Answer :
Let's analyze the problem and the method used by Carla and Jonah to determine if quadrilateral CDEF has parallel sides. The coordinates of the quadrilateral's vertices are [tex]\( C(2, 3) \)[/tex], [tex]\( D(1, 2) \)[/tex], [tex]\( E(4, 1) \)[/tex], and [tex]\( F(5, 3) \)[/tex].
### Carla's Method:
Carla sets up the following equations to calculate the slopes:
1. Slope [tex]\( m_{CD} \)[/tex]:
[tex]\[ m_{CD} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1 \][/tex]
2. Slope [tex]\( m_{DE} \)[/tex]:
[tex]\[ m_{DE} = \frac{1 - 2}{4 - 1} = \frac{-1}{3} = -\frac{1}{3} \][/tex]
### Jonah's Method:
Jonah sets up the following equations to calculate the slopes:
1. Slope [tex]\( m_{CO} \)[/tex]:
[tex]\[ m_{CO} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1 \][/tex]
2. Slope [tex]\( m_{EF} \)[/tex]:
[tex]\[ m_{EF} = \frac{3 - 1}{5 - 4} = \frac{2}{1} = 2 \][/tex]
### Analysis:
- Carla's approach involves calculating the slopes of segments [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex].
- Jonah's approach involves calculating the slopes of segments [tex]\( CO \)[/tex] (which is a mistake and should be [tex]\( CE \)[/tex]) and [tex]\( EF \)[/tex].
In quadrilateral CDEF, for opposite sides to be parallel, the slopes of those sides should be equal.
Let's compare Carla's slopes:
- If Carla was checking for parallel opposite sides, she should have compared:
- Slope [tex]\( CD \)[/tex] with slope [tex]\( EF \)[/tex]
- Slope [tex]\( DE \)[/tex] with slope [tex]\( CF \)[/tex]
However, from her calculations:
- [tex]\( m_{CD} = 1 \)[/tex]
- [tex]\( m_{DE} = -\frac{1}{3} \)[/tex]
These slopes are not equal, and thus, Carla concludes that the opposite sides (using her approach) are not parallel. This is correct because she checks the necessary slopes for parallelism of opposite sides.
### Conclusion:
Given the analysis, we can confirm that:
- Carla is on the right track because she is finding the slopes of opposite sides to check for parallelism. Her approach was correct in principle, but according to the solution provided, the quadrilateral does not have parallel sides since her intended calculation does not show equal slopes for opposite sides.
It turns out that the answer to the problem is that no sides of the quadrilateral are parallel.
The correct conclusion is:
- Carla is on the right track because she is finding the slopes of the opposite sides to check for parallel sides.
However, the final outcome is that from Carla's calculation, the quadrilateral CDEF does not have any parallel sides. Therefore, the answer is given as:
```
0
```
So, based on our reasoning and analysis, Carla's approach was technically justified but did not yield parallel sides in this particular case.
### Carla's Method:
Carla sets up the following equations to calculate the slopes:
1. Slope [tex]\( m_{CD} \)[/tex]:
[tex]\[ m_{CD} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1 \][/tex]
2. Slope [tex]\( m_{DE} \)[/tex]:
[tex]\[ m_{DE} = \frac{1 - 2}{4 - 1} = \frac{-1}{3} = -\frac{1}{3} \][/tex]
### Jonah's Method:
Jonah sets up the following equations to calculate the slopes:
1. Slope [tex]\( m_{CO} \)[/tex]:
[tex]\[ m_{CO} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1 \][/tex]
2. Slope [tex]\( m_{EF} \)[/tex]:
[tex]\[ m_{EF} = \frac{3 - 1}{5 - 4} = \frac{2}{1} = 2 \][/tex]
### Analysis:
- Carla's approach involves calculating the slopes of segments [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex].
- Jonah's approach involves calculating the slopes of segments [tex]\( CO \)[/tex] (which is a mistake and should be [tex]\( CE \)[/tex]) and [tex]\( EF \)[/tex].
In quadrilateral CDEF, for opposite sides to be parallel, the slopes of those sides should be equal.
Let's compare Carla's slopes:
- If Carla was checking for parallel opposite sides, she should have compared:
- Slope [tex]\( CD \)[/tex] with slope [tex]\( EF \)[/tex]
- Slope [tex]\( DE \)[/tex] with slope [tex]\( CF \)[/tex]
However, from her calculations:
- [tex]\( m_{CD} = 1 \)[/tex]
- [tex]\( m_{DE} = -\frac{1}{3} \)[/tex]
These slopes are not equal, and thus, Carla concludes that the opposite sides (using her approach) are not parallel. This is correct because she checks the necessary slopes for parallelism of opposite sides.
### Conclusion:
Given the analysis, we can confirm that:
- Carla is on the right track because she is finding the slopes of opposite sides to check for parallelism. Her approach was correct in principle, but according to the solution provided, the quadrilateral does not have parallel sides since her intended calculation does not show equal slopes for opposite sides.
It turns out that the answer to the problem is that no sides of the quadrilateral are parallel.
The correct conclusion is:
- Carla is on the right track because she is finding the slopes of the opposite sides to check for parallel sides.
However, the final outcome is that from Carla's calculation, the quadrilateral CDEF does not have any parallel sides. Therefore, the answer is given as:
```
0
```
So, based on our reasoning and analysis, Carla's approach was technically justified but did not yield parallel sides in this particular case.
