Question

partitioning 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=
y=

Explanation:

Step1: Identify coordinates of A and B

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

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 the 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 \( x_1=-4 \), \( x_2=2 \), \( m = 3 \), \( n = 10 \):
\( x=\frac{3\times2 + 10\times(-4)}{3 + 10}=\frac{6 - 40}{13}=\frac{-34}{13}\approx - 2.6 \)

Step4: Calculate y-coordinate

Substitute \( y_1=8 \), \( y_2=-4 \), \( m = 3 \), \( n = 10 \):
\( y=\frac{3\times(-4)+10\times8}{3 + 10}=\frac{-12 + 80}{13}=\frac{68}{13}\approx5.2 \)

Answer:

\( x\approx - 2.6 \), \( y\approx5.2 \)