Here the figure is made up of a quadrilateral and a semi circle.
ABCD is the quadrilateral here. We will find the sides of the quadrilateral by using the distance formula.
If (x₁, y₁) and (x₂, y₂) are two points given, then the distance between two points by using distance formula is,
[tex] d =\sqrt({ x_{1}-x_{2})^2 +( y_{1} -y_{2} )^2} [/tex]
The co-ordinate of A is (-1,2) and co-ordinate of B is (-2,-1).
So the length of side AB = [tex] \sqrt{(-1-(-2))^2+(2-(-1))^2} [/tex]
=[tex] \sqrt{(-1+2)^2+(2+1)^2} [/tex](As negative times negative is positive)
= [tex] \sqrt{(1)^2+(3)^2} =\sqrt{1+9} =\sqrt{10} [/tex]
The co-ordinate of C is (4,-3) and D is (5,0)
The length of side CD
= [tex] \sqrt(4-5)^2+(-3-0)^2} [/tex]
= [tex] \sqrt{(-1)^2+(-3)^2} [/tex]
= [tex] \sqrt{1+9} =\sqrt{10} [/tex]
So the sides AB and CD are equal.
The length of side AD
= [tex] \sqrt{(-1-5)^2+(2-0)^2} [/tex]
= [tex] \sqrt{(-6)^2+(2)^2} =\sqrt{36+4} =\sqrt{40} [/tex]
The length of side BC
= [tex] \sqrt{(-2-4)^2+(-1-(-3))^2} [/tex]
= [tex] \sqrt{(-2-4)^2+(-1+3)^2} [/tex]
= [tex] \sqrt{(-6)^2+(2)^2}=\sqrt{36+4} =\sqrt{40} [/tex]
So the lengths of the sides AD and BC are equal.
So the quadrilateral is a rectangle whose length is [tex] \sqrt{40} [/tex] and width is [tex] \sqrt{10} [/tex].
Area of a rectangle = length × width
= [tex] (\sqrt{40}) (\sqrt{10}) [/tex]
= [tex] \sqrt{(40)(10)}=\sqrt{400} [/tex]
= [tex] 20 unit^2 [/tex]
Now the diameter of the semicircle is the side AD = [tex] \sqrt{40} [/tex]
So, the radius of the semi-circle = [tex] \frac{\sqrt{40}}{2} [/tex]
= [tex] \frac{\sqrt{(4)(10)}}{2} [/tex]
= [tex] \frac{(\sqrt{4})(\sqrt{10})}{2} [/tex]
= [tex] \frac{2\sqrt{10}}{2} [/tex] = [tex] \sqrt{10} [/tex]
Area of semi-circle = [tex] \frac{1}{2} \pi r^2 [/tex], where r is the radius.
= [tex] \frac{1}{2} \pi (\sqrt{10})^2 [/tex]
= [tex] \frac{1}{2} \pi (10) [/tex]
= [tex] \frac{(\pi)(10)}{2} [/tex]
= [tex] \frac{10\pi}{2} = 5\pi [/tex] = [tex] 15.7 unit^2 [/tex] ( Approximately taken to the nearest tenth)
Total area of the figure = [tex] (20+15.7) unit^2 [/tex] = [tex] 35.7 unit^2 [/tex]
We have got the required answer.
Option a is correct here.
