 The midpoint of start overline, AB, end overline AB is M, left parenthesis, 3, comma, minus, 4, right parenthesisM(3,−4). If the coordinates of AA are left parenthesis, 7, comma, minus, 3, right parenthesis(7,−3), what are the coordinates of BB?

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.

The midpoint of line AB is M(3,−4). If the coordinates of A are (7,−3), what are the coordinates of B?

Solve a coordinate geometry problem by finding the coordinates of point B, given the midpoint M(3,−4) and point A(7,−3). Learn to apply the midpoint formula with a step-by-step solution. Suitable for Grades 8-10.

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