Question

find the distance between the pair of points.

\\((-4, -12)\\) and \\((-15, -19)\\)

the distance is \\(\square\\).
(round to the nearest thousandth as needed.)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Distance Formula",
"Coordinate Distance"
],
"new_concepts": [],
"current_concepts": [
"Distance Formula",
"Coordinate Distance"
]
}
</pre_analysis>

<reasoning>

Apply the distance formula

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
\[
d = \sqrt{(-15 - (-4))^2 + (-19 - (-12))^2}
\]

Simplify the expression

\[
d = \sqrt{(-11)^2 + (-7)^2}
\]
\[
d = \sqrt{121 + 49}
\]
\[
d = \sqrt{170}
\]

Calculate decimal value

\[
d \approx 13.0384048...
\]
\[
d \approx 13.038
\]
</reasoning>

<answer>
Find the distance between the pair of points.
\((-4, -12)\) and \((-15, -19)\)

The distance is <blank>13.038</blank>.
(Round to the nearest thousandth as needed.)
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Distance Formula",
"Coordinate Distance"
],
"new_concepts": [],
"current_concepts": [
"Distance Formula",
"Coordinate Distance"
]
}
</pre_analysis>

<reasoning>

Apply the distance formula

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
\[
d = \sqrt{(-15 - (-4))^2 + (-19 - (-12))^2}
\]

Simplify the expression

\[
d = \sqrt{(-11)^2 + (-7)^2}
\]
\[
d = \sqrt{121 + 49}
\]
\[
d = \sqrt{170}
\]

Calculate decimal value

\[
d \approx 13.0384048...
\]
\[
d \approx 13.038
\]
</reasoning>

<answer>
Find the distance between the pair of points.
\((-4, -12)\) and \((-15, -19)\)

The distance is <blank>13.038</blank>.
(Round to the nearest thousandth as needed.)
</answer>

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