https://r2.erweima.ai/mathbot/image/17260976828221ra2t20o.jpg

The problem in the image asks to find the coordinates of the other endpoint of a segment, given the midpoint \( M(-7, 8) \) and one endpoint \( Q(-19, 30) \).

To find the other endpoint, we can use the midpoint formula, which is given by:

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

Where:
- \( M(x, y) \) are the coordinates of the midpoint.
- \( (x_1, y_1) \) are the coordinates of one endpoint (which is \( Q(-19, 30) \)).
- \( (x_2, y_2) \) are the coordinates of the other endpoint (which we need to find).

### Step-by-step Solution:

1. Let the other endpoint be \( P(x_2, y_2) \).
2. Plug the values for the midpoint \( M(-7, 8) \) and the known endpoint \( Q(-19, 30) \) into the midpoint formula.

   - For the \( x \)-coordinates:  
   \[
   -7 = \frac{-19 + x_2}{2}
   \]

   - For the \( y \)-coordinates:  
   \[
   8 = \frac{30 + y_2}{2}
   \]

3. Solve for \( x_2 \) and \( y_2 \):

   - Multiply both sides of the first equation by 2:
   \[
   -14 = -19 + x_2
   \]
   Add 19 to both sides:
   \[
   x_2 = 5
   \]

   - Multiply both sides of the second equation by 2:
   \[
   16 = 30 + y_2
   \]
   Subtract 30 from both sides:
   \[
   y_2 = -14
   \]

### Conclusion:
The coordinates of the other endpoint \( P \) are \( (5, -14) \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the midpoint between two given endpoints?
2. What is the geometric significance of the midpoint in a line segment?
3. How can we use the distance formula to verify the length of a segment?
4. If both endpoints of a segment are given, how can you find its length?
5. Can the midpoint formula be used in 3D geometry? How would it work?

**Tip:** When using the midpoint formula, ensure the coordinates are accurately substituted to avoid small errors that can lead to incorrect results.

The coordinates of midpoint M and endpoint Q of a segment are M(-7, 8) and Q(-19, 30). Find the coordinates of the other endpoint.

Learn how to find the coordinates of the other endpoint of a line segment, given the midpoint M(-7, 8) and one endpoint Q(-19, 30). This solution uses the midpoint formula to calculate the unknown endpoint. Suitable for students in Grades 9-11.

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