Math Problem Statement

is this correct?

Is this proof correct?

Solution

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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Midpoints
Segment Addition
Transitive Property

Formulas

Segment Addition: PQ + QR = PR
Definition of Midpoint: Midpoint divides a segment into two equal parts

Theorems

Transitive Property
Segment Addition Postulate
Division Property

Suitable Grade Level

Grades 8-10