Question

find the error justin knows that point r is the midpoint of \\(\overline{qs}\\), and he knows that this means that \\(qr = rs\\). he says that \\(pr = pq + qr\\) by the segment addition postulate. so, \\(pr = pq + rs\\) by substitution. do you agree with justin’s reasoning? explain your reasoning. diagram with points p, q, r, s and dropdowns: \select choice\ the segment addition postulate, \select choice\, and points p, q, and r \select choice\ collinear.

Explanation:

Step1: Recall Segment Addition Postulate

The Segment Addition Postulate states that if three points \( A \), \( B \), \( C \) are collinear and \( B \) is between \( A \) and \( C \), then \( AB + BC = AC \).

Step2: Analyze Justin's Reasoning

Justin claims \( PR = PQ + QR \) by the Segment Addition Postulate. But for the Segment Addition Postulate to apply, points \( P \), \( Q \), \( R \) must be collinear (lie on a straight line). From the diagram, \( PQ \) and \( QR \) are not on a straight line (the angle at \( Q \) shows a bend), so \( P \), \( Q \), \( R \) are not collinear. Thus, the Segment Addition Postulate does not apply to \( P \), \( Q \), \( R \). Even though \( QR = RS \) (since \( R \) is the midpoint of \( \overline{QS} \)), the first step (\( PR = PQ + QR \)) is invalid because the postulate requires collinearity, which is not present here.

Answer:

I do not agree with Justin’s reasoning. The Segment Addition Postulate requires the three points (here, \( P \), \( Q \), \( R \)) to be collinear. From the diagram, \( P \), \( Q \), and \( R \) are not collinear (there is a bend at \( Q \)), so the Segment Addition Postulate does not apply to \( P \), \( Q \), \( R \). Thus, \( PR = PQ + QR \) is not a valid application of the postulate, and the subsequent substitution is also invalid because the initial equation is wrong.