Answer:
[tex](14+8.125 \pi)\ units^{2}[/tex]
Step-by-step explanation:
we know that
The area of the figure is equal to the area of a triangle plus the area of semicircle
Step 1
Find the area of triangle
The area of triangle is
[tex]A=bh/2[/tex]
we have
[tex]b=(3-(-4))=7\ units[/tex]
[tex]h=(2-(-2))=4\ units[/tex]
substitute
[tex]A=(7*4)/2=14\ units^{2}[/tex]
Step 2
The area of semicircle is equal to
[tex]A=\frac{1}{2} \pi r^{2}[/tex]
Find the radius
The radius is the half distance between points [tex](-4,-2)[/tex] and [tex](3,2)[/tex]
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
substitute the values
[tex]d=\sqrt{(2+2)^{2}+(3+4)^{2}}[/tex]
[tex]d=\sqrt{(4)^{2}+(7)^{2}}[/tex]
[tex]d=\sqrt{65}\ units[/tex] -------> is the diameter
the radius is equal to
[tex]r=(\frac{1}{2})\sqrt{65}\ units[/tex]
Find the area of semicircle
[tex]A=\frac{1}{2} \pi ((\frac{1}{2})\sqrt{65})^{2}[/tex]
[tex]A=\frac{65}{8} \pi\ units^{2}[/tex]
[tex]A=8.125 \pi\ units^{2}[/tex]
Step 3
Find the area pf the figure
[tex]14\ units^{2}+8.125 \pi\ units^{2}[/tex]
[tex](14+8.125 \pi)\ units^{2}[/tex]
