To find 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. The section formula helps us determine the coordinates of a point that divides a line segment into a particular ratio.
Given points [tex]\(A(x_1, y_1) = (-8, -9)\)[/tex] and [tex]\(B(x_2, y_2) = (24, -1)\)[/tex], and given that the fraction [tex]\(\frac{3}{8}\)[/tex] represents the part of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex], we can find the coordinates [tex]\((x, y)\)[/tex] of the desired point as follows:
1. Calculate the total difference in the x-coordinates, [tex]\(x_2 - x_1\)[/tex], and the y-coordinates, [tex]\(y_2 - y_1\)[/tex]:
[tex]\[
x_2 - x_1 = 24 - (-8) = 24 + 8 = 32
\][/tex]
[tex]\[
y_2 - y_1 = -1 - (-9) = -1 + 9 = 8
\][/tex]
2. Multiply these differences by the given fraction [tex]\(\frac{3}{8}\)[/tex] to find the distances along the x-axis and y-axis from [tex]\(A\)[/tex] to the point:
[tex]\[
\text{x-distance} = \frac{3}{8} \times 32 = \frac{96}{8} = 12
\][/tex]
[tex]\[
\text{y-distance} = \frac{3}{8} \times 8 = \frac{24}{8} = 3
\][/tex]
3. Add these distances to the coordinates of [tex]\(A\)[/tex] to get the coordinates of the new point:
[tex]\[
x = x_1 + \text{x-distance} = -8 + 12 = 4
\][/tex]
[tex]\[
y = y_1 + \text{y-distance} = -9 + 3 = -6
\][/tex]
So, 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].
Thus, the correct answer is:
A. [tex]\((4, -6)\)[/tex]
