To find the coordinates of the point [tex]\( P \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that [tex]\( P \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we can use the section formula. Given points [tex]\( A \left(\frac{-29}{4}, \frac{-3}{2}\right) \)[/tex] and [tex]\( B \left(\frac{25}{4}, \frac{-1}{2}\right) \)[/tex], let's calculate the coordinates of [tex]\( P \)[/tex].
1. Identify the coordinates of points A and B:
- Point [tex]\( A \)[/tex]: [tex]\(\left(\frac{-29}{4}, \frac{-3}{2}\right)\)[/tex]
- Point [tex]\( B \)[/tex]: [tex]\(\left(\frac{25}{4}, \frac{-1}{2}\right)\)[/tex]
2. Section formula:
The section formula for a point [tex]\( P \)[/tex] dividing a line segment [tex]\( AB \)[/tex] internally in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
P = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\][/tex]
In our case, point [tex]\( P \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], so the ratio [tex]\( m:n \)[/tex] is [tex]\( 1:3 \)[/tex] (as [tex]\( \frac{1}{4} \)[/tex] is equivalent to dividing into 1 part and 3 parts).
3. Apply the formula:
[tex]\[
x_P = \frac{1 \cdot x_B + 3 \cdot x_A}{4}
= \frac{1 \cdot \frac{25}{4} + 3 \cdot \frac{-29}{4}}{4}
= \frac{\frac{25}{4} + \frac{-87}{4}}{4}
= \frac{\frac{-62}{4}}{4}
= \frac{-62}{16}
= -3.875
\][/tex]
[tex]\[
y_P = \frac{1 \cdot y_B + 3 \cdot y_A}{4}
= \frac{1 \cdot \frac{-1}{2} + 3 \cdot \frac{-3}{2}}{4}
= \frac{\frac{-1}{2} + \frac{-9}{2}}{4}
= \frac{\frac{-10}{2}}{4}
= \frac{-10}{8}
= -1.25
\][/tex]
4. Coordinates of point [tex]\( P \)[/tex]:
So, the coordinates of point [tex]\( P \)[/tex] are [tex]\((-3.875, -1.25)\)[/tex].
