Question

michael is finding the area of parallelogram abcd. to do this, he follows the steps in the table.

\

$$\begin{tabular}{|l|l|} \\hline step 1 & draw a rectangle around parallelogram abcd. \\\\ \\hline step 2 & find the area of the rectangle. \\\\ \\hline step 3 & find the area of the four right triangles created in the corners of the rectangle. \\\\ \\hline step 4 & subtract the area of the right triangles from the area of the rectangle. \\\\ \\hline \\end{tabular}$$

to solve the problem using these steps, what are the dimensions of the rectangle he should draw?

• 2 units by 4 units
• 3 units by 2 units
• 4 units by 4 units

Explanation:

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<reasoning>

Identify the coordinates of the vertices

Using the Area of Polygons knowledge point
\[

$$\begin{aligned} &A = (1, -1)\\ &B = (3, 2)\\ &C = (1, 3)\\ &D = (-1, 0) \end{aligned}$$

\]

Determine the bounding box dimensions

Using the Composite Area knowledge point
\[

$$\begin{aligned} &x_{\min} = -1,\quad x_{\max} = 3 \implies \text{Width} = 3 - (-1) = 4\\ &y_{\min} = -1,\quad y_{\max} = 3 \implies \text{Height} = 3 - (-1) = 4 \end{aligned}$$

\]

State the dimensions of the rectangle

Using the Composite Area knowledge point
\[
\text{Dimensions} = 4\text{ units by } 4\text{ units}
\]
</reasoning>

<answer>
<mcq-option>2 units by 4 units</mcq-option>
<mcq-option>3 units by 2 units</mcq-option>
<mcq-correct>4 units by 4 units</mcq-correct>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[1, -1]],
"properties": {
"name": "A",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "right"
}
},
{
"type": "point",
"params": [[3, 2]],
"properties": {
"name": "B",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "right"
}
},
{
"type": "point",
"params": [[1, 3]],
"properties": {
"name": "C",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "right"
}
},
{
"type": "point",
"params": [[-1, 0]],
"properties": {
"name": "D",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "left"
}
},
{
"type": "polygon",
"params": [[1, -1], [3, 2], [1, 3], [-1, 0]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.2)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "polygon",
"params": [[-1, -1], [3, -1], [3, 3], [-1, 3]],
"properties": {
"fillColor": "rgba(85, 131, 242, 0.1)",
"strokeColor": "#5583F2",
"strokeWidth": 1.5,
"dash": 2
}
}
]
}
</plot>

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Answer:

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<reasoning>

Identify the coordinates of the vertices

Using the Area of Polygons knowledge point
\[

$$\begin{aligned} &A = (1, -1)\\ &B = (3, 2)\\ &C = (1, 3)\\ &D = (-1, 0) \end{aligned}$$

\]

Determine the bounding box dimensions

Using the Composite Area knowledge point
\[

$$\begin{aligned} &x_{\min} = -1,\quad x_{\max} = 3 \implies \text{Width} = 3 - (-1) = 4\\ &y_{\min} = -1,\quad y_{\max} = 3 \implies \text{Height} = 3 - (-1) = 4 \end{aligned}$$

\]

State the dimensions of the rectangle

Using the Composite Area knowledge point
\[
\text{Dimensions} = 4\text{ units by } 4\text{ units}
\]
</reasoning>

<answer>
<mcq-option>2 units by 4 units</mcq-option>
<mcq-option>3 units by 2 units</mcq-option>
<mcq-correct>4 units by 4 units</mcq-correct>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[1, -1]],
"properties": {
"name": "A",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "right"
}
},
{
"type": "point",
"params": [[3, 2]],
"properties": {
"name": "B",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "right"
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},
{
"type": "point",
"params": [[1, 3]],
"properties": {
"name": "C",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "right"
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},
{
"type": "point",
"params": [[-1, 0]],
"properties": {
"name": "D",
"size": 4,
"color": "#8C55F2",
"withLabel": true,
"labelPosition": "left"
}
},
{
"type": "polygon",
"params": [[1, -1], [3, 2], [1, 3], [-1, 0]],
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"fillColor": "rgba(140, 85, 242, 0.2)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
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"type": "polygon",
"params": [[-1, -1], [3, -1], [3, 3], [-1, 3]],
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"fillColor": "rgba(85, 131, 242, 0.1)",
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"dash": 2
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}
</plot>

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