Question

ng a directed line segment in the coordinate plane
what are the x- and y-coordinates of point c, which partitions the directed line segment from a to b into the ratio 3 : 10? round to the nearest tenth, if necessary.
x = -2.6
y = options: -5.2, 2.9, 5.2, 8.9

Explanation:

Step1: Identify coordinates of A and B

From the graph, \( A(-5, 8) \) and \( B(2, -4) \). The ratio is \( m:n = 3:10 \).

Step2: Use section formula

The section formula for a point \( (x, y) \) dividing the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2 + ny_1}{m + n} \).

Step3: Calculate x-coordinate

Substitute \( x_1=-5 \), \( x_2=2 \), \( m = 3 \), \( n = 10 \):
\( x=\frac{3(2)+10(-5)}{3 + 10}=\frac{6-50}{13}=\frac{-44}{13}\approx - 3.38 \)? Wait, but the given x is -2.6. Wait, maybe I misread A's coordinates. Wait, looking at the graph, A is at (-4, 8)? Wait, the grid: x-axis, A is at x=-4 (since from y-axis, left 4 units), y=8. B is at x=2, y=-4. Let's recheck.

Wait, correct coordinates: A(-4, 8), B(2, -4). Then:
\( x=\frac{3(2)+10(-4)}{3 + 10}=\frac{6-40}{13}=\frac{-34}{13}\approx - 2.6 \) (matches the given x).

Step4: Calculate y-coordinate

Substitute \( y_1=8 \), \( y_2=-4 \), \( m = 3 \), \( n = 10 \):
\( y=\frac{3(-4)+10(8)}{3 + 10}=\frac{-12 + 80}{13}=\frac{68}{13}\approx 5.23 \)? Wait, no, wait the ratio is from A to B, so direction matters. Wait, maybe I mixed m and n. Wait, the ratio is 3:10, so m is the part near A, n near B? Wait, section formula: if dividing from A to B in ratio m:n, then \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2 + ny_1}{m + n} \). Wait, maybe A is (-5, 8)? No, the grid: each square is 1 unit. A is at x=-4 (since between x=-5 and x=-3, so x=-4), y=8. B is at x=2, y=-4.

Wait, let's recalculate y:
\( y=\frac{3(-4)+10(8)}{13}=\frac{-12 + 80}{13}=\frac{68}{13}\approx 5.2 \)? Wait, but the options include 5.2. Wait, the given y options: -5.2, 2.9, 5.2, 8.9. Wait, my calculation gives ~5.2. Wait, maybe I had the ratio reversed? If the ratio is 3:10 from A to B, then m=3 (towards B), n=10 (towards A)? No, section formula: when dividing in ratio m:n (A to B), the formula is \( \frac{mx_B + nx_A}{m + n} \). So with A(-4,8), B(2,-4), m=3, n=10:

\( y=\frac{3(-4)+10(8)}{13}=\frac{-12 + 80}{13}=\frac{68}{13}\approx 5.2 \). So the correct y is 5.2.

Answer:

\( y = 5.2 \)