Certainly! Let's work through the problem step-by-step.
We are asked to find the coordinates of the point [tex]\( P \)[/tex] that is located [tex]\(\frac{5}{6}\)[/tex] of the distance along the line segment from point [tex]\( R \)[/tex] to point [tex]\( Q \)[/tex].
Given:
- Coordinates of point [tex]\( R \)[/tex] are [tex]\( R(0, 0) \)[/tex].
- Coordinates of point [tex]\( Q \)[/tex] are [tex]\( Q(6, 12) \)[/tex].
To find the coordinates of point [tex]\( P \)[/tex], which lies [tex]\(\frac{5}{6}\)[/tex] of the way from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex], we can use the section formula. The section formula for internal division of a line segment [tex]\( AB \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
\left( x_P, y_P \right) = \left( \frac{m x_B + n x_A}{m+n}, \frac{m y B + n y_A}{m+n} \right)
\][/tex]
In our problem, point [tex]\( P \)[/tex] divides [tex]\( RQ \)[/tex] in the ratio [tex]\( 5:1 \)[/tex] since [tex]\(\frac{5}{6}\)[/tex] implies that for every 5 parts of the total 6 parts of the distance, [tex]\( P \)[/tex] is at the 5th part.
Thus, the formula simplifies:
[tex]\[
\left( x_P, y_P \right) = \left( R_x + \frac{5}{6} (Q_x - R_x), R_y + \frac{5}{6} (Q_y - R_y) \right)
\][/tex]
Now plug in the coordinates of [tex]\( R \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
x_P = 0 + \frac{5}{6} (6 - 0)
\][/tex]
[tex]\[
x_P = 0 + \frac{5}{6} \times 6
\][/tex]
[tex]\[
x_P = 5
\][/tex]
For [tex]\( y_P \)[/tex]:
[tex]\[
y_P = 0 + \frac{5}{6} (12 - 0)
\][/tex]
[tex]\[
y_P = 0 + \frac{5}{6} \times 12
\][/tex]
[tex]\[
y_P = 10
\][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[
(5.0, 10.0)
\][/tex]
Rounding to the nearest tenth, the coordinates remain [tex]\( (5.0, 10.0) \)[/tex].
So, the coordinates of point [tex]\( P \)[/tex] are [tex]\( \boxed{(5.0, 10.0)} \)[/tex].
