Finding Coordinates of a point on a line in Coordinate Grid.

Question

To solve this question should I use Pythagoras rule like this? Please, any other method to get the coordinates?

Answer 1

Write the parametric equation of the line segment from $B$ to $C$: $$(7(1-t)+9t,8(1-t)+4t)$$ The point corresponding to a fixed $t$ is $t$ of the way from $B$ to $C$. It follows that if $BC=\frac23BE$ then $BE=\frac32BC$, i.e. we substitute $t=1.5$ into the abobe equation and get $E=(10,2)$.

Answer 2

Your picture is not correct. Point E should have been outside of BC

Otherwise you are on right track

Answer 3

First, you’ve gotten the order of the points incorrect. For $C$ to be $2/3$ of the way from $B$ to $E$ it must lie between those two points.

That aside, there’s no need to use the Pythagorean theorem for this. If the lengths of two segments of the same line are in a certain proportion, then the differences of their endpoint coordinates are in the same proportion. So, if $BC=\frac23BE$, then $x_C-x_B=\frac23(x_E-x_B)$ and $y_C-y_B=\frac23(y_E-y_B)$. Plug in the known coordinate values and solve for the two unknown coordinates.