Question

point a is located at (5, 10) and point b is located at (20, 25). what point partitions the directed line segment \\(\overline{ab}\\) into a 3 : 7 ratio? \\(\big(9\frac{1}{2}, 14\frac{1}{2}\big)\\) \\(\big(9\frac{1}{2}, 20\frac{1}{2}\big)\\) \\(\big(15\frac{1}{2}, 14\frac{1}{2}\big)\\) \\(\big(15\frac{1}{2}, 20\frac{1}{2}\big)\\)

Explanation:

Step1: Recall section formula

For a directed segment $\overline{AB}$ with $A(x_1,y_1)$, $B(x_2,y_2)$, partitioned in ratio $m:n$, the point $(x,y)$ is:
$$x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}$$
Here $A(5,10)$, $B(20,25)$, $m=3$, $n=7$.

Step2: Calculate x-coordinate

Substitute values for $x$:
$$x = \frac{3 \times 20 + 7 \times 5}{3+7} = \frac{60 + 35}{10} = \frac{95}{10} = 9\frac{1}{2}$$

Step3: Calculate y-coordinate

Substitute values for $y$:
$$y = \frac{3 \times 25 + 7 \times 10}{3+7} = \frac{75 + 70}{10} = \frac{145}{10} = 14\frac{1}{2}$$

Answer:

$\boldsymbol{(9\frac{1}{2}, 14\frac{1}{2})}$