Exercises: Find the Point Partitioning a Directed Line Segment

$(2, 3)$ — average the two $x$-coordinates and average the two $y$-coordinates, since the middle point is halfway along in each direction.

$(5, -5)$ — find the change in $x$ and the change in $y$ across the segment, then halve each of those changes to land at the middle.

$(10, -10)$ — subtract the first point's coordinates from the second point's coordinates to get the change across the segment, then use that change as the midpoint.

$(4, 5)$ — add the coordinates of the two endpoints together, then halve only the $y$-coordinate of that sum.

$50$ — multiply the known leg by $\frac{5}{2}$, using the scale factor upside down so the matching side comes out longer than the one you started from.

$8$ — multiply the known leg by the scale factor, since corresponding sides of similar figures are related by that constant multiplier.

$18$ — subtract the difference between the two numbers in the scale factor, treating $5 - 2 = 3$ as an amount to remove from the leg length.

$10$ — halve the known leg, reading the scale factor $\frac{2}{5}$ as though it meant "cut in half twice as evenly."

$(6, 5)$ — split the segment evenly, since a ratio with a $1$ in it marks off one equal share on each side of the point.

$\left(\frac{14}{3}, \frac{11}{3}\right)$ — travel $\frac{1}{3}$ of the way from $A$ toward $B$, reading the ratio $1:3$ directly as the fraction one-third.

$(4, 3)$ — travel $\frac{1}{4}$ of the way from $A$ toward $B$, since the ratio $1:3$ divides the segment into $1 + 3 = 4$ equal parts.

$(8, 7)$ — weight $A$ by $1$ and $B$ by $3$, putting the first ratio number on the starting point and the second on the ending point.

$(7, 4)$ — weight $C$ by $2$ and $D$ by $3$, matching the first ratio number to the first point named in the problem.

$\left(\frac{23}{3}, 5\right)$ — move $\frac{2}{3}$ of the way from $C$ toward $D$, reading the two ratio numbers as a numerator and a denominator.

$\left(6, \frac{5}{2}\right)$ — take the midpoint of the segment, since $2$ and $3$ are close enough in size that the point lands near the center.

$(5, 1)$ — move $\frac{2}{5}$ of the way from $C$ toward $D$, since $2 + 3 = 5$ equal parts make up the whole segment.

$(-2, -3)$ — take the midpoint of the segment, since a partition point must always fall somewhere in the middle region between the two endpoints.

$(-6, 1)$ — weight $J$ by $5$ and $K$ by $1$, attaching the larger ratio number to the point that is written first.

$(52, -57)$ — move $5$ times the length of the segment past $J$, reading the ratio $5:1$ as the multiplier $\frac{5}{1}$.

$(2, -7)$ — move $\frac{5}{6}$ of the way from $J$ toward $K$, since the ratio $5:1$ cuts the segment into $6$ equal parts.