To determine the coordinates of point B, we know the coordinates of point A and the midpoint of the line segment joining points A and B. Given:
- Point A: \(\((A_x, A_y) = (3, 5)\)\)
- Midpoint: \(\((\text{mid}_x, \text{mid}_y) = (5, 7)\)\)
We use the midpoint formula, which states that the coordinates of the midpoint \(\((\text{mid}_x, \text{mid}_y)\)\) of a line segment joining points \(\((A_x, A_y)\)\) and \(\((B_x, B_y)\)\) can be calculated as:
\(\[
\text{mid}_x = \frac{A_x + B_x}{2}
\]\)
\(\[
\text{mid}_y = \frac{A_y + B_y}{2}
\]\)
We know \(\(\text{mid}_x = 5\)\) and \(\(\text{mid}_y = 7\)\). Let's solve for \(\(B_x\)\) and \(\(B_y\)\):
1. For the x-coordinate:
\(\[
5 = \frac{3 + B_x}{2}
\]\)
Multiply both sides by 2 to solve for \(\(B_x\)\):
\(\[
10 = 3 + B_x
\]\)
Subtract 3 from both sides:
\(\[
B_x = 7
\]\)
2. For the y-coordinate:
\(\[
7 = \frac{5 + B_y}{2}
\]\)
Multiply both sides by 2 to solve for \(\(B_y\)\):
\(\[
14 = 5 + B_y
\]\)
Subtract 5 from both sides:
\(\[
B_y = 9
\]\)
Therefore, the coordinates of point B are:
\(\[
(B_x, B_y) = (7, 9)
\]\)
