Question

After exploring the properties of quadrilaterals, Jackie concluded that quadrilateral ABCD formed by vertices A(2, 1), B(5, -1), C(3, -4), and D(0, -2) is a square.
Part A
Determine the slope of each side of the quadrilateral. What do the slopes indicate about the sides of the quadrilateral? Show your work.
Part B
Determine the length of each side of the quadrilateral. What do the lengths indicate about the sides of the quadrilateral? Show your work.
Part C
Was Jackie's conclusion correct? Explain your reasoning.

Answer 1

Jackie's conclusion about the quadrilateral is correct because the slopes are opposite reciprocals, and the side lengths are congruent

The slope of each side

The vertices are given as:

A(2, 1), B(5, -1), C(3, -4), and D(0, -2)

The slope is calculated as:

$m = \frac{y_2 - y_1}{x_2 -x_1}$

So, we have

$AB = \frac{-1 -1}{5-2}$

$AB = -\frac{2}{3}$

$BC = \frac{-4 + 1}{3 -5}$

$BC = \frac{3}{2}$

$CD = \frac{-2 +4}{0-3}$

$CD = -\frac{2}{3}$

$DA = \frac{1 + 2}{2 - 0}$

$DA = \frac{3}{2}$

The slope shows that the adjacent sides of the quadrilaterals are perpendicular to one another because the slopes are opposite reciprocals

The distance of each side

The distance is calculated as:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 -y_1)^2$

So, we have:

$AB = \sqrt{(2 - 5)^2 + (1 + 1)^2}$

$AB = \sqrt{13}$

$BC = \sqrt{(5 - 3)^2 + (-1 + 4)^2}$

$BC = \sqrt{13}$

$CD = \sqrt{(3 - 0)^2 + (-4 + 2)^2}$

$CD = \sqrt{13}$

$DA = \sqrt{(0 - 2)^2 + (-2 -1)^2}$

$DA = \sqrt{13}$

The lengths indicate that the side lengths of the quadrilaterals are congruent

The conclusion

Because the slopes are opposite reciprocals, and the side lengths are equal; then we can conclude that Jackie's conclusion is correct

Read more about quadrilaterals at:

https://brainly.com/question/16691874