Question

what is the area of parallelogram abcd?

13 square units
14 square units
15 square units
16 square units

Answer:

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<reasoning>

Identify the coordinates of the vertices

Using the grid, we find the coordinates of the vertices of parallelogram \(ABCD\):

• \(A = (3, 6)\)
• \(B = (6, 5)\)
• \(C = (5, 1)\)
• \(D = (2, 2)\)

Calculate the area using bounding box subtraction

Using the Composite Area knowledge point:
We enclose the parallelogram in a bounding box with horizontal and vertical sides.

• The minimum \(x\)-coordinate is \(2\) (at \(D\)) and the maximum \(x\)-coordinate is \(6\) (at \(B\)). Thus, the width is \(6 - 2 = 4\).
• The minimum \(y\)-coordinate is \(1\) (at \(C\)) and the maximum \(y\)-coordinate is \(6\) (at \(A\)). Thus, the height is \(6 - 1 = 5\).
• The area of this bounding box is:

\[
\text{Area}_{\text{box}} = 4 \times 5 = 20
\]

Now, we subtract the areas of the four right triangles in the corners of the bounding box:

1. Top-left triangle (above \(D\) and left of \(A\)):

• Width = \(3 - 2 = 1\)
• Height = \(6 - 2 = 4\)
• \(\text{Area}_1 = \frac{1}{2} \times 1 \times 4 = 2\)

1. Top-right triangle (above \(B\) and right of \(A\)):

• Width = \(6 - 3 = 3\)
• Height = \(6 - 5 = 1\)
• \(\text{Area}_2 = \frac{1}{2} \times 3 \times 1 = 1.5\)

1. Bottom-right triangle (below \(B\) and right of \(C\)):

• Width = \(6 - 5 = 1\)
• Height = \(5 - 1 = 4\)
• \(\text{Area}_3 = \frac{1}{2} \times 1 \times 4 = 2\)

1. Bottom-left triangle (below \(D\) and left of \(C\)):

• Width = \(5 - 2 = 3\)
• Height = \(2 - 1 = 1\)
• \(\text{Area}_4 = \frac{1}{2} \times 3 \times 1 = 1.5\)

Subtracting these corner areas from the bounding box area:
\[
\text{Area} = 20 - (2 + 1.5 + 2 + 1.5) = 20 - 7 = 13
\]

Verify using the Shoelace Formula

Using the Parallelogram Area in Coordinate Plane knowledge point:
We apply the Shoelace Formula to the vertices listed in counterclockwise order: \(A(3,6)\), \(D(2,2)\), \(C(5,1)\), \(B(6,5)\).
\[

$$\begin{aligned} \text{Area} &= \frac{1}{2} |(3 \cdot 2 + 2 \cdot 1 + 5 \cdot 5 + 6 \cdot 6) - (6 \cdot 2 + 2 \cdot 5 + 1 \cdot 6 + 5 \cdot 3)| \\ &= \frac{1}{2} |(6 + 2 + 25 + 36) - (12 + 10 + 6 + 15)| \\ &= \frac{1}{2} |69 - 43| \\ &= \frac{1}{2} \times 26 = 13 \end{aligned}$$

\]
Both methods yield an area of \(13\) square units.
</reasoning>

<answer>
<mcq-correct>13 square units</mcq-correct>
<mcq-option>14 square units</mcq-option>
<mcq-option>15 square units</mcq-option>
<mcq-option>16 square units</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "polygon",
"params": [[3, 6], [6, 5], [5, 1], [2, 2]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.3)",
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "ABCD"
}
},
{
"type": "point",
"params": [[3, 6]],
"properties": {
"name": "A(3,6)",
"color": "#8C55F2",
"size": 4,
"withLabel": true
}
},
{
"type": "point",
"params": [[6, 5]],
"properties": {
"name": "B(6,5)",
"color": "#8C55F2",
"size": 4,
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 1]],
"properties": {
"name": "C(5,1)",
"color": "#8C55F2",
"size": 4,
"withLabel": true
}
},
{
"type": "point",
"params": [[2, 2]],
"properties": {
"name": "D(2,2)",
"color": "#8C55F2",
"size": 4,
"withLabel": true
}
}
]
}
</plot>

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"Parallelogram Area in Coordinate Plane"
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