To find the coordinates of point [tex]\( P \)[/tex] which lies [tex]\(\frac{5}{6}\)[/tex] of the length of the segment from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex], we need to use the fact that point [tex]\( P \)[/tex] divides the segment [tex]\( RQ \)[/tex] into a specific ratio.
The coordinates of [tex]\( R \)[/tex] are [tex]\( (0, 0) \)[/tex] and the coordinates of [tex]\( Q \)[/tex] are [tex]\( (6, 6) \)[/tex].
Given that [tex]\( P \)[/tex] is [tex]\(\frac{5}{6}\)[/tex] of the way from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex], we use the section formula for the internal division of a line segment. The formula for the coordinates of a point that divides the line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
For this problem, we consider [tex]\( P \)[/tex] divides [tex]\( RQ \)[/tex] in the ratio [tex]\( 5:1 \)[/tex] (since [tex]\(\frac{5}{6}\)[/tex] is the same as [tex]\( \frac{5}{5+1} \)[/tex]).
Substituting the coordinates and the ratio into the formula:
1. Calculate the x-coordinate of [tex]\( P \)[/tex]:
[tex]\[
Px = \frac{5 \cdot 6 + 1 \cdot 0}{5+1} = \frac{30 + 0}{6} = \frac{30}{6} = 5.0
\][/tex]
2. Calculate the y-coordinate of [tex]\( P \)[/tex]:
[tex]\[
Py = \frac{5 \cdot 6 + 1 \cdot 0}{5+1} = \frac{30 + 0}{6} = \frac{30}{6} = 5.0
\][/tex]
Thus, the coordinates of [tex]\( P \)[/tex] are [tex]\((5.0, 5.0)\)[/tex].
Since the coordinates [tex]\( (5.0, 5.0) \)[/tex] are already in the nearest tenth, there is no need for further rounding.
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\(\boxed{(5.0, 5.0)}\)[/tex].
