To find the coordinates of the other endpoint $S(x, y)$ of the segment $RS$ given the endpoint $R(x_1, y_1)$ and the midpoint $M(x_m, y_m)$, we use the midpoint formula:

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

Given:

• $R(x_1, y_1) = (x_1, y_1)$
• $M(x_m, y_m) = (x_m, y_m)$

Let the coordinates of the other endpoint $S$ be $(x_2, y_2)$.

Steps to find $S(x_2, y_2)$:

1. Solve for $x_2$: $x_m = \frac{x_1 + x_2}{2}$ Multiply both sides by 2 to solve for $x_2$: $2x_m = x_1 + x_2$ Subtract $x_1$ from both sides: $x_2 = 2x_m - x_1$

2. Solve for $y_2$: $y_m = \frac{y_1 + y_2}{2}$ Multiply both sides by 2 to solve for $y_2$: $2y_m = y_1 + y_2$ Subtract $y_1$ from both sides: $y_2 = 2y_m - y_1$

Final Answer:

The coordinates of the other endpoint $S$ are $\boxed{(2x_m - x_1, 2y_m - y_1)}$.

Would you like to see an example using specific coordinates, or is there anything else you'd like to know?

Here are 5 related questions:

1. What if the midpoint and one endpoint are on a diagonal line—how does that affect the calculation?
2. How can you verify that your calculated endpoint $S$ is correct?
3. Can you apply the same method to find a missing endpoint in 3D space?
4. How do these concepts extend to finding the center of mass in physics?
5. What is the significance of the midpoint in the context of geometric constructions?

Tip: Always double-check your final answer by substituting back into the original midpoint formula to ensure consistency.