The midpoint of $\overline{JK}$ is $M\left(6,\ 3\right)$ . One endpoint is $J\left(14,\ 9\right)$ . Find the coordinates of endpoint $K$ .

We are given:

- The midpoint $ M(6, 3) $,
- One endpoint $ J(14, 9) $,
- We need to find the coordinates of the other endpoint $ K(x, y) $.

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

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

Using the known midpoint and the coordinates of $ J $, we set up the following equations:

\[
\frac{14 + x}{2} = 6
\]
\[
\frac{9 + y}{2} = 3
\]

### Solving for $ x $:
\[
\frac{14 + x}{2} = 6
\]
Multiply both sides by 2:
\[
14 + x = 12
\]
Subtract 14 from both sides:
\[
x = -2
\]

### Solving for $ y $:
\[
\frac{9 + y}{2} = 3
\]
Multiply both sides by 2:
\[
9 + y = 6
\]
Subtract 9 from both sides:
\[
y = -3
\]

Thus, the coordinates of the other endpoint $ K $ are $ (-2, -3) $.

### Final answer:
The coordinates of endpoint $ K $ are $ (-2, -3) $.

Would you like further details or have any questions?

### Related Questions:
1. How would you find the distance between points $ J $ and $ K $?
2. What is the slope of the line segment $ \overline{JK} $?
3. How can you verify that $ M(6, 3) $ is indeed the midpoint of $ \overline{JK} $?
4. Can you generalize this method for 3-dimensional coordinates?
5. How would this problem change if the midpoint formula were applied in the context of a parametric equation?

### Tip:
Always double-check your calculations when using formulas to avoid small mistakes that can lead to incorrect results.

The midpoint of $\overline{JK}$ is $M(6, 3)$. One endpoint is $J(14, 9)$. Find the coordinates of endpoint $K$.

Learn how to find the coordinates of the other endpoint of a line segment when the midpoint and one endpoint are given. This problem covers key geometric concepts such as the midpoint formula and solving for unknown coordinates. Suitable for students in Grades 8-10.

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