The midpoint of start overline, AB, end overlineAB is M, left parenthesis, minus, 1, comma, 5, right parenthesis*M*(−1,5). If the coordinates of A*A* are left parenthesis, minus, 7, comma, 7, right parenthesis(−7,7), what are the coordinates of B*B*?

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

\[
M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

Given:
- Midpoint \( M(-1, 5) \)
- Coordinates of \( A(-7, 7) \)

Let the coordinates of \( B \) be \( (x_2, y_2) \). Plugging the values into the midpoint formula:

\[
\frac{-7 + x_2}{2} = -1
\]
\[
\frac{7 + y_2}{2} = 5
\]

Now, solve these equations:

1. Solve for \( x_2 \):

\[
\frac{-7 + x_2}{2} = -1
\]
Multiply both sides by 2:
\[
-7 + x_2 = -2
\]
Add 7 to both sides:
\[
x_2 = 5
\]

2. Solve for \( y_2 \):

\[
\frac{7 + y_2}{2} = 5
\]
Multiply both sides by 2:
\[
7 + y_2 = 10
\]
Subtract 7 from both sides:
\[
y_2 = 3
\]

Therefore, the coordinates of \( B \) are \( \mathbf{(5, 3)} \).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the midpoint formula change if applied in three-dimensional space?
2. What is the distance between points \( A(-7, 7) \) and \( B(5, 3) \)?
3. If you know the coordinates of the midpoint and one endpoint, how can you find the length of the line segment?
4. How does the midpoint formula relate to the concept of the centroid of a triangle?
5. Can the midpoint formula be used to find the center of a circle? If so, how?

**Tip:** Always double-check your signs when using the midpoint formula, as a small mistake in sign can lead to an incorrect answer.

Learn how to find the coordinates of point B given the midpoint and coordinates of point A using the midpoint formula. This problem involves basic coordinate geometry and is suitable for high school students.

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