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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 perpendicular sides.

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]\[
\begin{array}{l}
m_{CD}=\frac{2-3}{1-2} \\
m_{EF}=\frac{3-1}{5-4}
\end{array}
\][/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 consecutive sides to check for perpendicular sides.
B. Carla is on the right track because she is finding the slopes of the opposite sides to check for perpendicular sides.
C. Jonah is on the right track because he is finding the slopes of the opposite sides to check for perpendicular sides.
D. Jonah is on the right track because he is finding the slopes of consecutive sides to check for perpendicular sides.



Answer :

GinnyAnswer
To determine whether the sides of quadrilateral [tex]\( CDEF \)[/tex] are perpendicular, we need to consider the slopes of the relevant sides.

Carla's Calculations:

1. She calculates the slope of side [tex]\( CD \)[/tex]:
[tex]\[ m_{CD} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1.0 \][/tex]

2. She calculates the slope of side [tex]\( DE \)[/tex]:
[tex]\[ m_{DE} = \frac{1 - 2}{4 - 1} = \frac{-1}{3} \approx -0.333 \][/tex]

She then checks if the product of these slopes is [tex]\(-1\)[/tex]:
[tex]\[ m_{CD} \cdot m_{DE} = 1.0 \cdot (-0.333) \approx -0.333 \][/tex]
Since [tex]\(-0.333 \neq -1\)[/tex], sides [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex] are not perpendicular.

Jonah's Calculations:

1. He calculates the slope of side [tex]\( CD \)[/tex]:
[tex]\[ m_{CD} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1.0 \][/tex]

2. He calculates the slope of side [tex]\( EF \)[/tex]:
[tex]\[ m_{EF} = \frac{3 - 1}{5 - 4} = \frac{2}{1} = 2.0 \][/tex]

He then checks if the product of these slopes is [tex]\(-1\)[/tex]:
[tex]\[ m_{CD} \cdot m_{EF} = 1.0 \cdot 2.0 = 2.0 \][/tex]
Since [tex]\( 2.0 \neq -1\)[/tex], sides [tex]\( CD \)[/tex] and [tex]\( EF \)[/tex] are not perpendicular.

Analysis of Who is Correct:

The goal is to check whether any pair of consecutive sides of quadrilateral [tex]\( CDEF \)[/tex] are perpendicular. Carla is checking the slopes of consecutive sides ([tex]\( CD \)[/tex] and [tex]\( DE \)[/tex]) to determine whether the sides are perpendicular. This approach is correct because checking consecutive sides is essential for assessing perpendicularity in a polygon's geometry.

Jonah, on the other hand, is comparing the slope of [tex]\( CD \)[/tex] with [tex]\( EF \)[/tex], which are not consecutive sides. In the context of checking for right angles in a polygon, such non-consecutive comparisons do not help in determining perpendicularity between sides of a quadrilateral.

Therefore, the correct answer is:

Carla is on the right track because she is finding the slopes of consecutive sides to check for perpendicular sides.

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