To determine the coordinates of the point that is [tex]\(\frac{3}{8}\)[/tex] of the way from point [tex]\(A(-8, -9)\)[/tex] to point [tex]\(B(24, -1)\)[/tex], we can use the section formula for dividing a line segment in a given ratio.
The section formula states that the coordinates of a point dividing the line segment joining [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] are given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Here, [tex]\(A(x_1, y_1) = (-8, -9)\)[/tex] and [tex]\(B(x_2, y_2) = (24, -1)\)[/tex], and the ratio [tex]\(m:n = \frac{3}{8}\)[/tex] implies [tex]\(m = 3\)[/tex] and [tex]\(n = 8 - 3 = 5\)[/tex].
Let's find the x-coordinate of the new point.
[tex]\[
x_{new} = \frac{m \cdot x_2 + n \cdot x_1}{m + n} = \frac{3 \cdot 24 + 5 \cdot (-8)}{3+5} = \frac{72 - 40}{8} = \frac{32}{8} = 4
\][/tex]
Next, let's find the y-coordinate of the new point.
[tex]\[
y_{new} = \frac{m \cdot y_2 + n \cdot y_1}{m + n} = \frac{3 \cdot (-1) + 5 \cdot (-9)}{3+5} = \frac{-3 - 45}{8} = \frac{-48}{8} = -6
\][/tex]
Therefore, the coordinates of the point that is [tex]\(\frac{3}{8}\)[/tex] of the way from [tex]\(A(-8, -9)\)[/tex] to [tex]\(B(24, -1)\)[/tex] are [tex]\((4, -6)\)[/tex].
So, the correct answer is:
B. [tex]\( (4, -6) \)[/tex]
