To find the point [tex]\( P \)[/tex] along the directed line segment from point [tex]\( A(-18, -2) \)[/tex] to point [tex]\( B(0, 6) \)[/tex] that divides the segment in the ratio 1:3, follow these steps:
1. Understand the given information:
- Point [tex]\( A = (-18, -2) \)[/tex]
- Point [tex]\( B = (0, 6) \)[/tex]
- Ratio [tex]\( = 1:3 \)[/tex]
2. Determine the formula for finding the point that divides a line segment in a given ratio:
If point [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] in the ratio [tex]\( m:n \)[/tex], the coordinates of [tex]\( P \)[/tex] are given by:
[tex]\[
P = \left( \frac{mB_x + nA_x}{m+n}, \frac{mB_y + nA_y}{m+n} \right)
\][/tex]
3. Substitute the given ratio [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex], and the coordinates of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] into the formula:
- Let [tex]\( A_x = -18 \)[/tex], [tex]\( A_y = -2 \)[/tex]
- Let [tex]\( B_x = 0 \)[/tex], [tex]\( B_y = 6 \)[/tex]
- The ratio [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex]
4. Calculate the x-coordinate of [tex]\( P \)[/tex]:
[tex]\[
P_x = \frac{1 \cdot 0 + 3 \cdot (-18)}{1+3} = \frac{0 + (-54)}{4} = \frac{-54}{4} = -13.5
\][/tex]
5. Calculate the y-coordinate of [tex]\( P \)[/tex]:
[tex]\[
P_y = \frac{1 \cdot 6 + 3 \cdot (-2)}{1+3} = \frac{6 + (-6)}{4} = \frac{0}{4} = 0
\][/tex]
6. Determine the point [tex]\( P \)[/tex]:
[tex]\[
P = (-13.5, 0)
\][/tex]
Therefore, the correct point [tex]\( P \)[/tex] that divides the segment from [tex]\( A(-18, -2) \)[/tex] to [tex]\( B(0, 6) \)[/tex] in the ratio 1:3 is:
[tex]\[ \boxed{(-13.5, 0.0)} \][/tex]
This matches the calculated result and precisely defines the location of point [tex]\( P \)[/tex] as [tex]\( (-13.5, 0.0) \)[/tex], even though it doesn't directly match with the provided options [tex]\( P(-9,2) \)[/tex], [tex]\( P\left(-\frac{9}{2}, 4\right) \)[/tex], or [tex]\( P\left(-\frac{27}{2}, 0\right) \)[/tex].
