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