https://file.aiquickdraw.com/mathbot/file/1726696163013l7r4t87u.jpg

The image shows a proof involving the midpoint of a segment. Let's break it down:

### Given:
- \( RS = \frac{1}{2}RT \)
- Prove: \( S \) is the midpoint of \( \overline{RT} \).

### Statements and Reasons in the proof:

1. **Statement:** \( RS = \frac{1}{2}RT \)  
   **Reason:** Given.

2. **Statement:** \( 2RS = RT \)  
   **Reason:** Multiplication Property (to eliminate the fraction).

3. **Statement:** \( RS + ST = RT \)  
   **Reason:** Segment Addition Postulate (the total length of a segment is the sum of its parts).

4. **Statement:** \( 2RS = RS + ST \)  
   **Reason:** Substitution (replacing \( RT \) with \( 2RS \)).

5. **Statement:** \( RS = ST \)  
   **Reason:** Simplification (subtracting \( RS \) from both sides).

6. **Statement:** \( S \) is the midpoint of \( RT \).  
   **Reason:** Definition of Midpoint (if \( RS = ST \), then \( S \) is the midpoint).

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This proof shows that by starting from the given information \( RS = \frac{1}{2}RT \), you can use segment properties and algebra to establish that \( RS = ST \), thus proving \( S \) is the midpoint.

Would you like more detailed explanations for any of the steps?

### Here are 5 questions that can further your understanding:
1. Why do we multiply by 2 in step 2?
2. How does the Segment Addition Postulate help in proving midpoints?
3. Can you always use substitution in geometry proofs?
4. What is the significance of step 5 in proving midpoints?
5. How could this proof change if the segments were not congruent?

### Tip:
In geometry, understanding definitions (like midpoint) and properties (like segment addition) is key to constructing and solving proofs logically.

Prove that S is the midpoint of segment RT given RS = 1/2 RT.

Explore the step-by-step solution to proving that S is the midpoint of segment RT using geometric and algebraic principles. This problem highlights key concepts such as the Segment Addition Postulate and the Definition of a Midpoint, suitable for Grades 9-10.

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