Question

essential question how can you find the perimeter and area of a polygon in a coordinate plane? exploration 1 finding the perimeter and area of a quadrilateral work with a partner. a. on a piece of centimeter graph paper, draw quadrilateral abcd in a coordinate plane. label the points a(1, 4), b(-3, 1), c(0, -3), and d(4, 0). b. find the perimeter of quadrilateral abcd.

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 $A(1,4)$ and $B(-3,1)$, $x_1 = 1,y_1 = 4,x_2=-3,y_2 = 1$. Then $AB=\sqrt{(-3 - 1)^2+(1 - 4)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$.

Step3: Calculate length of side BC

For points $B(-3,1)$ and $C(0,-3)$, $x_1=-3,y_1 = 1,x_2 = 0,y_2=-3$. Then $BC=\sqrt{(0+3)^2+(-3 - 1)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step4: Calculate length of side CD

For points $C(0,-3)$ and $D(4,0)$, $x_1 = 0,y_1=-3,x_2 = 4,y_2 = 0$. Then $CD=\sqrt{(4 - 0)^2+(0 + 3)^2}=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5$.

Step5: Calculate length of side DA

For points $D(4,0)$ and $A(1,4)$, $x_1 = 4,y_1 = 0,x_2 = 1,y_2 = 4$. Then $DA=\sqrt{(1 - 4)^2+(4 - 0)^2}=\sqrt{(-3)^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step6: Calculate perimeter

The perimeter $P$ of quadrilateral $ABCD$ is $P=AB + BC+CD + DA=5 + 5+5 + 5 = 20$.

Answer:

20