Question

what is the area of parallelogram abcd?

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

Explanation:

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: \(A(3,6)\), \(D(2,2)\), \(C(5,1)\), \(B(6,5)\).

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Both methods yield an area of \(13\) square units.

Answer:

• 13 square units (Correct answer)
• 14 square units
• 15 square units
• 16 square units