Question

which statement proves that parallelogram klmn is a rhombus?

the midpoint of both diagonals is (4, 4).
the length of \\(km\\) is \\(\sqrt{72}\\) and the length of \\(nl\\) is \\(\sqrt{8}\\).
the slopes of \\(\overline{lm}\\) and \\(\overline{kn}\\) are both \\(\frac{1}{2}\\) and \\(nk = ml = \sqrt{20}\\).
the slope of \\(\overline{km}\\) is 1 and the slope of \\(\overline{nl}\\) is -1.

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Perpendicular Lines",
"Linear Equations"
],
"new_concepts": [
"Rhombus Properties",
"Diagonals of a Rhombus"
],
"current_concepts": [
"Perpendicular Lines",
"Linear Equations",
"Rhombus Properties",
"Diagonals of a Rhombus"
]
}
</pre_analysis>

<reasoning>

Identify the properties of a rhombus and its diagonals

\[
\text{A parallelogram is a rhombus if and only if its diagonals are perpendicular.}
\]

Calculate the slopes of the diagonals KM and NL

\[

$$\begin{aligned} &K(7,7), \quad M(1,1) \implies \text{Slope of } \overline{KM} = \frac{1 - 7}{1 - 7} = 1 \\ &N(3,5), \quad L(5,3) \implies \text{Slope of } \overline{NL} = \frac{3 - 5}{5 - 3} = -1 \end{aligned}$$

\]

Determine the perpendicular relationship

\[
\text{Slope of } \overline{KM} \times \text{Slope of } \overline{NL} = 1 \times (-1) = -1 \implies \overline{KM} \perp \overline{NL}
\]
</reasoning>

<answer>
<mcq-option>The midpoint of both diagonals is (4, 4).</mcq-option>
<mcq-option>The length of KM is \(\sqrt{72}\) and the length of NL is \(\sqrt{8}\).</mcq-option>
<mcq-option>The slopes of \(\overline{LM}\) and \(\overline{KN}\) are both \(\frac{1}{2}\) and \(NK = ML = \sqrt{20}\).</mcq-option>
<mcq-correct>The slope of \(\overline{KM}\) is 1 and the slope of \(\overline{NL}\) is \(-1\).</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Rhombus Properties"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Perpendicular Lines",
"Linear Equations"
],
"new_concepts": [
"Rhombus Properties",
"Diagonals of a Rhombus"
],
"current_concepts": [
"Perpendicular Lines",
"Linear Equations",
"Rhombus Properties",
"Diagonals of a Rhombus"
]
}
</pre_analysis>

<reasoning>

Identify the properties of a rhombus and its diagonals

\[
\text{A parallelogram is a rhombus if and only if its diagonals are perpendicular.}
\]

Calculate the slopes of the diagonals KM and NL

\[

$$\begin{aligned} &K(7,7), \quad M(1,1) \implies \text{Slope of } \overline{KM} = \frac{1 - 7}{1 - 7} = 1 \\ &N(3,5), \quad L(5,3) \implies \text{Slope of } \overline{NL} = \frac{3 - 5}{5 - 3} = -1 \end{aligned}$$

\]

Determine the perpendicular relationship

\[
\text{Slope of } \overline{KM} \times \text{Slope of } \overline{NL} = 1 \times (-1) = -1 \implies \overline{KM} \perp \overline{NL}
\]
</reasoning>

<answer>
<mcq-option>The midpoint of both diagonals is (4, 4).</mcq-option>
<mcq-option>The length of KM is \(\sqrt{72}\) and the length of NL is \(\sqrt{8}\).</mcq-option>
<mcq-option>The slopes of \(\overline{LM}\) and \(\overline{KN}\) are both \(\frac{1}{2}\) and \(NK = ML = \sqrt{20}\).</mcq-option>
<mcq-correct>The slope of \(\overline{KM}\) is 1 and the slope of \(\overline{NL}\) is \(-1\).</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Rhombus Properties"
]
}
</post_analysis>