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GinnyAnswer
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]

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