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 slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a quadrilateral \(CDEF\) with vertices \(C(2,3)\), \(D(1,2)\), \(E(4,1)\), and \(F(5,3)\), to check for parallel sides, we need to check the slopes of opposite sides.

Step2: Analyze Carla's work

Carla calculates \(m_{CD}=\frac{2 - 3}{1 - 2}\) (slope of side \(CD\)) and \(m_{DE}=\frac{1 - 2}{4 - 1}\) (slope of side \(DE\)). She is calculating the slopes of consecutive sides, which is not the way to check for parallel sides in a quadrilateral.

Step3: Analyze Jonah's work

Jonah calculates \(m_{CD}=\frac{2 - 3}{1 - 2}\) (slope of side \(CD\)) and \(m_{EF}=\frac{3 - 1}{5 - 4}\) (slope of side \(EF\)). In a quadrilateral \(CDEF\), \(CD\) and \(EF\) are opposite sides. If the slopes of opposite sides are equal, then the sides are parallel.

Answer:

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