Finding coordinates of points using distance between points.

Question

I got a question in my homework which I can't solve. Here is the question: (I am not a native speaker so please explain step by step and clearly.)

Point $C$ internally divides the segment containing the endpoints $A(3,6)$ and $B(18,31)$ such that $3(AB)=5(AC)$. Find the coordinates of Point $C$.

Ans is $(12,21)$ but I don't know how to solve it.

Answer 1

1. Find the length $L$ of the line segment using endpoint coordinates.
2. Your equation implies $x = \overline{AC} = 3L/5$, compute that.
3. Vector along which you move from $A$ to $B$ is $\vec{\Delta} = \vec{B} - \vec{A} = (15,25)$. Can you now use this to construct $\vec{C} = \vec{A} + k\vec{\Delta}$?

Answer 2

Use similar triangles: $AC:AB :: AE:AD :: AG:AF$.

This means that you can find the $x$- and $y$ coordinates of $C$ separately by looking only at the corresponding coordinates of $A$ and $B$. Taking the horizontal direction first, $AD = 18-3 = 15$. Find the correct portion of this for $AE$ and add this number to the $x$-coordinate of $A$. The computation for the $y$-coordinate of $C$ goes the same way. You can do this all at once using vectors: If we let $\lambda$ be the required proportion, then $$\vec C = \vec A+\lambda(\vec B-\vec A) = (1-\lambda)\vec A+\lambda\vec B.$$