To find the coordinates of point [tex]\( P \)[/tex] on the line segment [tex]\(\overline{CD}\)[/tex] such that the point [tex]\( P \)[/tex] divides the segment from [tex]\(D\)[/tex] in the ratio [tex]\(\frac{5}{8}\)[/tex], we can use the section formula in coordinate geometry. This formula helps us determine the coordinates of a point that divides a line segment joining two points in a given ratio.
Given:
- Coordinates of [tex]\(C = (-8, 4)\)[/tex]
- Coordinates of [tex]\(D = (6, -6)\)[/tex]
- Ratio in which [tex]\(P\)[/tex] divides the segment from [tex]\(D\)[/tex] is [tex]\( \frac{5}{8} \)[/tex]
We use the section formula:
[tex]\[
P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
where
- [tex]\((x_1, y_1)\)[/tex] are the coordinates of [tex]\(C\)[/tex]
- [tex]\((x_2, y_2)\)[/tex] are the coordinates of [tex]\(D\)[/tex]
- [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are the parts into which the line segment is divided
Here, the section formula will be adapted because the problem specifies the point [tex]\(P\)[/tex] to be [tex]\( \frac{5}{8} \)[/tex] of the length from [tex]\( D \)[/tex]. This implies [tex]\(m = 5\)[/tex] and [tex]\( n = 3\)[/tex] because:
[tex]\[ \frac{m}{m+n} = \frac{5}{8} \][/tex]
Now, substituting [tex]\(m = 5\)[/tex] and [tex]\(n = 3\)[/tex] into the coordinates:
1. Compute the x-coordinate of [tex]\(P\)[/tex]:
[tex]\[
p_x = \frac{5 \cdot (-8) + 3 \cdot 6}{5 + 3} = \frac{(-40) + 18}{8} = \frac{-22}{8} = -2.75
\][/tex]
2. Compute the y-coordinate of [tex]\(P\)[/tex]:
[tex]\[
p_y = \frac{5 \cdot 4 + 3 \cdot (-6)}{8} = \frac{20 + (-18)}{8} = \frac{2}{8} = 0.25
\][/tex]
Therefore, the coordinates of point [tex]\(P\)[/tex] are [tex]\((-2.75, 0.25)\)[/tex].
Thus, the correct answer is:
- [tex]\((-2.75, 0.25)\)[/tex]
This matches the point we computed!
