To determine the correct approach between Carla and Jonah for identifying if quadrilateral CDEF has perpendicular sides, let's analyze the slopes Carla and Jonah calculated, and check for perpendicularity.
The coordinates of points are given as:
- [tex]\( C(2, 3) \)[/tex]
- [tex]\( D(1, 2) \)[/tex]
- [tex]\( E(4, 1) \)[/tex]
- [tex]\( F(5, 3) \)[/tex]
### Carla's Approach:
Carla calculates the slopes of the sides [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex].
1. Calculate the slope of [tex]\( CD \)[/tex]:
[tex]\[
m_{CD} = \frac{D_y - C_y}{D_x - C_x} = \frac{2 - 3}{1 - 2} = \frac{-1}{-1} = 1.0
\][/tex]
2. Calculate the slope of [tex]\( DE \)[/tex]:
[tex]\[
m_{DE} = \frac{E_y - D_y}{E_x - D_x} = \frac{1 - 2}{4 - 1} = \frac{-1}{3} \approx -0.3333
\][/tex]
Check for perpendicularity between [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex]:
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]:
[tex]\[
m_{CD} \times m_{DE} = 1.0 \times -0.3333 = -0.3333 \neq -1
\][/tex]
So, [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex] are not perpendicular.
### Jonah's Approach:
Jonah calculates the slopes of the sides [tex]\( CD \)[/tex] and [tex]\( EF \)[/tex].
1. Calculate the slope of [tex]\( CD \)[/tex] (same as calculated by Carla):
[tex]\[
m_{CD} = 1.0
\][/tex]
2. Calculate the slope of [tex]\( EF \)[/tex]:
[tex]\[
m_{EF} = \frac{F_y - E_y}{F_x - E_x} = \frac{3 - 1}{5 - 4} = \frac{2}{1} = 2.0
\][/tex]
Check for perpendicularity between [tex]\( CD \)[/tex] and [tex]\( EF \)[/tex]:
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]:
[tex]\[
m_{CD} \times m_{EF} = 1.0 \times 2.0 = 2.0 \neq -1
\][/tex]
So, [tex]\( CD \)[/tex] and [tex]\( EF \)[/tex] are not perpendicular.
### Conclusion:
Neither the slopes of [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex] nor the slopes of [tex]\( CD \)[/tex] and [tex]\( EF \)[/tex] indicate perpendicular sides. Therefore, neither Carla's check (consecutive sides) nor Jonah's check (one consecutive and one opposite side) result in perpendicular slopes. Since both pairs are not perpendicular, Jonah gets recognition for being correct in using opposite sides, spotting no perpendicularity correctly.
Thus, the correct conclusion is:
Jonah is on the right track because he is finding the slopes of opposite sides to check for perpendicular sides.
