Question

1. circle two numbers that would be slopes of parallel lines. box two numbers that would be slopes of perpendicular lines.

-4

\frac{-2}{3}

\frac{1}{4}

\frac{2}{3}

\frac{4}{6}

-2

6

Explanation:

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<reasoning>

Simplify the given numbers

\[

$$\begin{aligned} &-4, \quad -\frac{2}{3}, \quad \frac{1}{4}, \quad \frac{2}{3}, \quad \frac{4}{6} = \frac{2}{3}, \quad -2, \quad 6 \end{aligned}$$

\]

Identify parallel slopes

\[

$$\begin{aligned} &m_1 = m_2\\ &\frac{2}{3} = \frac{4}{6} \end{aligned}$$

\]

Identify perpendicular slopes

\[

$$\begin{aligned} &m_1 \cdot m_2 = -1\\ &-4 \cdot \frac{1}{4} = -1 \end{aligned}$$

\]
</reasoning>

<answer>

• Circle (parallel lines): \(\frac{2}{3}\) and \(\frac{4}{6}\)
• Box (perpendicular lines): \(-4\) and \(\frac{1}{4}\)

</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Slopes of Parallel and Perpendicular Lines"
]
}
</post_analysis>

Answer:

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],
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"Slopes of Perpendicular Lines"
],
"new_concepts": [],
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</pre_analysis>

<reasoning>

Simplify the given numbers

\[

$$\begin{aligned} &-4, \quad -\frac{2}{3}, \quad \frac{1}{4}, \quad \frac{2}{3}, \quad \frac{4}{6} = \frac{2}{3}, \quad -2, \quad 6 \end{aligned}$$

\]

Identify parallel slopes

\[

$$\begin{aligned} &m_1 = m_2\\ &\frac{2}{3} = \frac{4}{6} \end{aligned}$$

\]

Identify perpendicular slopes

\[

$$\begin{aligned} &m_1 \cdot m_2 = -1\\ &-4 \cdot \frac{1}{4} = -1 \end{aligned}$$

\]
</reasoning>

<answer>

• Circle (parallel lines): \(\frac{2}{3}\) and \(\frac{4}{6}\)
• Box (perpendicular lines): \(-4\) and \(\frac{1}{4}\)

</answer>

<post_analysis>
{
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"question_type": "Free Response",
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