To find the coordinates of the point that is [tex]\(\frac{3}{5}\)[/tex] of the way from point [tex]\(A(-9,3)\)[/tex] to point [tex]\(B(21,-2)\)[/tex], we can follow these steps:
1. Identify the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- Point [tex]\(A\)[/tex] has coordinates [tex]\((-9, 3)\)[/tex].
- Point [tex]\(B\)[/tex] has coordinates [tex]\((21, -2)\)[/tex].
2. Calculate the fraction [tex]\(\frac{3}{5}\)[/tex] of the distance between [tex]\(A\)[/tex] and [tex]\(B\)[/tex] along each axis:
- Since we want a point [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex], we can use the formula for finding a point dividing a line segment in a given ratio.
- The general formula for finding a point that divides the line joining [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] is given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
- Here, the ratio is [tex]\(\frac{3}{5}\)[/tex], so [tex]\(m = 3\)[/tex] and [tex]\(n = 2\)[/tex].
3. Apply the formula to find the x-coordinate and y-coordinate:
- The x-coordinate:
[tex]\[
x = x_1 + \frac{3}{5}(x_2 - x_1) = -9 + \frac{3}{5}(21 - (-9)) = -9 + \frac{3}{5}(30) = -9 + 18 = 9
\][/tex]
- The y-coordinate:
[tex]\[
y = y_1 + \frac{3}{5}(y_2 - y_1) = 3 + \frac{3}{5}(-2 - 3) = 3 + \frac{3}{5}(-5) = 3 + (-3) = 0
\][/tex]
4. Combine the coordinates to find the point:
- The point that is [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] has coordinates [tex]\((9, 0)\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{(9, 0)} \][/tex]
