Question

1. a) on a grid, draw the quadrilateral with each set of vertices. for each quadrilateral, determine the lengths of its sides and its perimeter. i) a(3, 8), b(5, 5), c(-1, 1), d(-3, 4) b) choose one triangle from part a. write to explain how you classified it

Explanation:

Step1: Recall distance formula

The distance formula 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: Calculate length of side AB

For points $A(3,8)$ and $B(5,5)$, $x_1 = 3,y_1 = 8,x_2 = 5,y_2 = 5$.
$AB=\sqrt{(5 - 3)^2+(5 - 8)^2}=\sqrt{2^2+( - 3)^2}=\sqrt{4 + 9}=\sqrt{13}$

Step3: Calculate length of side BC

For points $B(5,5)$ and $C(-1,1)$, $x_1 = 5,y_1 = 5,x_2=-1,y_2 = 1$.
$BC=\sqrt{(-1 - 5)^2+(1 - 5)^2}=\sqrt{(-6)^2+( - 4)^2}=\sqrt{36+16}=\sqrt{52}=2\sqrt{13}$

Step4: Calculate length of side CD

For points $C(-1,1)$ and $D(-3,4)$, $x_1=-1,y_1 = 1,x_2=-3,y_2 = 4$.
$CD=\sqrt{(-3+1)^2+(4 - 1)^2}=\sqrt{(-2)^2+3^2}=\sqrt{4 + 9}=\sqrt{13}$

Step5: Calculate length of side DA

For points $D(-3,4)$ and $A(3,8)$, $x_1=-3,y_1 = 4,x_2 = 3,y_2 = 8$.
$DA=\sqrt{(3 + 3)^2+(8 - 4)^2}=\sqrt{6^2+4^2}=\sqrt{36 + 16}=\sqrt{52}=2\sqrt{13}$

Step6: Calculate the perimeter

Perimeter $P=AB + BC+CD + DA=\sqrt{13}+2\sqrt{13}+\sqrt{13}+2\sqrt{13}=6\sqrt{13}$

Answer:

The lengths of the sides are $AB=\sqrt{13}$, $BC = 2\sqrt{13}$, $CD=\sqrt{13}$, $DA=2\sqrt{13}$ and the perimeter is $6\sqrt{13}$