Math Problem Statement
is this correct?
Solution
Yes, the proof appears to be correct. Here’s a detailed breakdown of each statement and its corresponding reason:
-
Statement: Q is the midpoint of PR
Reason: Given
This is given as part of the problem statement. -
Statement: R is the midpoint of QS
Reason: Given
This is also given in the problem. -
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. -
Statement: PQ = RS
Reason: Transitive property
By the transitive property of equality, if PQ = QR and QR = RS, then PQ = RS. -
Statement: PQ + PQ = PQ + RS
Reason: Addition
Adding PQ to both sides of the equation PQ = RS to prepare for the next steps. -
Statement: 2PQ = PQ + RS
Reason: Substitution
The sum of two equal quantities (PQ + PQ) is 2PQ. -
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. -
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. -
Statement: 2PQ = QS
Reason: Transitive property
By the transitive property, since 2PQ = QR + RS and QR + RS = QS, it follows that 2PQ = QS. -
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:
- How does the transitive property work in segment proofs?
- What is the definition of a midpoint in geometry?
- How does the segment addition postulate apply in geometric proofs?
- Why is it necessary to use the division property in the last step?
- 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.
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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
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