PARKING Stephanie is drawing the layout of a parking lot on a coordinate grid and needs to verify that two

sides meet in a right angle. In her sketch, one side of the parking lot has endpoints A(0,0) and B(50,0). The

second side of the parking lot has endpoints B(50,0) and C(18, 24). How can Stephanie prove that the two

sides of the parking lot meet in a right angle? Select all that apply.

® Find the slopes of AC and CB. If the product of the slopes is -1, then the sides meet in a right angle.

B Find the slopes of AB and AC. If the product of the slopes is -1, then the sides meet in a right angle.

© Find the slopes of CB and AB. If the slope of AB is the opposite of the slope of AC, then the sides meet in

a right angle.

Find AB, BC, and AC. If AB + BC = AC, then the sides meet in a right angle.

Find AB, BC, and AC. IfAB2+ BC = AC?, then the sides meet in a right angle.

From Analytical Geometry, two lines are perpendicular to each other if and only if [tex]m_{1}\cdot m_{2} = -1[/tex], where [tex]m_{1}[/tex] and [tex]m_{2}[/tex] are the slopes of each line, respectively. According to this, the following expression must be observed:

[tex]m_{AB} \cdot m_{BC} = -1[/tex] (1)

Each slope can be determine by definition of the slope of a Secant Line:

[tex]m_{AB} = \frac{y_{B}-y_{A}}{x_{B}-x_{A}}[/tex] (2)

[tex]m_{BC} = \frac{y_{C}-y_{B}}{x_{C}-x_{B}}[/tex] (3)

Where:

[tex]x_{A}[/tex], [tex]x_{B}[/tex], [tex]x_{C}[/tex] - x-Coordinates of points A, B and C.

[tex]y_{A}[/tex], [tex]y_{B}[/tex], [tex]y_{C}[/tex] - y-Coordinates of points A, B and C.

If we know that [tex]x_{A} = 0[/tex], [tex]y_{A} = 0[/tex], [tex]x_{B} = 50[/tex], [tex]y_{B} = 0[/tex], [tex]x_{C} = 18[/tex] and [tex]y_{C} = 24[/tex], then the relationship between slopes is:

[tex]m_{AB} = \frac{0-0}{50-0}[/tex]

[tex]m_{AB} = 0[/tex]

[tex]m_{BC} = \frac{24-0}{18-50}[/tex]

[tex]m_{BC} = -\frac{3}{4}[/tex]

[tex]m_{AB}\cdot m_{BC} = 0\cdot \left(-\frac{3}{4} \right)[/tex]

[tex]m_{AB}\cdot m_{BC} = 0[/tex]

Which means that both line segments are not perpendicular to each other.

In a nutshell, we need to find the slopes of AB and CB, if the product between both slopes is -1. Then, the sides meet in a right angle.