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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 parallel 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 the opposite sides to check for parallel sides.

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

C. Jonah is on the right track because he is finding the slopes of the opposite sides to check for parallel sides.

D. Jonah is on the right track because he is finding the slopes of consecutive sides to check for parallel sides.

Sagot :

GinnyAnswer
To determine whether quadrilateral CDEF has parallel sides, we need to calculate and compare the slopes of its sides. Parallel lines have equal slopes. Let's go through the process step-by-step.

Given points:
- [tex]\( C(2, 3) \)[/tex]
- [tex]\( D(1, 2) \)[/tex]
- [tex]\( E(4, 1) \)[/tex]
- [tex]\( F(5, 3) \)[/tex]

### Step 1: Calculate the slopes of the sides of quadrilateral CDEF

1. Slope of [tex]\( CD \)[/tex]
[tex]\[ m_{CD} = \frac{y_D - y_C}{x_D - x_C} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1.0 \][/tex]

2. Slope of [tex]\( DE \)[/tex]
[tex]\[ m_{DE} = \frac{y_E - y_D}{x_E - x_D} = \frac{1 - 2}{4 - 1} = \frac{-1}{3} \approx -0.333 \][/tex]

3. Slope of [tex]\( EF \)[/tex]
[tex]\[ m_{EF} = \frac{y_F - y_E}{x_F - x_E} = \frac{3 - 1}{5 - 4} = \frac{2}{1} = 2.0 \][/tex]

4. Slope of [tex]\( CF \)[/tex]
[tex]\[ m_{CF} = \frac{y_F - y_C}{x_F - x_C} = \frac{3 - 3}{5 - 2} = \frac{0}{3} = 0.0 \][/tex]

### Step 2: Analyze the slopes according to the students' setups

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

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

### Step 3: Determine if any sides are parallel
- [tex]\( m_{CD} = 1.0 \)[/tex] and [tex]\( m_{CF} = 0.0 \)[/tex], [tex]\( CD \)[/tex] and [tex]\( CF \)[/tex] are not parallel
- [tex]\( m_{DE} \approx -0.333 \)[/tex] and [tex]\( m_{EF} = 2.0 \)[/tex], [tex]\( DE \)[/tex] and [tex]\( EF \)[/tex] are not parallel

### Conclusion
From the calculations, we can see that none of the opposite or consecutive sides of quadrilateral CDEF are parallel. Therefore, the conclusion is:

Neither Carla nor Jonah is on the right track. The slopes they calculated do not indicate that any of the sides are parallel.
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