Question

point a is located at $(4,1)$ and point $b$ is located at $(9,13)$.
what are the coordinates of the point that partitions the directed line segment $overline{ab}$ in a $4:1$ ratio?
enter your answer as a decimal or integer in the boxes.
( , )

Explanation:

Step1: Recall section formula

For a directed segment $\overline{AB}$ partitioned in ratio $m:n$, the point $(x,y)$ is given by:
$$x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}$$
Here, $A(x_1,y_1)=(4,1)$, $B(x_2,y_2)=(9,13)$, $m=4$, $n=1$.

Step2: Calculate x-coordinate

Substitute values into x formula:
$$x = \frac{4 \times 9 + 1 \times 4}{4+1} = \frac{36 + 4}{5} = \frac{40}{5} = 8$$

Step3: Calculate y-coordinate

Substitute values into y formula:
$$y = \frac{4 \times 13 + 1 \times 1}{4+1} = \frac{52 + 1}{5} = \frac{53}{5} = 10.6$$

Answer:

$(8, 10.6)$