Question

find the distance between the two points.

\\((-6,6)\\) and \\((39, -194)\\)

the distance is \\(\square\\).
(type an exact answer, using radicals as needed.)

Explanation:

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<reasoning>

Identify the coordinates of the two points

\[
(x_1, y_1) = (-6, 6),\quad (x_2, y_2) = (39, -194)
\]

Apply the distance formula

\[

$$\begin{aligned} d &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ d &= \sqrt{(39 - (-6))^2 + (-194 - 6)^2} \\ d &= \sqrt{(45)^2 + (-200)^2} \end{aligned}$$

\]

Calculate the final distance value

\[

$$\begin{aligned} d &= \sqrt{2025 + 40000} \\ d &= \sqrt{42025} \\ d &= 205 \end{aligned}$$

\]
</reasoning>

<answer>
Find the distance between the two points.
\((-6,6)\) and \((39, -194)\)

The distance is <blank>205</blank>.
(Type an exact answer, using radicals as needed.)
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Distance Formula"
]
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Answer:

<pre_analysis>
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"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
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"concepts_used": [
"Distance Formula",
"Coordinate Distance"
],
"new_concepts": [],
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}
</pre_analysis>

<reasoning>

Identify the coordinates of the two points

\[
(x_1, y_1) = (-6, 6),\quad (x_2, y_2) = (39, -194)
\]

Apply the distance formula

\[

$$\begin{aligned} d &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ d &= \sqrt{(39 - (-6))^2 + (-194 - 6)^2} \\ d &= \sqrt{(45)^2 + (-200)^2} \end{aligned}$$

\]

Calculate the final distance value

\[

$$\begin{aligned} d &= \sqrt{2025 + 40000} \\ d &= \sqrt{42025} \\ d &= 205 \end{aligned}$$

\]
</reasoning>

<answer>
Find the distance between the two points.
\((-6,6)\) and \((39, -194)\)

The distance is <blank>205</blank>.
(Type an exact answer, using radicals as needed.)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Distance Formula"
]
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