Question

point a is located at (2, 8) and point b is located at (8, 5). what point partitions the directed line segment \\(\overline{ab}\\) into a 1 : 3 ratio? \\(\circ\\) \\(\left(2\frac{1}{2}, 3\frac{1}{4}\
ight)\\) \\(\circ\\) \\(\left(3\frac{1}{2}, 7\frac{1}{4}\
ight)\\) \\(\circ\\) \\(\left(6\frac{1}{2}, 5\frac{1}{4}\
ight)\\) \\(\circ\\) \\(\left(3\frac{1}{3}, 4\frac{1}{3}\
ight)\\)

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 $A(2,8)$, $B(8,5)$, $m=1$, $n=3$.

Step2: Calculate x-coordinate

Substitute values for $x$:
$$x = \frac{1 \times 8 + 3 \times 2}{1+3} = \frac{8+6}{4} = \frac{14}{4} = 3\frac{1}{2}$$

Step3: Calculate y-coordinate

Substitute values for $y$:
$$y = \frac{1 \times 5 + 3 \times 8}{1+3} = \frac{5+24}{4} = \frac{29}{4} = 7\frac{1}{4}$$

Answer:

B. $(3\frac{1}{2}, 7\frac{1}{4})$