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.
