Question

in the diagram, \\(kl = 2\sqrt{2}\\), \\(lm = \sqrt{5}\\), and \\(mn = \sqrt{2}\\). what is the perimeter of isosceles trapezoid klmn?

\\(\sqrt{2}\\) units

\\(\sqrt{5}\\) units

\\(3\sqrt{2} + 2\sqrt{5}\\) units

\\(4\sqrt{2} + 2\sqrt{5}\\) units

Explanation:

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<reasoning>

Identify the given side lengths

Using the Isosceles Trapezoid Definition knowledge point
We are given three side lengths of the trapezoid \(KLMN\):

• \(KL = 2\sqrt{2}\)
• \(LM = \sqrt{5}\)
• \(MN = \sqrt{2}\)

Determine the missing side length

Using the Isosceles Trapezoid Definition knowledge point
An isosceles trapezoid has two non-parallel sides (legs) that are equal in length.

• The parallel bases are \(KL\) and \(MN\) because their slopes are equal:

\[
\text{Slope of } KL = \frac{-4 - (-2)}{-2 - (-4)} = \frac{-2}{2} = -1
\]
\[
\text{Slope of } MN = \frac{-2 - (-1)}{-1 - (-2)} = \frac{-1}{1} = -1
\]

• Since \(KL \parallel MN\), the legs are \(LM\) and \(NK\).
• In an isosceles trapezoid, the legs are congruent:

\[
NK = LM = \sqrt{5}
\]

Calculate the perimeter

Using the Perimeter in Coordinate Plane knowledge point
The perimeter is the sum of all four side lengths:
\[
\text{Perimeter} = KL + LM + MN + NK
\]
\[
\text{Perimeter} = 2\sqrt{2} + \sqrt{5} + \sqrt{2} + \sqrt{5}
\]
\[
\text{Perimeter} = (2\sqrt{2} + \sqrt{2}) + (\sqrt{5} + \sqrt{5}) = 3\sqrt{2} + 2\sqrt{5}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt{2}\) units</mcq-option>
<mcq-option>(B) \(\sqrt{5}\) units</mcq-option>
<mcq-correct>(C) \(3\sqrt{2} + 2\sqrt{5}\) units</mcq-correct>
<mcq-option>(D) \(4\sqrt{2} + 2\sqrt{5}\) units</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given side lengths

Using the Isosceles Trapezoid Definition knowledge point
We are given three side lengths of the trapezoid \(KLMN\):

• \(KL = 2\sqrt{2}\)
• \(LM = \sqrt{5}\)
• \(MN = \sqrt{2}\)

Determine the missing side length

Using the Isosceles Trapezoid Definition knowledge point
An isosceles trapezoid has two non-parallel sides (legs) that are equal in length.

• The parallel bases are \(KL\) and \(MN\) because their slopes are equal:

\[
\text{Slope of } KL = \frac{-4 - (-2)}{-2 - (-4)} = \frac{-2}{2} = -1
\]
\[
\text{Slope of } MN = \frac{-2 - (-1)}{-1 - (-2)} = \frac{-1}{1} = -1
\]

• Since \(KL \parallel MN\), the legs are \(LM\) and \(NK\).
• In an isosceles trapezoid, the legs are congruent:

\[
NK = LM = \sqrt{5}
\]

Calculate the perimeter

Using the Perimeter in Coordinate Plane knowledge point
The perimeter is the sum of all four side lengths:
\[
\text{Perimeter} = KL + LM + MN + NK
\]
\[
\text{Perimeter} = 2\sqrt{2} + \sqrt{5} + \sqrt{2} + \sqrt{5}
\]
\[
\text{Perimeter} = (2\sqrt{2} + \sqrt{2}) + (\sqrt{5} + \sqrt{5}) = 3\sqrt{2} + 2\sqrt{5}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt{2}\) units</mcq-option>
<mcq-option>(B) \(\sqrt{5}\) units</mcq-option>
<mcq-correct>(C) \(3\sqrt{2} + 2\sqrt{5}\) units</mcq-correct>
<mcq-option>(D) \(4\sqrt{2} + 2\sqrt{5}\) units</mcq-option>
</answer>

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