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The problem describes that the midpoint of the segment TS is the origin (0,0), and point T is located in Quadrant II. We are tasked with determining which quadrant contains point S.

### Step-by-Step Solution:

1. **Understanding the Midpoint Formula**:  
   The midpoint \( M \) of a line segment connecting two points \( T(x_T, y_T) \) and \( S(x_S, y_S) \) is given by:
   \[
   M = \left( \frac{x_T + x_S}{2}, \frac{y_T + y_S}{2} \right)
   \]
   In this case, the midpoint \( M \) is the origin, \( (0,0) \).

2. **Set up the midpoint condition**:  
   Since the midpoint is \( (0, 0) \), we can write:
   \[
   \frac{x_T + x_S}{2} = 0 \quad \text{and} \quad \frac{y_T + y_S}{2} = 0
   \]
   Solving these equations:
   \[
   x_T + x_S = 0 \quad \text{and} \quad y_T + y_S = 0
   \]
   This simplifies to:
   \[
   x_S = -x_T \quad \text{and} \quad y_S = -y_T
   \]

3. **Determine coordinates of point S**:  
   The coordinates of point S are the negatives of point T’s coordinates.

4. **Quadrant II coordinates**:  
   In Quadrant II, point T has the coordinates \( (x_T, y_T) \) such that:
   \[
   x_T < 0 \quad \text{and} \quad y_T > 0
   \]
   Thus, for point S, the coordinates are:
   \[
   x_S = -x_T > 0 \quad \text{and} \quad y_S = -y_T < 0
   \]
   Therefore, point S has a positive x-coordinate and a negative y-coordinate.

5. **Conclusion**:  
   Point S must be located in **Quadrant IV**, where \( x > 0 \) and \( y < 0 \).

Would you like further details on this explanation, or do you have any other questions?

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Here are 5 related questions for further exploration:

1. How would the answer change if point T were in Quadrant I?
2. What are the coordinates of point S if point T is at (-3, 4)?
3. What if the midpoint was not at the origin? How would the solution differ?
4. Can you derive the formula for the midpoint of a segment in 3D space?
5. How does the concept of symmetry help in understanding the quadrants in this problem?

**Tip**: In geometry problems involving symmetry, remember that the midpoint evenly divides a line segment, so the coordinates of the endpoints will always mirror each other across the midpoint.

The midpoint of TS is the origin. Point T is located in Quadrant II. Which quadrant contains Point S?

Solve a geometry problem involving the midpoint of a line segment where Point T is in Quadrant II and the midpoint is at the origin. Learn how to determine which quadrant contains Point S using the midpoint formula and symmetry principles. Suitable for students in Grades 9-10.

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