Question

the endpoints of $overline{ab}$ are $a(-8, -1)$ and $b(-3, 4)$. point $c$ lies on $overline{ab}$ and is $\frac{2}{5}$ of the way from $a$ to $b$. what are the coordinates of point $c$? explain how you found your answer.

$\bigcirc$ a. $c$ is the midpoint of $overline{ab}$, so use the midpoint formula. the coordinates of point $c$ are $square$.

$\bigcirc$ b. first, find $\frac{2}{5}$ of the horizontal and vertical distances from $a$ to $b$. then, move this amount from $b$ to $c$. the coordinates of point $c$ are $square$.

$\bigcirc$ c. use the distance formula to find $\frac{2}{5}$ the length of $overline{ab}$. the coordinates of point $c$ are $square$.

$\bigcirc$ d. first, find $\frac{2}{5}$ of the horizontal and vertical distances from $a$ to $b$. then, move this amount from $a$ to $c$. the coordinates of point $c$ are $square$.

Explanation:

Step1: Find horizontal/vertical distances

Horizontal distance (Δx) from A to B: $-3 - (-8) = 5$
Vertical distance (Δy) from A to B: $4 - (-1) = 5$

Step2: Calculate 2/5 of distances

2/5 of Δx: $\frac{2}{5} \times 5 = 2$
2/5 of Δy: $\frac{2}{5} \times 5 = 2$

Step3: Move from A to C

A's coordinates: $(-8, -1)$
C's x-coordinate: $-8 + 2 = -6$
C's y-coordinate: $-1 + 2 = 1$

For the options:

• Option A is wrong (C is not the midpoint).
• Option B is wrong (moves from B, but we need from A).
• Option C is wrong (Distance Formula isn’t needed here).
• Option D is correct (process matches our calculation).

Answer:

D. First, find $\frac{2}{5}$ of the horizontal and vertical distances from A to B. Then, move this amount from A to C. The coordinates of point C are $(-6, 1)$.