Question

find the perimeter of a quadrilateral with vertices at c (2, 4), d (1, 1), e (5, 0), and f (3, 4). round your answer to the nearest hundredth when necessary.

12.76 units
16.25 units
20.12 units
24.85 units

Explanation:

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<reasoning>

Calculate the lengths of the four sides using the distance formula

\[

$$\begin{aligned} CD &= \sqrt{(1-2)^2 + (1-4)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.162 \\ DE &= \sqrt{(5-1)^2 + (0-1)^2} = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123 \\ EF &= \sqrt{(3-5)^2 + (4-0)^2} = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.472 \\ FC &= \sqrt{(2-3)^2 + (4-4)^2} = \sqrt{(-1)^2 + 0^2} = \sqrt{1} = 1 \end{aligned}$$

\]

Sum the side lengths to find the perimeter

\[

$$\begin{aligned} \text{Perimeter} &= CD + DE + EF + FC \\ &= \sqrt{10} + \sqrt{17} + \sqrt{20} + 1 \\ &\approx 3.162 + 4.123 + 4.472 + 1 \\ &= 12.757 \approx 12.76 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) 12.76 units</mcq-correct>
<mcq-option>(B) 16.25 units</mcq-option>
<mcq-option>(C) 20.12 units</mcq-option>
<mcq-option>(D) 24.85 units</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[2, 4]],
"properties": {
"name": "C",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[1, 1]],
"properties": {
"name": "D",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 0]],
"properties": {
"name": "E",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[3, 4]],
"properties": {
"name": "F",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "polygon",
"params": [[2, 4], [1, 1], [5, 0], [3, 4]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.15)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
]
}
</plot>

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Answer:

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<reasoning>

Calculate the lengths of the four sides using the distance formula

\[

$$\begin{aligned} CD &= \sqrt{(1-2)^2 + (1-4)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.162 \\ DE &= \sqrt{(5-1)^2 + (0-1)^2} = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123 \\ EF &= \sqrt{(3-5)^2 + (4-0)^2} = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.472 \\ FC &= \sqrt{(2-3)^2 + (4-4)^2} = \sqrt{(-1)^2 + 0^2} = \sqrt{1} = 1 \end{aligned}$$

\]

Sum the side lengths to find the perimeter

\[

$$\begin{aligned} \text{Perimeter} &= CD + DE + EF + FC \\ &= \sqrt{10} + \sqrt{17} + \sqrt{20} + 1 \\ &\approx 3.162 + 4.123 + 4.472 + 1 \\ &= 12.757 \approx 12.76 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) 12.76 units</mcq-correct>
<mcq-option>(B) 16.25 units</mcq-option>
<mcq-option>(C) 20.12 units</mcq-option>
<mcq-option>(D) 24.85 units</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[2, 4]],
"properties": {
"name": "C",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[1, 1]],
"properties": {
"name": "D",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 0]],
"properties": {
"name": "E",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[3, 4]],
"properties": {
"name": "F",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "polygon",
"params": [[2, 4], [1, 1], [5, 0], [3, 4]],
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"strokeWidth": 2
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</plot>

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