To determine the coordinates of the point that is [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\(A(-9, 3)\)[/tex] to [tex]\(B(21, -2)\)[/tex], we need to follow these steps:
1. Identify the coordinates of points A and B:
- [tex]\(A = (-9, 3)\)[/tex]
- [tex]\(B = (21, -2)\)[/tex]
2. Calculate the ratio which is [tex]\(\frac{3}{5}\)[/tex]:
- This indicates that the point we are looking for is [tex]\(\frac{3}{5}\)[/tex] of the total distance from A to B.
3. Determine the changes in the x-coordinate and y-coordinate from A to B:
- Change in x-coordinate ([tex]\(\Delta x\)[/tex]) = [tex]\(B_x - A_x = 21 - (-9) = 21 + 9 = 30\)[/tex]
- Change in y-coordinate ([tex]\(\Delta y\)[/tex]) = [tex]\(B_y - A_y = -2 - 3 = -5\)[/tex]
4. Apply the ratio to the changes in the coordinates:
- For the x-coordinate, the point is [tex]\(\frac{3}{5}\)[/tex] of the way: [tex]\[\text{New } x = A_x + \left(\frac{3}{5}\right) \times \Delta x = -9 + \left(\frac{3}{5}\right) \times 30\][/tex]
[tex]\[= -9 + 18 = 9\][/tex]
- For the y-coordinate, the point is [tex]\(\frac{3}{5}\)[/tex] of the way: [tex]\[\text{New } y = A_y + \left(\frac{3}{5}\right) \times \Delta y = 3 + \left(\frac{3}{5}\right) \times (-5)\][/tex]
[tex]\[= 3 - 3 = 0\][/tex]
5. Combine the new coordinates:
- Hence, the coordinates of the point that is [tex]\(\frac{3}{5}\)[/tex] of the way from A to B are [tex]\((9, 0)\)[/tex].
So, the correct answer is [tex]\(D\)[/tex].
