To find the coordinates of the other endpoint $S(x_1, y_1)$, given that the midpoint is $M(7, -1)$ and one endpoint is $T(6, 4)$, we can use the midpoint formula.

The midpoint formula is:

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

where $M(x_m, y_m)$ is the midpoint and $T(x_2, y_2)$ is the other endpoint.

We are given:

• $M(7, -1)$
• $T(6, 4)$

From the midpoint formula, we have the following system of equations:

$\frac{x_1 + 6}{2} = 7 \quad \text{(1)}$ $\frac{y_1 + 4}{2} = -1 \quad \text{(2)}$

Solving for $x_1$ and $y_1$:

For equation (1):

$\frac{x_1 + 6}{2} = 7$ Multiply both sides by 2: $x_1 + 6 = 14$ Subtract 6 from both sides: $x_1 = 14 - 6 = 8$

For equation (2):

$\frac{y_1 + 4}{2} = -1$ Multiply both sides by 2: $y_1 + 4 = -2$ Subtract 4 from both sides: $y_1 = -2 - 4 = -6$

Conclusion:

The coordinates of the other endpoint $S$ are $S(8, -6)$.

Let me know if you would like further clarification or have any additional questions!

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Here are some related questions to explore:

1. What is the midpoint of two points given their coordinates?
2. How can you derive the midpoint formula from basic geometric principles?
3. How do you find the distance between two points on a coordinate plane?
4. What is the formula to find the coordinates of the centroid of a triangle?
5. How can you use the midpoint formula in 3D geometry?

Tip: Always ensure you understand the midpoint formula, as it's widely applicable in problems involving symmetry or geometric properties.