To find the coordinates of 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] of the way from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex], follow these steps:
1. Identify the coordinates of points [tex]\( R \)[/tex] and [tex]\( Q \)[/tex]:
- Let point [tex]\( R \)[/tex] have coordinates [tex]\( (R_x, R_y) = (0, 0) \)[/tex].
- Let point [tex]\( Q \)[/tex] have coordinates [tex]\( (Q_x, Q_y) = (6, 6) \)[/tex].
2. Determine the ratio:
- The ratio given is [tex]\(\frac{5}{6}\)[/tex].
3. Use the section formula:
- The formula to find the coordinates of a point [tex]\( P \)[/tex] that divides the segment joining [tex]\( R \)[/tex] and [tex]\( Q \)[/tex] in the ratio [tex]\(\frac{m}{n}\)[/tex] is:
[tex]\[
P_x = R_x + \left(\frac{m}{m+n}\right) (Q_x - R_x)
\][/tex]
[tex]\[
P_y = R_y + \left(\frac{m}{m+n}\right) (Q_y - R_y)
\][/tex]
- In this case, [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex], thus the ratio [tex]\(\frac{m}{m+n}\)[/tex] is [tex]\(\frac{5}{6}\)[/tex].
4. Calculate the coordinates of [tex]\( P \)[/tex]:
[tex]\[
P_x = 0 + \left(\frac{5}{6}\right) (6 - 0) = \left(\frac{5}{6}\right) \times 6 = 5.0
\][/tex]
[tex]\[
P_y = 0 + \left(\frac{5}{6}\right) (6 - 0) = \left(\frac{5}{6}\right) \times 6 = 5.0
\][/tex]
5. Round the coordinates to the nearest tenth:
- The coordinates [tex]\( 5.0 \)[/tex] are already at the required precision, so no further rounding is needed.
Therefore, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (5.0, 5.0) \)[/tex].
