin{aligned} &\angle GEC = 90^\circ \quad (\text{tangent } GE \perp \text{radius } EC) \\ &GE = 10 \\ &EC = 7.5 \quad (\text{radius } r) \end{aligned}$$

\]

Calculate the hypotenuse GC

\[

$$\begin{aligned} &GC = \sqrt{GE^2 + EC^2} \\ &GC = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25} = 12.5 \end{aligned}$$

\]

Calculate the length of segment GF

\[

$$\begin{aligned} &CF = r = 7.5 \\ &GF = GC + CF = 12.5 + 7.5 = 20.0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>12.5 units</mcq-option>
<mcq-option>15.0 units</mcq-option>
<mcq-option>17.5 units</mcq-option>
<mcq-correct>20.0 units</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Tangent Line to Circle"
]
}
</post_analysis>