To determine the coordinates of the point that is [tex]\(\frac{3}{10}\)[/tex] of the way from point [tex]\(A\)[/tex] to point [tex]\(B\)[/tex], we need to apply the concept of the section formula. Here's a detailed, step-by-step solution:
Given:
- Coordinates of point [tex]\(A\)[/tex] are [tex]\( (2, 3) \)[/tex]
- Coordinates of point [tex]\(B\)[/tex] are [tex]\( (8, 15) \)[/tex]
We are to find a point [tex]\(P\)[/tex] that is [tex]\(\frac{3}{10}\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex].
1. Find the fraction along the x- and y-coordinates:
- The difference in the x-coordinates of [tex]\(B\)[/tex] and [tex]\(A\)[/tex] is:
[tex]\[
\Delta x = B_x - A_x = 8 - 2 = 6
\][/tex]
- The difference in the y-coordinates of [tex]\(B\)[/tex] and [tex]\(A\)[/tex] is:
[tex]\[
\Delta y = B_y - A_y = 15 - 3 = 12
\][/tex]
2. Multiply these differences by the fraction [tex]\(\frac{3}{10}\)[/tex]:
[tex]\[
\Delta x_{\text{fraction}} = \frac{3}{10} \times 6 = 1.8
\][/tex]
[tex]\[
\Delta y_{\text{fraction}} = \frac{3}{10} \times 12 = 3.6
\][/tex]
3. Add these results to the coordinates of point [tex]\(A\)[/tex]:
[tex]\[
P_x = A_x + \Delta x_{\text{fraction}} = 2 + 1.8 = 3.8
\][/tex]
[tex]\[
P_y = A_y + \Delta y_{\text{fraction}} = 3 + 3.6 = 6.6
\][/tex]
Therefore, the coordinates of the point that is [tex]\(\frac{3}{10}\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] are:
[tex]\[
(3.8, 6.6)
\][/tex]
