Question

in the diagram, \\(zy = wx = 2\sqrt{10}\\). what is the perimeter of rectangle wxyz?

\\(\sqrt{10}\\) units
\\(2\sqrt{10}\\) units
\\(6\sqrt{10}\\) units
\\(8\sqrt{10}\\) units

Explanation:

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<reasoning>

Identify given dimensions

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &\text{Rectangle } WXYZ \text{ has opposite sides of equal length.}\\ &ZY = WX = 2\sqrt{10} \end{aligned}$$

\]

Calculate remaining side lengths

Using the Pythagorean Theorem and Perimeter in Coordinate Plane knowledge points
\[

$$\begin{aligned} &W = (-6, 6),\quad Z = (-9, 5)\\ &WZ = \sqrt{(-6 - (-9))^2 + (6 - 5)^2} = \sqrt{3^2 + 1^2} = \sqrt{10}\\ &XY = WZ = \sqrt{10} \end{aligned}$$

\]

Compute the total perimeter

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} P &= WX + XY + YZ + ZW\\ P &= 2\sqrt{10} + \sqrt{10} + 2\sqrt{10} + \sqrt{10}\\ P &= 6\sqrt{10} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt{10}\) units</mcq-option>
<mcq-option>(B) \(2\sqrt{10}\) units</mcq-option>
<mcq-correct>(C) \(6\sqrt{10}\) units</mcq-correct>
<mcq-option>(D) \(8\sqrt{10}\) units</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify given dimensions

Using the Properties of Rectangles knowledge point
\[

$$\begin{aligned} &\text{Rectangle } WXYZ \text{ has opposite sides of equal length.}\\ &ZY = WX = 2\sqrt{10} \end{aligned}$$

\]

Calculate remaining side lengths

Using the Pythagorean Theorem and Perimeter in Coordinate Plane knowledge points
\[

$$\begin{aligned} &W = (-6, 6),\quad Z = (-9, 5)\\ &WZ = \sqrt{(-6 - (-9))^2 + (6 - 5)^2} = \sqrt{3^2 + 1^2} = \sqrt{10}\\ &XY = WZ = \sqrt{10} \end{aligned}$$

\]

Compute the total perimeter

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} P &= WX + XY + YZ + ZW\\ P &= 2\sqrt{10} + \sqrt{10} + 2\sqrt{10} + \sqrt{10}\\ P &= 6\sqrt{10} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt{10}\) units</mcq-option>
<mcq-option>(B) \(2\sqrt{10}\) units</mcq-option>
<mcq-correct>(C) \(6\sqrt{10}\) units</mcq-correct>
<mcq-option>(D) \(8\sqrt{10}\) units</mcq-option>
</answer>

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