𝑀(− 3; 4 is the midpoint of line with; and . 1 2 ) 𝑁𝑂 𝑁(− 2; 3) 𝑂(𝑠; 𝑡) Determine the value of 𝑠 and 𝑡.

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.

Learn how to determine the coordinates of point O using the midpoint formula. Given that point M(-3, 4) is the midpoint of line segment NO, where N(-2, 3) and O(s, t), solve for the values of s and t. This problem involves coordinate geometry and the application of the midpoint formula.

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