Question

10 consider a polygon with vertices $(-4,2)$, $(1,2)$, $(4,-2)$, $(-4,-2)$

part a
classify the figure using the most specific term: trapezoid

part b
what is the area of the figure?
area: 26 square units

part c
what is the perimeter of the figure?
perimeter: 22 units

Explanation:

Step1: Calculate the lengths of the parallel sides (bases)

The formula for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For the points \((-4,2)\) and \((1,2)\), since \(y_1 = y_2=2\), \(b_1=\vert-4 - 1\vert=5\). For the points \((-4,-2)\) and \((4,-2)\), since \(y_1 = y_2=-2\), \(b_2=\vert-4 - 4\vert=8\).

Step2: Calculate the height (h)

The height is the vertical distance between the two parallel sides. Using the \(y -\) values of the two parallel lines (\(y = 2\) and \(y=-2\)), \(h=\vert2-(-2)\vert = 4\).

Step3: Calculate the area (A) of the trapezoid

The formula for the area of a trapezoid is \(A=\frac{(b_1 + b_2)h}{2}\). Substitute \(b_1 = 5\), \(b_2=8\) and \(h = 4\) into the formula: \(A=\frac{(5 + 8)\times4}{2}=\frac{13\times4}{2}=26\).

Step4: Calculate the lengths of the non - parallel sides

For the side between \((1,2)\) and \((4,-2)\): \(d_1=\sqrt{(4 - 1)^2+(-2 - 2)^2}=\sqrt{9 + 16}=\sqrt{25}=5\). For the side between \((-4,2)\) and \((-4,-2)\): \(d_2=\vert2-(-2)\vert = 4\).

Step5: Calculate the perimeter (P)

The perimeter of a trapezoid is \(P=b_1 + b_2+d_1 + d_2\). Substitute \(b_1 = 5\), \(b_2=8\), \(d_1 = 5\) and \(d_2=4\) into the formula: \(P=5 + 8+5 + 4=22\).

Answer:

Area: \(26\) square units. Perimeter: \(22\) units.