BC has a midpoint at M(4, 14). Point B is at (5, 10). Find the coordinates of point C.

Write the coordinates as decimals or integers.

To find the coordinates of point \( C \), we can use the midpoint formula. The formula for the midpoint \( M(x_m, y_m) \) of a segment with endpoints \( B(x_1, y_1) \) and \( C(x_2, y_2) \) is:

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

Given:
- \( M(4, 14) \) (midpoint)
- \( B(5, 10) \) (point B)

Let \( C(x_2, y_2) \) be the coordinates of point C.

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

\[
\frac{5 + x_2}{2} = 4 \quad \text{(for the x-coordinate)}
\]
\[
\frac{10 + y_2}{2} = 14 \quad \text{(for the y-coordinate)}
\]

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

### Solving for \( x_2 \):
\[
\frac{5 + x_2}{2} = 4
\]
Multiply both sides by 2:
\[
5 + x_2 = 8
\]
Subtract 5 from both sides:
\[
x_2 = 3
\]

### Solving for \( y_2 \):
\[
\frac{10 + y_2}{2} = 14
\]
Multiply both sides by 2:
\[
10 + y_2 = 28
\]
Subtract 10 from both sides:
\[
y_2 = 18
\]

Thus, the coordinates of point \( C \) are \( (3, 18) \).

Would you like to review any other details or have any questions?

Here are five related questions to explore:
1. How do you find the midpoint if the coordinates of both endpoints are known?
2. What is the distance between points B and C using the distance formula?
3. Can the midpoint formula be used in three dimensions?
4. What are the properties of a line segment when its midpoint and endpoints are given?
5. How do you find the equation of the line passing through points B and C?

**Tip:** Always double-check your calculations when solving midpoint problems to avoid simple arithmetic errors.

Learn how to find the coordinates of point C using the midpoint formula. Given the midpoint M(4, 14) and point B(5, 10), this problem involves basic coordinate geometry concepts. Suitable for students in Grades 7-9.

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