Question

what is the area of the parallelogram? (-5,2) (-1,2) (-6,-4) (-2,-4)

Explanation:

Step1: Calculate the base length

The base of the parallelogram can be found using the distance formula for two points on the same horizontal line (since \(y\)-coordinates are the same for \((-5,2)\) and \((-1,2)\)). The formula for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) when \(y_1 = y_2\) is \(d=\vert x_2 - x_1\vert\).
Here, \(x_1=-5\), \(x_2 = - 1\), so \(b=\vert-1-(-5)\vert=\vert-1 + 5\vert=4\).

Step2: Calculate the height

The height of the parallelogram can be found using the distance formula for two points on the vertical line (using the \(y\)-coordinates of \((-5,2)\) and \((-6,-4)\)). The formula for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) when \(x_1=x_2\) (not the case here, but we can use the \(y\)-values for height as the base is horizontal). The height \(h\) is the vertical distance. If we take a point on the base \((x,y) = (-5,2)\) and a corresponding point on the opposite side \((x',y')=(-6,-4)\), the vertical distance (height) is calculated as \(h=\vert2-(-4)\vert=\vert2 + 4\vert=6\).

Step3: Use the area formula for a parallelogram

The area formula for a parallelogram is \(A = b\times h\).
Substitute \(b = 4\) and \(h=6\) into the formula: \(A=4\times6\).

Answer:

\(24\)