Question

marcus says the distance between points c(2, 8) and d(2, -3) is 5 units because you just count on the graph. samira says she needs to use the distance formula. who is correct and why? explain your reasoning using precise mathematical vocabulary. when is it appropriate to (count vs) use the distance formula?

Explanation:

Step1: Identify point coordinates

Points: $C(2, 8)$, $D(2, -3)$

Step2: Check for vertical line

Since $x$-coordinates are equal ($2=2$), the line is vertical.

Step3: Calculate vertical distance

Distance = $|y_1 - y_2| = |8 - (-3)| = |11| = 11$

Step4: Evaluate Marcus's claim

Marcus calculated $8-3=5$, ignoring the negative sign, so he is wrong.

Step5: Explain counting vs. formula

Counting works for vertical/horizontal points (integer coordinates), but the distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ works for all points. For vertical points, it simplifies to $|y_2-y_1|$.

Answer:

Sandra is correct, and Marcus is incorrect.

1. The distance between $C(2, 8)$ and $D(2, -3)$ is 11 units, calculated as $|8 - (-3)| = 11$. Marcus made an error by not accounting for the negative $y$-coordinate of point $D$, leading to an incorrect distance of 5.
2. Counting grid units on a graph is only appropriate when the two points lie on a horizontal or vertical line (shared $x$- or $y$-coordinate) with integer coordinates. The distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ is the precise, general method that works for all pairs of points, including non-axis-aligned ones, and simplifies to the absolute difference of coordinates for horizontal/vertical lines.