Question

(04.01 mc)
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 a right angle.
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 right angles.
• carla is on the right track because she is finding the slopes of consecutive sides to check for right angles.
• jonah is on the right track because he is finding the slopes of the opposite sides to check for right angles.
• jonah is on the right track because he is finding the slopes of consecutive sides to check for right angles.

Explanation:

Step1: Recall slope formula

The slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$.

Step2: Check Carla's slope calculations

For $m_{CD}$ (C(2,3), D(1,2)):
$m_{CD}=\frac{2-3}{1-2}=\frac{-1}{-1}=1$ (correct formula use).
For $m_{DE}$ (D(1,2), E(4,1)):
$m_{DE}=\frac{1-2}{4-1}=\frac{-1}{3}$ (correct formula use).
Carla is calculating slopes of consecutive sides (CD & DE). To check for right angles in a quadrilateral, we verify if consecutive sides are perpendicular (product of slopes = -1).

Step3: Check Jonah's slope calculations

For $m_{CD}$ (C(2,3), D(1,2)):
$m_{CD}=\frac{2-3}{1-2}=1$ (correct formula, but he is calculating slopes of opposite sides (CD & EF) which is for parallelism, not right angles).

Answer:

Carla is on the right track because she is finding the slopes of consecutive sides to check for right angles.