Question

carla and jonah are working together to determine if quadrilateral cdef with coordinates c(2, 3), d(1, 2), e(4, 1), and f(5, 3) has parallel sides.
carla sets up the following equations
$m_{cd} = \frac{2 - 3}{1 - 2}$
$m_{de} = \frac{1 - 2}{4 - 1}$
jonah sets up the following equations
$m_{cd} = \frac{2 - 3}{1 - 2}$
$m_{ef} = \frac{3 - 1}{5 - 4}$
who is on track to get the correct answer, and why?
carla is on the right track because she is finding the slopes of the opposite sides to check for parallel sides
carla is on the right track because she is finding the slopes of consecutive sides to check for parallel sides
jonah is on the right track because he is finding the slopes of the opposite sides to check for parallel sides.
jonah is on the right track because he is finding the slopes of consecutive sides to check for parallel sides

Explanation:

Step1: Recall the property of parallel sides

Parallel sides of a quadrilateral have equal slopes. To check for parallel sides, we need to find the slopes of opposite sides.

Step2: Analyze Carla's equations

Carla finds \(m_{CD}=\frac{2 - 3}{1 - 2}\) (slope of side \(CD\)) and \(m_{DE}=\frac{1 - 2}{4 - 1}\) (slope of side \(DE\)). These are slopes of consecutive sides, not opposite sides.

Step3: Analyze Jonah's equations

Jonah finds \(m_{CD}=\frac{2 - 3}{1 - 2}\) (slope of side \(CD\)) and \(m_{EF}=\frac{3 - 1}{5 - 4}\) (slope of side \(EF\)). \(CD\) and \(EF\) are opposite sides.

Answer:

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