Let's find the coordinates of the point [tex]\( P \)[/tex] on the directed line segment from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex] such that [tex]\( P \)[/tex] is [tex]\(\frac{5}{6}\)[/tex] the length of the line segment from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex]. We are given the coordinates of points [tex]\( R \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[ R = (0, 0) \][/tex]
[tex]\[ Q = (6, 9) \][/tex]
To find the coordinates of point [tex]\( P \)[/tex], we use the section formula for internal division of a line segment. The formula to find the coordinates of a point [tex]\( P \)[/tex] that divides the line segment joining points [tex]\( R(x_1, y_1) \)[/tex] and [tex]\( Q(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is:
[tex]\[
P \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
In this problem, the ratio is given as [tex]\(\frac{5}{6}\)[/tex]. We can treat this ratio as [tex]\(5 : 1\)[/tex] where [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex]. The coordinates of [tex]\( R \)[/tex] are [tex]\((0, 0)\)[/tex] and the coordinates of [tex]\( Q \)[/tex] are [tex]\((6, 9)\)[/tex].
Substituting these values into the formula, we get:
[tex]\[
P \left( \frac{5 \cdot 6 + 1 \cdot 0}{5 + 1}, \frac{5 \cdot 9 + 1 \cdot 0}{5 + 1} \right)
\][/tex]
Simplify the expressions inside the parentheses:
[tex]\[
P \left( \frac{30 + 0}{6}, \frac{45 + 0}{6} \right)
\][/tex]
This simplifies to:
[tex]\[
P \left( \frac{30}{6}, \frac{45}{6} \right)
\][/tex]
Further simplifying, we get:
[tex]\[
P \left( 5, 7.5 \right)
\][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (5.0, 7.5) \)[/tex].
