Question

what is the perimeter of kite wxyz?

\\(2\sqrt{53} + 2\\) units
\\(2\sqrt{53} + 5\\) units
\\(2\sqrt{53} + 10\\) units
\\(2\sqrt{53} + 14\\) units

Explanation:

<pre_analysis>
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"concepts_used": [
"Kite Definition",
"Distance Formula"
],
"new_concepts": [],
"current_concepts": [
"Kite Definition",
"Distance Formula",
"Perimeter of Polygons"
]
}
</pre_analysis>

<reasoning>

Calculate the lengths of the shorter adjacent sides

\[

$$\begin{aligned} &W = (-3, 3), \quad X = (2, 3), \quad Z = (-3, -2)\\ &WX = \sqrt{(2 - (-3))^2 + (3 - 3)^2} = \sqrt{5^2 + 0^2} = 5\\ &WZ = \sqrt{(-3 - (-3))^2 + (-2 - 3)^2} = \sqrt{0^2 + (-5)^2} = 5 \end{aligned}$$

\]

Calculate the lengths of the longer adjacent sides

\[

$$\begin{aligned} &X = (2, 3), \quad Y = (4, -4), \quad Z = (-3, -2)\\ &XY = \sqrt{(4 - 2)^2 + (-4 - 3)^2} = \sqrt{2^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53}\\ &ZY = \sqrt{(4 - (-3))^2 + (-4 - (-2))^2} = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \end{aligned}$$

\]

Sum the side lengths to find the perimeter

\[
\text{Perimeter} = WX + WZ + XY + ZY = 5 + 5 + \sqrt{53} + \sqrt{53} = 2\sqrt{53} + 10
\]
</reasoning>

<answer>
<mcq-option>(A) \(2\sqrt{53} + 2\) units</mcq-option>
<mcq-option>(B) \(2\sqrt{53} + 5\) units</mcq-option>
<mcq-correct>(C) \(2\sqrt{53} + 10\) units</mcq-correct>
<mcq-option>(D) \(2\sqrt{53} + 14\) units</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Perimeter of Polygons"
]
}
</post_analysis>

Answer:

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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
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"concepts_used": [
"Kite Definition",
"Distance Formula"
],
"new_concepts": [],
"current_concepts": [
"Kite Definition",
"Distance Formula",
"Perimeter of Polygons"
]
}
</pre_analysis>

<reasoning>

Calculate the lengths of the shorter adjacent sides

\[

$$\begin{aligned} &W = (-3, 3), \quad X = (2, 3), \quad Z = (-3, -2)\\ &WX = \sqrt{(2 - (-3))^2 + (3 - 3)^2} = \sqrt{5^2 + 0^2} = 5\\ &WZ = \sqrt{(-3 - (-3))^2 + (-2 - 3)^2} = \sqrt{0^2 + (-5)^2} = 5 \end{aligned}$$

\]

Calculate the lengths of the longer adjacent sides

\[

$$\begin{aligned} &X = (2, 3), \quad Y = (4, -4), \quad Z = (-3, -2)\\ &XY = \sqrt{(4 - 2)^2 + (-4 - 3)^2} = \sqrt{2^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53}\\ &ZY = \sqrt{(4 - (-3))^2 + (-4 - (-2))^2} = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \end{aligned}$$

\]

Sum the side lengths to find the perimeter

\[
\text{Perimeter} = WX + WZ + XY + ZY = 5 + 5 + \sqrt{53} + \sqrt{53} = 2\sqrt{53} + 10
\]
</reasoning>

<answer>
<mcq-option>(A) \(2\sqrt{53} + 2\) units</mcq-option>
<mcq-option>(B) \(2\sqrt{53} + 5\) units</mcq-option>
<mcq-correct>(C) \(2\sqrt{53} + 10\) units</mcq-correct>
<mcq-option>(D) \(2\sqrt{53} + 14\) units</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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"Perimeter of Polygons"
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