Answer: 14+8.125π units²
Step-by-step explanation:
By the given diagram,
The diameter of the semicircle = The line segment having the end points (-4,-2) and (3,2),
[tex]=\sqrt{(3+4)^2+(2+2)^2}[/tex]
[tex]=\sqrt{7^2+4^2}[/tex]
[tex]=\sqrt{49+16}[/tex]
[tex]=\sqrt{65}[/tex] unit,
Thus, the radius of the semicircle = √65/2 unit,
⇒ [tex]\text{Area of the semicircle}=\frac{1}{2}\pi(\frac{\sqrt{65}}{2})^2[/tex]
[tex]= \frac{1}{2}\pi(\frac{65}{4})[/tex]
[tex]=\frac{65\pi}{8}[/tex] square unit.
Now, by the given diagram,
The area of the triangle having the vertices (-4,-2), (3,2) and (-4,2) ( By the coordinate form of area of a triangle formula )
[tex]=\frac{1}{2}[-4(2-2)+3(2+2)-4(-2-2)][/tex]
[tex]=\frac{1}{2}\times 28[/tex]
[tex]=14[/tex] square unit,
Hence, the total area of the given figure = Area of semicircle having diameter √65 + area of triangle having vertices (-4,-2), (3,2) and (-4,2)
[tex]=\frac{65\pi}{8}+14[/tex]
[tex]=(8.125\pi+14)[/tex] square unit.
⇒ Fourth option is correct.
