To find the coordinates of point $S$ given the midpoint $M$ of $ST$ and the coordinates of $T$, we can use the midpoint formula.

The midpoint $M$ of a line segment $ST$ with endpoints $S(x_1, y_1)$ and $T(x_2, y_2)$ is given by:

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

In this problem:

• The midpoint $M$ has coordinates $(9.5, 2)$.
• The coordinates of $T$ are $(4, 1)$.

Let $S$ have coordinates $(x, y)$. Plugging into the midpoint formula:

$\left( \frac{x + 4}{2}, \frac{y + 1}{2} \right) = (9.5, 2)$

Now, we solve for $x$ and $y$ by setting up two equations:

1. $\frac{x + 4}{2} = 9.5$
2. $\frac{y + 1}{2} = 2$

Solving for $x$:

$\frac{x + 4}{2} = 9.5$ Multiply both sides by 2: $x + 4 = 19$ Subtract 4 from both sides: $x = 15$

Solving for $y$:

$\frac{y + 1}{2} = 2$ Multiply both sides by 2: $y + 1 = 4$ Subtract 1 from both sides: $y = 3$

Conclusion

The coordinates of point $S$ are $(15, 3)$.

Would you like further explanation on the steps, or do you have additional questions?

Here are some related questions:

1. What is the midpoint formula, and how is it derived?
2. How would you find a missing endpoint if only the midpoint and one endpoint are known?
3. What are some real-life applications of the midpoint formula?
4. How can you verify if a point is indeed the midpoint of a segment?
5. How would you use this method in three-dimensional space?

Tip: When finding a missing endpoint, always remember to double the midpoint values before subtracting the known endpoint's coordinates.