What are 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] the length of the line segment from [tex]R[/tex] to [tex]Q[/tex]? Round to the nearest tenth, if necessary.

[tex]\square[/tex]
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Answer :

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] the length of the line segment from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex], follow these steps:

1. Identify the coordinates of points [tex]\( R \)[/tex] and [tex]\( Q \)[/tex]:
- [tex]\( R \)[/tex] is at [tex]\((0, 0)\)[/tex].
- [tex]\( Q \)[/tex] is at [tex]\((6, 8)\)[/tex].

2. Determine the given ratio:
- The ratio is [tex]\(\frac{5}{6}\)[/tex].

3. Calculate the coordinates of point [tex]\( P \)[/tex]:
- The formula to find the coordinates of point [tex]\( P \)[/tex] which divides the segment [tex]\( RQ \)[/tex] in the ratio [tex]\( \frac{m}{n} \)[/tex] can be given by:
[tex]\[ P = \left( R_x + \frac{m}{m+n} (Q_x - R_x), \; R_y + \frac{m}{m+n} (Q_y - R_y) \right) \][/tex]
- Plugging in the values:
[tex]\[ P_x = 0 + \frac{5}{6} \cdot (6 - 0) = \frac{5}{6} \cdot 6 = 5 \][/tex]
[tex]\[ P_y = 0 + \frac{5}{6} \cdot (8 - 0) = \frac{5}{6} \cdot 8 \approx 6.6667 \][/tex]

4. Round the coordinates to the nearest tenth:
- [tex]\( P_x = 5.0 \)[/tex]
- [tex]\( P_y = 6.7 \)[/tex]

Hence, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ (5.0, 6.7) \][/tex]

These coordinates represent 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 from [tex]\( R \)[/tex] to [tex]\( Q \)[/tex].