Question

Find the area of the parallelogram in the coordinate plane.

A (7,5)
D(-9,-2)
Units
B(6,5)
C(4-2)

Answer 1

Check the picture below.

`\textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h~~=height\\ a,b=\stackrel{parallel~sides}{bases~\hfill }\\[-0.5em] \hrulefill\\ a=1\\ b=13\\ h=7 \end{cases}\implies A=\cfrac{7(1+13)}{2}\implies A=49`

Answer 2

To find the area of a parallelogram in the coordinate plane, we need to determine the base and the height of the parallelogram.

Using the given coordinates, we can find the length of one side of the parallelogram as the distance between points A and B.

The length of AB = sqrt((6 - 7)^2 + (5 - 5)^2) = sqrt((-1)^2 + 0^2) = sqrt(1) = 1

The height of the parallelogram can be found as the distance between point D and the line passing through points A and B. We can use the formula for the distance between a point and a line to find the perpendicular distance.

The equation of the line passing through A and B can be found using the point-slope form:

y - 5 = (5 - 5)/(7 - 6) * (x - 7)

y - 5 = 0 * (x - 7)

y - 5 = 0

y = 5

The perpendicular distance from point D(-9, -2) to the line y = 5 is the difference in their y-coordinates:

Perpendicular distance = |-2 - 5| = 7

Now, we have the base length AB = 1 and the height = 7.

The area of the parallelogram is given by the formula: Area = base * height.

Area = 1 * 7 = 7 square units.