To find the coordinates of point $B$, we use the midpoint formula. The midpoint $M(x_m, y_m)$ of a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by:

$M(x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

We are given the following:

• Midpoint $M(3, -4)$
• Point $A(7, -3)$
• We need to find the coordinates of point $B(x_2, y_2)$.

Using the midpoint formula, we can set up two equations, one for the $x$-coordinates and one for the $y$-coordinates:

$\frac{7 + x_2}{2} = 3 \quad \text{(for the x-coordinates)}$ $\frac{-3 + y_2}{2} = -4 \quad \text{(for the y-coordinates)}$

Solving for $x_2$:

$\frac{7 + x_2}{2} = 3$ Multiply both sides by 2: $7 + x_2 = 6$ Subtract 7 from both sides: $x_2 = 6 - 7 = -1$

Solving for $y_2$:

$\frac{-3 + y_2}{2} = -4$ Multiply both sides by 2: $-3 + y_2 = -8$ Add 3 to both sides: $y_2 = -8 + 3 = -5$

So, the coordinates of point $B$ are $(-1, -5)$.

Would you like further details, or do you have any questions?

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Here are 5 related questions for deeper understanding:

1. What would happen if the midpoint was located at the origin?
2. How can you verify the solution using distance formulas?
3. What if point $A$ was at the origin—what would $B$'s coordinates be?
4. How do the midpoint and the slope of the line relate geometrically?
5. Can the midpoint formula be used for three-dimensional points?

Tip: Always double-check midpoint calculations by substituting back the result into the formula.