Answer:
This area can be seen as a semicircle plus a triangle. So, we will find the area of each figure and then add both to get the total area.
The area of a triangle is given by the following formula:
[tex]A_{triangle}=\frac{1}{2}b.h[/tex] (1)
Where [tex]b[/tex] is the base and [tex]h[/tex] is the height.
[tex]A_{triangle}=\frac{1}{2}(3)(7)[/tex]
[tex]A_{triangle}=10.5 units[/tex] (2)
Now we are going to find the area of the cemicircle, which is the half of the area of a circle:
[tex]A_{cemicircle}=\frac{1}{2}\pi r^{2}[/tex] (3)
Where [tex]r[/tex] is the radius, in order to find it we have to calculate the diameter of this semicircle first, and we will do it as follows:
We know the points of the ends of the diameter, which are:
[tex]P_{1}:(-5,-2)[/tex] and [tex]P_{2}:(2,1)[/tex]
We have to use the Pithagorean theorem to calculate the distance between both points (taking into account the x-component and the y-component of each one)
[tex]c^{2} =a^{2}+ b^{2}[/tex]
[tex]c=\sqrt{a^{2}+b^{2}}[/tex]
[tex]c=\sqrt{(-5-2)^{2}+(-2-1)^{2}}[/tex]
[tex]c=\sqrt{58}[/tex]>>>>This is the diameter of the semicircle
Then, the radius is:
[tex]r=\frac{c}{2}=\frac{\sqrt{58}}{2}[/tex]
Now we can use the formula written in equation (3):
[tex]A_{cemicircle}=\frac{1}{2}\pi (\frac{\sqrt{58}}{2})^{2}[/tex]
[tex]A_{cemicircle}=7.25\pi units[/tex] (4)
Adding (2) and (4):
[tex]A_{triangle}+A_{cemicircle}=10.5 units+7.25\pi units[/tex]
[tex]A_{triangle}+A_{cemicircle}=10.5+7.25\pi units^{2}[/tex]>>>>This is the answer
