Question

use the diagram to find the perimeter and the area of △cde. round your answers to the nearest hundredth. the perimeter is about units. the area is square units.

Explanation:

Step1: Calculate the lengths of the sides using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$

For side $CD$ where $C(4,-1)$ and $D(4,-5)$:
$d_{CD}=\sqrt{(4 - 4)^2+(-5+1)^2}=\sqrt{0 + 16}=4$
For side $DE$ where $D(4,-5)$ and $E(2,3)$:
$d_{DE}=\sqrt{(2 - 4)^2+(3 + 5)^2}=\sqrt{(-2)^2+8^2}=\sqrt{4 + 64}=\sqrt{68}\approx 8.25$
For side $CE$ where $C(4,-1)$ and $E(2,3)$:
$d_{CE}=\sqrt{(2 - 4)^2+(3 + 1)^2}=\sqrt{(-2)^2+4^2}=\sqrt{4 + 16}=\sqrt{20}\approx 4.47$

Step2: Calculate the perimeter

$P=d_{CD}+d_{DE}+d_{CE}=4 + 8.25+4.47 = 16.72$

Step3: Calculate the area using Heron's formula $A=\sqrt{s(s - a)(s - b)(s - c)}$ where $s=\frac{a + b + c}{2}$ and $a = 4$, $b\approx8.25$, $c\approx4.47$

$s=\frac{4 + 8.25+4.47}{2}=\frac{16.72}{2}=8.36$
$A=\sqrt{8.36(8.36 - 4)(8.36 - 8.25)(8.36 - 4.47)}=\sqrt{8.36\times4.36\times0.11\times3.89}\approx\sqrt{15.67}\approx 3.96$

Answer:

The area is about $3.96$ square units.
The perimeter is about $16.72$ units.