Question

in exercises 13 - 16, find the area of the polygon with the given vertices. (see example 3.) 13. e(3,1),f(3, - 2),g( - 2, - 2) 14. j( - 3,4),k(4,4),l(3, - 3) 15. w(0,0),x(0,3),y( - 3,3),z( - 3,0) 16. n( - 2,1),p(3,1),q(3, - 1),r( - 2, - 1) in exercises 17 - 24, use the diagram. 17. find the perimeter of △cde. 18. find the perimeter of rectangle bcef. 19. find the perimeter of △abf. 20. find the perimeter of quadrilateral abcd. 21. find the area of △cde. 22. find the area of rectangle bcef. 23. find the area of △abf

Explanation:

Step1: Recall distance formula

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Find side - lengths of $\triangle CDE$

For side $CD$ with $C(4,-1)$ and $D(4,-5)$:
$x_1 = 4,y_1=-1,x_2 = 4,y_2=-5$. Then $CD=\sqrt{(4 - 4)^2+(-5+1)^2}=\sqrt{0 + 16}=4$.
For side $DE$ with $D(4,-5)$ and $E(2,-3)$:
$DE=\sqrt{(2 - 4)^2+(-3 + 5)^2}=\sqrt{(-2)^2+2^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$.
For side $CE$ with $C(4,-1)$ and $E(2,-3)$:
$CE=\sqrt{(2 - 4)^2+(-3 + 1)^2}=\sqrt{(-2)^2+(-2)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$.

Step3: Calculate perimeter of $\triangle CDE$

The perimeter $P$ of $\triangle CDE$ is $P=CD + DE+CE=4 + 2\sqrt{2}+2\sqrt{2}=4 + 4\sqrt{2}$.

Answer:

$4 + 4\sqrt{2}$