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. c. are adjacent sides of quadrilateral abcd perpendicular to each other? how can you tell? d. what is the definition of a square? is quadrilateral abcd a square? justify your answer. find the area of quadrilateral abcd.

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: Calculate length of $AB$

For $A(1,4)$ and $B(-3,1)$, $d_{AB}=\sqrt{(-3 - 1)^2+(1 - 4)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$.

Step3: Calculate length of $BC$

For $B(-3,1)$ and $C(0,-3)$, $d_{BC}=\sqrt{(0+3)^2+(-3 - 1)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step4: Calculate length of $CD$

For $C(0,-3)$ and $D(4,0)$, $d_{CD}=\sqrt{(4 - 0)^2+(0 + 3)^2}=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5$.

Step5: Calculate length of $DA$

For $D(4,0)$ and $A(1,4)$, $d_{DA}=\sqrt{(1 - 4)^2+(4 - 0)^2}=\sqrt{(-3)^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step6: Find perimeter

Perimeter $P=d_{AB}+d_{BC}+d_{CD}+d_{DA}=5 + 5+5+5 = 20$.

Step7: Recall slope formula

The slope of the line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step8: Calculate slope of $AB$

$m_{AB}=\frac{1 - 4}{-3 - 1}=\frac{-3}{-4}=\frac{3}{4}$.

Step9: Calculate slope of $BC$

$m_{BC}=\frac{-3 - 1}{0+3}=\frac{-4}{3}$.

Step10: Check perpendicularity

Since $m_{AB}\times m_{BC}=\frac{3}{4}\times\frac{-4}{3}=- 1$, $AB\perp BC$. Similarly, we can check other - adjacent sides. All adjacent sides are perpendicular as the product of slopes of adjacent sides is - 1.

Step11: Recall square definition

A square is a quadrilateral with four equal - length sides and four right angles.

Step12: Determine if it's a square

Since all sides are equal ($AB = BC=CD=DA = 5$) and adjacent sides are perpendicular, $ABCD$ is a square.

Step13: Calculate area

For a square with side length $s$, area $A=s^2$. Here $s = 5$, so $A = 25$.

Answer:

b. Perimeter: 20
c. Yes, by checking the product of slopes of adjacent sides (product is - 1 for all adjacent sides).
d. A square is a quadrilateral with four equal - length sides and four right angles. Yes, it is a square because all sides are equal and adjacent sides are perpendicular. Area: 25