Question

What are the coordinates of the point on the directed line segment from (-10,-8) to (-8,8) that partitions the segment into a ratio of 1 to 3?

Answer 1

In order to calculate the coordinates of the point, first let's calculate the change in x and change in y between the line points:

$\begin{gathered} \Delta x=-8-(-10)=-8+10=2\\ \\ \Delta y=8-(-8)=8+8=16 \end{gathered}$

Now, since the ratio is 1 to 3, we will divide this change into 4 parts, this way we can use 1 part to the left and 3 parts to the right of the required point:

$\begin{gathered} \frac{\Delta x}{4}=\frac{2}{4}=0.5\\ \\ \frac{\Delta y}{4}=\frac{16}{4}=4 \end{gathered}$

Now, to find the required coordinates, let's add this change to the coordinates of the first point:

$\begin{gathered} A(-10,-8) \\ P_x=A_x+\frac{\Delta x}{4}=-10+0.5=-9.5 \\ P_y=A_y+\frac{\Delta y}{4}=-8+4=-4 \end{gathered}$

Therefore the answer is (-9.5, -4).