To determine the values of $s$ and $t$ for the point $O(s, t)$, given that $M(-3, 4)$ is the midpoint of the line segment $NO$, where $N(-2, 3)$ is one endpoint, we can use the midpoint formula.

The midpoint formula states that the coordinates of the midpoint $M(x_m, y_m)$ between two points $N(x_1, y_1)$ and $O(x_2, y_2)$ are given by:

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

Given:

• $M(-3, 4)$ (midpoint)
• $N(-2, 3)$ (one endpoint)
• $O(s, t)$ (the other endpoint)

We can set up the following equations based on the midpoint formula:

$-3 = \frac{-2 + s}{2} \quad \text{(for the x-coordinates)}$ $4 = \frac{3 + t}{2} \quad \text{(for the y-coordinates)}$

Now, solve these equations for $s$ and $t$.

Solving for $s$:

$-3 = \frac{-2 + s}{2}$ Multiply both sides by 2: $-6 = -2 + s$ Add 2 to both sides: $s = -4$

Solving for $t$:

$4 = \frac{3 + t}{2}$ Multiply both sides by 2: $8 = 3 + t$ Subtract 3 from both sides: $t = 5$

Final Answer:

The coordinates of point $O$ are $s = -4$ and $t = 5$.

Do you want further details or have any questions?

Here are 5 related questions to explore further:

1. How would the midpoint change if one of the coordinates of point $N$ changed?
2. What if the midpoint was different but the endpoints $N$ and $O$ were the same?
3. How does the midpoint formula work in three-dimensional space?
4. Can the midpoint formula be used with variables instead of numbers?
5. How would you find the distance between points $N$ and $O$ after finding $s$ and $t$?

Tip: When solving for unknown coordinates using the midpoint formula, ensure that you set up and solve separate equations for the x and y coordinates to avoid confusion.