To determine the coordinates of the point that is [tex]\(\frac{1}{3}\)[/tex] of the way from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex], let's start by denoting the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Let [tex]\( A \)[/tex] be at [tex]\((0, 0)\)[/tex].
- Let [tex]\( B \)[/tex] be at [tex]\((3, 3)\)[/tex].
Step-by-Step Solution:
1. Calculate the change in coordinates between [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
The horizontal change (difference in x-coordinates) is:
[tex]\[
\Delta x = B_x - A_x = 3 - 0 = 3
\][/tex]
The vertical change (difference in y-coordinates) is:
[tex]\[
\Delta y = B_y - A_y = 3 - 0 = 3
\][/tex]
2. Determine the fraction of these changes that corresponds to [tex]\(\frac{1}{3}\)[/tex]:
The x-distance that corresponds to [tex]\(\frac{1}{3}\)[/tex] of the way is:
[tex]\[
\frac{1}{3} \times \Delta x = \frac{1}{3} \times 3 = 1
\][/tex]
The y-distance that corresponds to [tex]\(\frac{1}{3}\)[/tex] of the way is:
[tex]\[
\frac{1}{3} \times \Delta y = \frac{1}{3} \times 3 = 1
\][/tex]
3. Calculate the coordinates of the point that is [tex]\(\frac{1}{3}\)[/tex] of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]:
Adding these distances to the coordinates of point [tex]\( A \)[/tex]:
[tex]\[
\text{New } x = A_x + \frac{1}{3} \times (B_x - A_x) = 0 + 1 = 1
\][/tex]
[tex]\[
\text{New } y = A_y + \frac{1}{3} \times (B_y - A_y) = 0 + 1 = 1
\][/tex]
Thus, the coordinates of the point that is [tex]\(\frac{1}{3}\)[/tex] of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] are [tex]\((1.0, 1.0)\)[/tex].
