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Carla and Jonah are working together to determine if quadrilateral [tex]$\operatorname{CDEF}$[/tex] with coordinates [tex]$C(2,3)$[/tex], [tex]$D(1,2)$[/tex], [tex]$E(4,1)$[/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 if the quadrilateral [tex]\( CDEF \)[/tex] has perpendicular sides, we need to check the slopes of consecutive sides. Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex].

First, let's look at Carla's and Jonah's work in calculating the slopes.

Carla's Slopes:
- Slope of [tex]\( CD \)[/tex]: [tex]\( m_{CD} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1.0 \)[/tex]
- Slope of [tex]\( DE \)[/tex]: [tex]\( m_{DE} = \frac{1 - 2}{4 - 1} = \frac{-1}{3} \approx -0.333 \)[/tex]

Jonah's Slopes:
- Slope of [tex]\( CD \)[/tex]: [tex]\( m_{CD} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1.0 \)[/tex]
- Slope of [tex]\( EF \)[/tex]: [tex]\( m_{EF} = \frac{3 - 1}{5 - 4} = \frac{2}{1} = 2.0 \)[/tex]

### Interpretation and Conclusion:

To check for perpendicularity, we need the slopes of consecutive sides. Jonah is calculating the slopes of [tex]\( CD \)[/tex] and [tex]\( EF \)[/tex], which are consecutive sides of the quadrilateral.

Now, let's examine whether the product of the slopes is [tex]\(-1\)[/tex]:
- [tex]\( m_{CD} \times m_{EF} = 1.0 \times 2.0 = 2.0 \)[/tex]

Since [tex]\( 2.0 \neq -1 \)[/tex], the sides [tex]\( CD \)[/tex] and [tex]\( EF \)[/tex] are not perpendicular. However, Jonah is still following the correct method for determining perpendicularity by finding the slopes of consecutive sides.

Thus, Jonah is on the right track because he is finding the slopes of consecutive sides to check for perpendicular sides.

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