Question

for #5-8, find the area of the polygon with the given vertices. 5. e(3, 1), f(3, -2), g(-2, -2)

Explanation:

Step1: Identify the polygon type

The vertices are \( E(3, 1) \), \( F(3, -2) \), \( G(-2, -2) \). Plotting these points, we see that \( EF \) is vertical (same x - coordinate), \( FG \) is horizontal (same y - coordinate), and \( EG \) is the hypotenuse. So, it's a right triangle.

Step2: Calculate the length of the vertical leg (\( EF \))

The length of a vertical line segment with endpoints \( (x,y_1) \) and \( (x,y_2) \) is \( |y_1 - y_2| \). For \( E(3,1) \) and \( F(3, - 2) \), \( EF=\vert1-(-2)\vert=\vert3\vert = 3 \).

Step3: Calculate the length of the horizontal leg (\( FG \))

The length of a horizontal line segment with endpoints \( (x_1,y) \) and \( (x_2,y) \) is \( |x_1 - x_2| \). For \( F(3,-2) \) and \( G(-2,-2) \), \( FG=\vert3-(-2)\vert=\vert5\vert = 5 \).

Step4: Calculate the area of the right triangle

The area of a right triangle is \( A=\frac{1}{2}\times\text{base}\times\text{height} \). Here, base \( = FG = 5 \) and height \( = EF = 3 \). So, \( A=\frac{1}{2}\times5\times3=\frac{15}{2}=7.5 \).

Answer:

The area of the polygon (right triangle) is \( 7.5 \) (or \( \frac{15}{2} \)).