Answer:
D. 16.2 units.
Step-by-step explanation:
We have been given an image of a kite on coordinate plane and we are asked to find the perimeter of our given kite.
Since we know that a kite has two disjoint pairs of congruent consecutive sides, so the perimeter of our given kite will be 2 times the sum of side XY and YZ.
To find the length of sides XY and YZ we will use distance formula.
\(\text{Distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Upon substituting coordinates of point X and Y in above formula we will get,
\(\text{Distance between point X and Y}=\sqrt{(3-5)^2+(4-1)^2}\)
\(\text{Distance between point X and Y}=\sqrt{(-2)^2+(3)^2}\)
\(\text{Distance between point X and Y}=\sqrt{4+9}\)
\(\text{Distance between point X and Y}=\sqrt{13}\)
Now similarly we will find the length of segment YZ.
\(\text{Distance between point Y and Z}=\sqrt{(5-3)^2+(1--3)^2}\)
\(\text{Distance between point Y and Z}=\sqrt{(2)^2+(1+3)^2}\)
\(\text{Distance between point Y and Z}=\sqrt{4+(4)^2}\)
\(\text{Distance between point Y and Z}=\sqrt{4+16}\)
\(\text{Distance between point Y and Z}=\sqrt{20}\)
\(\text{Distance between point Y and Z}=2\sqrt{5}\)
\(\text{Perimeter of kite WXYZ}=2(XY+YZ)\)
\(\text{Perimeter of kite WXYZ}=2(\sqrt{13}+2\sqrt{5})\)
\(\text{Perimeter of kite WXYZ}=2(3.6055512754639893+4.4721359549995794)\)
\(\text{Perimeter of kite WXYZ}=2(8.0776872304635687)\)
\(\text{Perimeter of kite WXYZ}=16.1553744609271374\approx 16.2\)
Therefore, the perimeter of the kite WXYZ is 16.2 units and option D is the correct choice.
