Question

partitioning a directed line segment in the coordinate plane
what are the coordinates of point p on the directed line segment from r to q such that p is \\(\frac{5}{6}\\) the length of the line segment from r to q? round to the nearest tenth, if necessary
(\\(\square\\), \\(\square\\))

Explanation:

Step1: Identify coordinates of R and Q

From the graph, \( R=(4, -1) \) and \( Q=(-5, 3) \). We use the section formula: if a point \( P(x, y) \) divides the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( P \) is \( \frac{5}{6} \) from \( R \) to \( Q \), so \( m = 5 \), \( n=6 - 5 = 1 \) (since total parts \( 6 \), \( P \) is \( 5 \) parts from \( R \), \( 1 \) part from \( P \) to \( Q \)). So ratio \( m:n = 5:1 \), \( (x_1, y_1)=(4, -1) \), \( (x_2, y_2)=(-5, 3) \).

Step2: Calculate x-coordinate of P

\( x=\frac{5\times(-5)+1\times4}{5 + 1}=\frac{-25 + 4}{6}=\frac{-21}{6}=-3.5 \)

Step3: Calculate y-coordinate of P

\( y=\frac{5\times3+1\times(-1)}{5 + 1}=\frac{15 - 1}{6}=\frac{14}{6}\approx2.3 \) (rounded to nearest tenth)

Answer:

\((-3.5, 2.3)\)