Question

\\(\overline{\text{fh}}\\) is tangent to circle g at point f.

what is the length of the radius, \\(r\\)?

10 units
12 units
20 units
24 units

Explanation:

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"concepts_used": [
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"Pythagorean Theorem"
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"current_concepts": [
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"Circle Geometry"
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<reasoning>

Identify the right triangle and its side lengths

\[

$$\begin{aligned} &\angle GFD = 90^\circ \quad (\text{since } \overline{FH} \text{ is tangent to circle } G \text{ at } F) \\ &FH = 35 \\ &FG = r \\ &HG = 25 + r \end{aligned}$$

\]

Apply the Pythagorean theorem to solve for r

\[

$$\begin{aligned} &FG^2 + FH^2 = HG^2 \\ &r^2 + 35^2 = (25 + r)^2 \\ &r^2 + 1225 = 625 + 50r + r^2 \\ &1225 = 625 + 50r \\ &50r = 600 \\ &r = 12 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>10 units</mcq-option>
<mcq-correct>12 units</mcq-correct>
<mcq-option>20 units</mcq-option>
<mcq-option>24 units</mcq-option>
</answer>

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

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"step_cot",
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"Circle Geometry"
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</pre_analysis>

<reasoning>

Identify the right triangle and its side lengths

\[

$$\begin{aligned} &\angle GFD = 90^\circ \quad (\text{since } \overline{FH} \text{ is tangent to circle } G \text{ at } F) \\ &FH = 35 \\ &FG = r \\ &HG = 25 + r \end{aligned}$$

\]

Apply the Pythagorean theorem to solve for r

\[

$$\begin{aligned} &FG^2 + FH^2 = HG^2 \\ &r^2 + 35^2 = (25 + r)^2 \\ &r^2 + 1225 = 625 + 50r + r^2 \\ &1225 = 625 + 50r \\ &50r = 600 \\ &r = 12 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>10 units</mcq-option>
<mcq-correct>12 units</mcq-correct>
<mcq-option>20 units</mcq-option>
<mcq-option>24 units</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
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"Circle Geometry"
]
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