The midpoint from Distance and Midpoint Formulas splits a segment in the ratio $1 : 1$. Sometimes you need a point that splits it in another ratio, like $2 : 3$. The side-splitting idea from The Triangle Proportionality Theorem shows how.
Why it works
Draw the right triangle under $\overline{AB}$, with legs along the grid. A point $P$ on $\overline{AB}$ with $AP : PB = a : b$ cuts the hypotenuse into $a + b$ equal parts and keeps $a$ of them. Lines through $P$ parallel to the legs cut the legs in the same ratio (the triangles are similar), so $P$ is $\frac{a}{a + b}$ of the way along in $x$ and in $y$. Its coordinates $(x, y)$ are $$x = x_1 + \tfrac{a}{a + b}(x_2 - x_1)$$ $$y = y_1 + \tfrac{a}{a + b}(y_2 - y_1)$$
Key idea
To divide $\overline{AB}$ so that $AP : PB = a : b$, go $\frac{a}{a + b}$ of the way from $A$ to $B$: add that fraction of the change in $x$ to $x_1$, and of the change in $y$ to $y_1$.
Worked example: On a number line
$A$ is at $-3$ and $B$ is at $17$. Find $P$ with $AP : PB = 1 : 4$.
- There are $1 + 4 = 5$ parts, and $P$ is $\frac{1}{5}$ of the way from $A$.
- The distance is $17 - (-3) = 20$, and $\frac{1}{5} \cdot 20 = 4$.
- $P = -3 + 4 = 1$.
Answer: $P = 1$
Worked example: On the coordinate plane
Find $P$ on $\overline{AB}$ with $A(1, 1)$, $B(11, 6)$ and $AP : PB = 2 : 3$.
- $P$ is $\frac{2}{5}$ of the way. The change is $\langle 10, 5 \rangle$, and $\frac{2}{5}$ of it is $\langle 4, 2 \rangle$.
- $P = (1 + 4, 1 + 2) = (5, 3)$.
Answer: $P(5, 3)$
Worked example: A fraction answer
Find $P$ on $\overline{AB}$ with $A(-2, 5)$, $B(4, -1)$ and $AP : PB = 3 : 1$.
- $P$ is $\frac{3}{4}$ of the way. The change is $\langle 6, -6 \rangle$; three quarters of it is $\left\langle \frac{9}{2}, -\frac{9}{2} \right\rangle$.
- $P = \left(-2 + \frac{9}{2}, 5 - \frac{9}{2}\right) = \left(\frac{5}{2}, \frac{1}{2}\right)$.
Answer: $P\left(\frac{5}{2}, \frac{1}{2}\right)$
Common mistake
Going $\frac{a}{b}$ of the way instead of $\frac{a}{a + b}$. With $AP : PB = 2 : 3$, the point is $\frac{2}{5}$ of the way, not $\frac{2}{3}$: the ratio compares the two pieces with each other, but you need a piece compared with the whole. Check: $AP$ and $PB$ should come out in the ratio you were given.