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].
