The midpoint *M* of *GH* has coordinates (16, –11.5). Point *H* has coordinates (14, –10). Find the coordinates of point *G*.




*Write the coordinates as decimals or integers.*

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

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

Given:
- \( M(16, -11.5) \) (midpoint)
- \( H(14, -10) \) (point H)

Let \( G(x_1, y_1) \) be the coordinates of point G.

Using the midpoint formula, we can set up the following equations:

\[
\frac{x_1 + 14}{2} = 16 \quad \text{(for the x-coordinate)}
\]
\[
\frac{y_1 - 10}{2} = -11.5 \quad \text{(for the y-coordinate)}
\]

Now, solve for \( x_1 \) and \( y_1 \):

### Solving for \( x_1 \):
\[
\frac{x_1 + 14}{2} = 16
\]
Multiply both sides by 2:
\[
x_1 + 14 = 32
\]
Subtract 14 from both sides:
\[
x_1 = 18
\]

### Solving for \( y_1 \):
\[
\frac{y_1 - 10}{2} = -11.5
\]
Multiply both sides by 2:
\[
y_1 - 10 = -23
\]
Add 10 to both sides:
\[
y_1 = -13
\]

Thus, the coordinates of point \( G \) are \( (18, -13) \).

Would you like to explore this further or have any questions?

Here are five related questions to consider:
1. How can you verify if the calculated point G is correct using the midpoint formula?
2. How do you determine the length of segment GH?
3. What is the slope of the line segment GH?
4. How would the process change if you had the coordinates of point G and needed to find H?
5. Can the midpoint formula be extended to three-dimensional coordinates?

**Tip:** When working with negative numbers in midpoint problems, carefully handle the signs to ensure accuracy.

Learn how to find the coordinates of point G using the midpoint formula. Given the coordinates of point H and the midpoint M of segment GH, solve for the coordinates of point G step-by-step. Suitable for high school students.

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