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.