Question

the point b lies on the segment \\( \overline{ac} \\) and the coordinates of b so that ab is \\( \frac{1}{4} \\) of ac

Explanation:

Step1: Identify coordinates of A and C

Point \( A(-20, 15) \), Point \( C(4, -1) \)

Step2: Use section formula (internal division)

The section formula for a point \( B(x, y) \) dividing the line segment joining \( A(x_1, y_1) \) and \( C(x_2, y_2) \) in the ratio \( m:n \) is:
\[
x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}
\]
Here, \( AB:BC = 1:3 \) (since \( AB = \frac{1}{4}AC \), so \( m = 1 \), \( n = 3 \))

Step3: Calculate x-coordinate of B

\[
x = \frac{1 \times 4 + 3 \times (-20)}{1 + 3} = \frac{4 - 60}{4} = \frac{-56}{4} = -14
\]

Step4: Calculate y-coordinate of B

\[
y = \frac{1 \times (-1) + 3 \times 15}{1 + 3} = \frac{-1 + 45}{4} = \frac{44}{4} = 11
\]

Answer:

\((-14, 11)\)