Question

how do the areas of the parallelograms compare?

• the area of parallelogram abcd is 4 square units greater than the area of parallelogram efgh.
• the area of parallelogram abcd is 2 square units greater than the area of parallelogram efgh.
• the area of parallelogram abcd is equal to the area of parallelogram efgh.
• the area of parallelogram abcd is 2 square units less than the area of parallelogram efgh.

Explanation:

Identify coordinates and dimensions of ABCD

We find the coordinates of the vertices of parallelogram \(ABCD\) from the grid:

• \(A = (4, 2)\)
• \(B = (7, 2)\)
• \(C = (4, 6)\)
• \(D = (1, 6)\)

Using the Area of Polygons concept, we find the base and height:

• The base \(AB\) lies on the horizontal line \(y = 2\).
• Length of base \(b_1 = 7 - 4 = 3\) units.
• The height \(h_1\) is the vertical distance between the parallel lines \(y = 2\) and \(y = 6\).
• Height \(h_1 = 6 - 2 = 4\) units.

Calculate the area of ABCD

Using the formula for the area of a parallelogram:

$$ \text{Area}(ABCD) = b_1 \times h_1 $$

$$ \text{Area}(ABCD) = 3 \times 4 = 12\text{ square units} $$

Identify coordinates and dimensions of EFGH

We find the coordinates of the vertices of parallelogram \(EFGH\) from the grid:

• \(E = (-2, 2)\)
• \(F = (-5, 2)\)
• \(G = (-6, 6)\)
• \(H = (-3, 6)\)

Using the Area of Polygons concept, we find the base and height:

• The base \(FE\) lies on the horizontal line \(y = 2\).
• Length of base \(b_2 = -2 - (-5) = 3\) units.
• The height \(h_2\) is the vertical distance between the parallel lines \(y = 2\) and \(y = 6\).
• Height \(h_2 = 6 - 2 = 4\) units.

Calculate the area of EFGH

Using the formula for the area of a parallelogram:

$$ \text{Area}(EFGH) = b_2 \times h_2 $$

$$ \text{Area}(EFGH) = 3 \times 4 = 12\text{ square units} $$

Compare the areas

We compare the calculated areas:

• \(\text{Area}(ABCD) = 12\text{ square units}\)
• \(\text{Area}(EFGH) = 12\text{ square units}\)

Since both areas are equal, the areas of the two parallelograms are equal.

Answer:

• The area of parallelogram ABCD is 4 square units greater than the area of parallelogram EFGH.
• The area of parallelogram ABCD is 2 square units greater than the area of parallelogram EFGH.
• The area of parallelogram ABCD is equal to the area of parallelogram EFGH. (Correct answer)
• The area of parallelogram ABCD is 2 square units less than the area of parallelogram EFGH.