Partitioning Segments and Perimeter in the Coordinate Plane

Picture the ratio $1:3$ as equal parts: one part from $A$ to $C$ and three parts from $C$ to $B$. That makes four parts in all, so $C$ is $\frac{1}{4}$ of the way from $A$ to $B$.

The same reasoning works for any positive $m$ and $n$. The statement $AC:CB = m:n$ means $AC = mk$ and $CB = nk$ for some positive number $k$. Since $C$ is between $A$ and $B$, the Segment Addition Postulate in its forward direction (if $C$ is between $A$ and $B$, then $AC + CB = AB$) gives the first equation below. The second equation compares $AC$ to the whole length. Its divisor $(m+n)k$ is positive, so it is not zero, and the common factor $k$ cancels.

Why does this rule work? Here is the proof.

Given: $A(x_1, y_1)$ and $B(x_2, y_2)$ are different points, $m > 0$, $n > 0$, $t = \frac{m}{m+n}$, and $C$ is the point the rule gives.

To prove: $C$ lies between $A$ and $B$, and $AC:CB = m:n$.

Since $m$ and $n$ are positive, $t > 0$ and $1 - t = \frac{(m+n) - m}{m+n} = \frac{n}{m+n} > 0$. By the rule, moving from $A$ to $C$ changes each coordinate by $t$ times its full change. In the lines below, the Distance Formula gives the first line, and squaring each product gives the second. The Distributive Property factors out $t^2$ in the third. The Product Property of Square Roots, $\sqrt{ab} = \sqrt{a}\sqrt{b}$ for nonnegative radicands (the numbers under the radical sign), gives the fourth. The last line uses $\sqrt{t^2} = t$, which holds because $t > 0$ and the square root symbol names the nonnegative root. It also uses the Distance Formula for $AB$.

Moving from $C$ to $B$ changes $x$ by $x_2 - (x_1 + t(x_2 - x_1)) = (x_2 - x_1) - t(x_2 - x_1) = (1 - t)(x_2 - x_1)$, and $y$ changes the same way. So the same steps, with $1 - t$ in place of $t$, give $CB = (1 - t) \cdot AB$. Since $A$ and $B$ are different points, $AB > 0$, so $AC$ and $CB$ are both positive. Their sum is $t \cdot AB + (1 - t) \cdot AB = AB$.

Now use the Segment Addition Postulate in its converse direction, with the if and then parts swapped: if positive lengths $AC$ and $CB$ add up to $AB$, then $C$ lies between $A$ and $B$.

To compare the lengths, divide both by the positive number $AB$, then multiply both terms by $m + n$: $AC:CB = t:(1 - t) = \frac{m}{m+n}:\frac{n}{m+n} = m:n$. So $C$ meets the definition of a partition point.

Now see the rule on a specific segment. In the figure below, the horizontal axis is the $x$-axis and the vertical axis is the $y$-axis. The side of the triangle from $A(1, 1)$ to $B(9, 5)$ is the directed segment. The side from $(1, 1)$ to $(9, 1)$ shows the change in $x$, which is $9 - 1 = 8$. The side from $(9, 1)$ to $(9, 5)$ shows the change in $y$, which is $5 - 1 = 4$.

The dots at $C(3, 2)$, $(5, 3)$ and $(7, 4)$ split the directed segment into four parts. Each part changes $x$ by $2$ and $y$ by $1$, so the Distance Formula gives every part the same length, $\sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$. These lengths come from the stated coordinates, not from how the picture looks. The next example finds $C$ from the coordinates alone.

Now for your second skill. The perimeter of a polygon is the sum of the lengths of its sides. List the vertices in order around the polygon. Each side joins two vertices that are next to each other in the list, and one more side joins the last vertex back to the first. For quadrilateral $DEFG$, the sides are $DE$, $EF$, $FG$ and $GD$.

Use the Distance Formula for each side, and simplify each radical by factoring out perfect squares. Then add. Only like radicals, which have the same radicand after simplifying, combine into one term. Keep the sum exact, and label any decimal as an approximation.

In the figure below, the horizontal axis is the $x$-axis and the vertical axis is the $y$-axis. The labeled dots are the vertices $D(-3, 0)$, $E(-1, 2)$, $F(2, -1)$ and $G(-2, -4)$, and the polygon's four sides are the lengths you will add in the next example. Take those lengths from the coordinates, not from how the picture looks.

The takeaway: To partition a directed segment in the ratio $m:n$, with $m$ and $n$ positive, begin at its start and add $\frac{m}{m+n}$ of the change in each coordinate. To find a polygon's perimeter, use the Distance Formula on every side, including the side that joins the last vertex back to the first, and add the exact lengths. In both skills, exact coordinates give exact answers, so keep them exact and label any decimal as an approximation.