Question

what point divides the directed line segment \\(\overline{ab}\\) into a 3 : 1 ratio?
\\(\circ\\) \\( (3, 2)\\)
\\(\circ\\) \\( (4, 2)\\)
\\(\circ\\) \\( (8, 2)\\)
\\(\circ\\) \\( (9, 2)\\)
(image of a coordinate plane with points a at (0,2) and b at (12,2))

Explanation:

Step1: Identify coordinates of A, B

From the graph: $A=(1, 2)$, $B=(13, 2)$

Step2: Apply section formula

For ratio $m:n=3:1$, the point $(x,y)$ is:
$x=\frac{m\cdot x_B + n\cdot x_A}{m+n}$, $y=\frac{m\cdot y_B + n\cdot y_A}{m+n}$

Step3: Calculate x-coordinate

$x=\frac{3\cdot13 + 1\cdot1}{3+1}=\frac{39+1}{4}=\frac{40}{4}=10$
Wait, correction: Wait, directed segment $\overline{AB}$ means from A to B, ratio 3:1 means the point is 3 parts from A, 1 part from B.
$x = x_A + \frac{3}{3+1}(x_B - x_A) = 1 + \frac{3}{4}(13-1)=1+\frac{3}{4}\cdot12=1+9=10$
Wait, no, check options: Wait, maybe ratio is 3:1 from A to B, so total parts 4.
Wait, recheck coordinates: A is (1,2), B is (13,2). The length AB is 12 units. 3:1 ratio means the point is $\frac{3}{4}$ from A to B.
$x=1 + \frac{3}{4}(13-1)=1+9=10$, but 10 is not an option. Wait, maybe ratio is 3:1 as from B to A? No, directed segment $\overline{AB}$ is A to B. Wait, maybe I misread A's coordinate: A is (0,2)? No, graph shows A at x=1, y=2. B at x=13, y=2.
Wait, wait options: (9,2) is there. Wait, maybe ratio is 1:3? No, question says 3:1. Wait, no: section formula for internal division: if point divides AB in ratio m:n (A to B), then $x=\frac{n x_A + m x_B}{m+n}$? No, no: standard formula: if point P divides AB such that AP:PB = m:n, then $P=(\frac{n x_A + m x_B}{m+n}, \frac{n y_A + m y_B}{m+n})$.
Ah! I mixed up. AP:PB=3:1, so m=3, n=1.
$x=\frac{1\cdot1 + 3\cdot13}{3+1}=\frac{1+39}{4}=\frac{40}{4}=10$, still 10. Wait, maybe the graph's B is at x=12? Let me check again: graph shows B at x=13? No, x-axis goes to 13, B is at 12? If B is (12,2):
$x=\frac{1\cdot1 +3\cdot12}{4}=\frac{1+36}{4}=\frac{37}{4}=9.25$, no. Wait, A is (1,2), if ratio is 3:1 as PB:AP=3:1, then AP:PB=1:3, then $x=\frac{3\cdot1 +1\cdot13}{4}=\frac{3+13}{4}=4$, which is option (4,2). But question says directed segment AB into 3:1 ratio, meaning AB is split into 3 parts and 1 part, so from A, 3 parts, then 1 part to B. So total length 4 parts. AB length is 12 (from x=1 to x=13), each part 3. So 3 parts from A is 1+33=10, but 10 is not an option. Wait, maybe A is (0,2)? Then x=0 + 9=9, which is (9,2). Oh! Maybe I misread A's x-coordinate: A is at x=0, y=2. Let's check: if A=(0,2), B=(12,2). AB length 12. 3:1 ratio, 3 parts from A: 0 + (3/4)12=9, so (9,2), which is an option. That must be it. I misread A's x-coordinate as 1, but it's 0.

Step4: Correct coordinate input, recalculate

$A=(0,2)$, $B=(12,2)$, ratio $AP:PB=3:1$
$x=0 + \frac{3}{3+1}(12-0)=0+9=9$
$y=2$ (since y-coordinate is same for all points on AB)

Answer:

(9, 2)