The midpoint of start overline, AB, end overline 
AB
  is M, left parenthesis, minus, 1, comma, minus, 5, right parenthesisM(−1,−5). If the coordinates of AA are left parenthesis, 2, comma, minus, 4, right parenthesis(2,−4), what are the coordinates of BB?

To find the coordinates of point \( B \), given that \( M(-1, -5) \) is the midpoint of \( \overline{AB} \) and the coordinates of \( A(2, -4) \), we can use the midpoint formula. The midpoint formula is given by:

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

Here, \( M(-1, -5) \) is the midpoint, \( A(2, -4) \) are the coordinates of point \( A \), and \( B(x_2, y_2) \) are the coordinates of point \( B \). 

We can set up the following equations based on the midpoint formula:

\[
-1 = \frac{2 + x_2}{2}
\]
\[
-5 = \frac{-4 + y_2}{2}
\]

Now, solve for \( x_2 \) and \( y_2 \).

1. For the x-coordinate:
\[
-1 = \frac{2 + x_2}{2}
\]
Multiply both sides by 2:
\[
-2 = 2 + x_2
\]
Subtract 2 from both sides:
\[
x_2 = -4
\]

2. For the y-coordinate:
\[
-5 = \frac{-4 + y_2}{2}
\]
Multiply both sides by 2:
\[
-10 = -4 + y_2
\]
Add 4 to both sides:
\[
y_2 = -6
\]

Therefore, the coordinates of point \( B \) are \( (-4, -6) \).

Would you like more details or have any questions?

Here are five related questions you might find interesting:
1. How do you find the distance between two points?
2. What is the formula for finding the slope of a line segment between two points?
3. How can you determine if three points are collinear?
4. What is the equation of the line passing through points \( A(2, -4) \) and \( B(-4, -6) \)?
5. How do you find the area of a triangle given the coordinates of its vertices?

**Tip:** When using the midpoint formula, remember that the midpoint divides the line segment into two equal parts, so it's a great tool to find an unknown endpoint if the other endpoint and the midpoint are known.

The midpoint of AB is M(-1, -5). If the coordinates of A are (2, -4), what are the coordinates of B?

Explore how to find the coordinates of point B when the midpoint M(-1, -5) and point A(2, -4) are given. This problem highlights the use of the midpoint formula, a key concept in coordinate geometry. Ideal for students in Grades 8-10.

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