https://file.aiquickdraw.com/mathbot/file/1727749527327so4rybqc.jpg

Yes, the proof appears to be correct. Here’s a detailed breakdown of each statement and its corresponding reason:

1. **Statement**: Q is the midpoint of PR  
   **Reason**: Given  
   This is given as part of the problem statement.

2. **Statement**: R is the midpoint of QS  
   **Reason**: Given  
   This is also given in the problem.

3. **Statement**: PQ = QR; QR = RS  
   **Reason**: Definition of midpoint  
   Since Q is the midpoint of PR, it divides PR into two equal segments, making PQ = QR. Similarly, since R is the midpoint of QS, it divides QS into two equal segments, making QR = RS.

4. **Statement**: PQ = RS  
   **Reason**: Transitive property  
   By the transitive property of equality, if PQ = QR and QR = RS, then PQ = RS.

5. **Statement**: PQ + PQ = PQ + RS  
   **Reason**: Addition  
   Adding PQ to both sides of the equation PQ = RS to prepare for the next steps.

6. **Statement**: 2PQ = PQ + RS  
   **Reason**: Substitution  
   The sum of two equal quantities (PQ + PQ) is 2PQ.

7. **Statement**: 2PQ = QR + RS  
   **Reason**: Substitution  
   Using the fact that PQ = QR from step 3, the PQ in the sum can be replaced by QR.

8. **Statement**: QR + RS = QS  
   **Reason**: Segment addition postulate  
   The segment addition postulate states that the sum of the segments QR and RS is the total segment QS.

9. **Statement**: 2PQ = QS  
   **Reason**: Transitive property  
   By the transitive property, since 2PQ = QR + RS and QR + RS = QS, it follows that 2PQ = QS.

10. **Statement**: PQ = ½ QS  
    **Reason**: Division  
    Dividing both sides of the equation 2PQ = QS by 2 gives PQ = ½ QS, which is the statement to be proven.

This proof logically follows the sequence of statements and reasoning, and all steps are justified. The answer is correct!

Would you like further details or have any questions?

Here are 5 related questions you might find useful:
1. How does the transitive property work in segment proofs?
2. What is the definition of a midpoint in geometry?
3. How does the segment addition postulate apply in geometric proofs?
4. Why is it necessary to use the division property in the last step?
5. How can you apply this method to other geometric proofs?

**Tip**: When proving geometric statements, always ensure that each step is justified by a previously established theorem or definition.

Analyze a geometric proof where midpoints and segment addition postulate are used to prove PQ = ½ QS. This solution covers key geometric concepts like midpoints, segment addition, and the transitive property, suitable for students in Grades 8-10.

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