Answered

Carla and Jonah are working together to determine if quadrilateral [tex]\(CDEF\)[/tex] with coordinates [tex]\(C(2,3), D(1,2), E(4,1)\)[/tex], and [tex]\(F(5,3)\)[/tex] has a right angle.

Carla sets up the following equations:
[tex]\[
\begin{array}{l}
m_{CD}=\frac{2-3}{1-2} \\
m_{DE}=\frac{1-2}{4-1}
\end{array}
\][/tex]

Jonah sets up the following equations:
[tex]\[
\begin{array}{l}
m_{CO}=\frac{2-3}{1-2} \\
m_{EF}=\frac{3-1}{5-4}
\end{array}
\][/tex]

Who is on track to get the correct answer, and why?

A. Carla is on the right track because she is finding the slopes of the opposite sides to check for right angles.
B. Carla is on the right track because she is finding the slopes of consecutive sides to check for right angles.
C. Jonah is on the right track because he is finding the slopes of the opposite sides to check for right angles.
D. Jonah is on the right track because he is finding the slopes of consecutive sides to check for right angles.



Answer :

GinnyAnswer
To determine whether quadrilateral [tex]\(CDEF\)[/tex] has a right angle, we need to check for perpendicular sides. Perpendicular sides have slopes whose product is [tex]\(-1\)[/tex].

Let's analyze the steps taken by both Carla and Jonah.

Carla's Setup:

1. Slope of [tex]\(CD\)[/tex]:
[tex]\[ m_{CD} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1.0 \][/tex]
This means the slope of segment [tex]\(CD\)[/tex] is [tex]\(1.0\)[/tex].

2. Slope of [tex]\(DE\)[/tex]:
[tex]\[ m_{DE} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 2}{4 - 1} = \frac{-1}{3} = -0.3333 \][/tex]
This means the slope of segment [tex]\(DE\)[/tex] is approximately [tex]\(-0.3333\)[/tex].

Carla is checking whether the product of the slopes of consecutive segments [tex]\(CD\)[/tex] and [tex]\(DE\)[/tex] is [tex]\(-1\)[/tex]:
[tex]\[ m_{CD} \times m_{DE} = 1.0 \times -0.3333 = -0.3333 \][/tex]
The result is not [tex]\(-1\)[/tex]. Thus, there's no right angle between [tex]\(CD\)[/tex] and [tex]\(DE\)[/tex], but Carla's method is valid as she is correctly checking the slopes of consecutive sides.

Jonah's Setup:

1. Slope of [tex]\(CO\)[/tex] (which is the same as [tex]\(CD\)[/tex]):
[tex]\[ m_{CO} = \frac{2 - 3}{1 - 2} = 1.0 \][/tex]
This is identical to [tex]\(m_{CD}\)[/tex], which we already know is 1.0.

2. Slope of [tex]\(EF\)[/tex]:
[tex]\[ m_{EF} = \frac{3 - 1}{5 - 4} = \frac{2}{1} = 2.0 \][/tex]
This means the slope of segment [tex]\(EF\)[/tex] is [tex]\(2.0\)[/tex].

Jonah is checking whether the product of the slopes is [tex]\(-1\)[/tex]:
[tex]\[ m_{CO} \times m_{EF} = 1.0 \times 2.0 = 2.0 \][/tex]
The result is not [tex]\(-1\)[/tex]. Thus, there's no right angle between [tex]\(CO\)[/tex] and [tex]\(EF\)[/tex], and Jonah's setup isn't appropriate for verifying the slopes of opposite sides correctly.

Conclusion:

Carla's method is correct because she is evaluating the slopes of consecutive sides to check for right angles. Checking consecutive slopes is essential for verifying if two sides form a right angle.

Hence, the correct answer is:
Carla is on the right track because she is finding the slopes of consecutive sides to check for right angles.