To determine the coordinates of the point that is [tex]\(\frac{3}{5}\)[/tex] of the way from point [tex]\(A\)[/tex] to point [tex]\(B\)[/tex], we can follow these steps:
1. Identify the coordinates of points A and B:
- Point [tex]\(A\)[/tex] has coordinates [tex]\(A(-9, 3)\)[/tex].
- Point [tex]\(B\)[/tex] has coordinates [tex]\(B(21, -2)\)[/tex].
2. Calculate the displacement from point A to point B:
- The change in the x-coordinate from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] is [tex]\(B_x - A_x = 21 - (-9) = 21 + 9 = 30\)[/tex].
- The change in the y-coordinate from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] is [tex]\(B_y - A_y = -2 - 3 = -5\)[/tex].
3. Determine the fraction of the distance we need to move along this displacement:
- We need to move [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex].
4. Calculate the coordinates of the point that is [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex]:
- For the x-coordinate:
[tex]\[
C_x = A_x + \left(\frac{3}{5} \times (B_x - A_x)\right) = -9 + \left(\frac{3}{5} \times 30\right)
\][/tex]
[tex]\[
C_x = -9 + \left(\frac{3}{5} \times 30\right) = -9 + 18 = 9
\][/tex]
- For the y-coordinate:
[tex]\[
C_y = A_y + \left(\frac{3}{5} \times (B_y - A_y)\right) = 3 + \left(\frac{3}{5} \times (-5)\right)
\][/tex]
[tex]\[
C_y = 3 + \left(\frac{3}{5} \times -5\right) = 3 + (-3) = 0
\][/tex]
5. Conclusion:
- The coordinates of the point that is [tex]\(\frac{3}{5}\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] are [tex]\((9, 0)\)[/tex].
Given the multiple-choice options:
- A. [tex]\((3, 1)\)[/tex]
- B. [tex]\((-7, 2)\)[/tex]
- C. [tex]\((9, 0)\)[/tex]
- D. [tex]\((9, 1)\)[/tex]
The correct answer is C. [tex]\((9, 0)\)[/tex].
