Question

congruence stations
q is the midpoint of segment pr.
solve for qr.
5x-3
11x-3
p q r

Explanation:

Step1: Use midpoint property

Since \( Q \) is the midpoint of \( PR \), \( PQ = QR \). So \( 5x - 3=11x - 3 \)? Wait, no, wait. Wait, actually, \( PQ \) is \( 5x - 3 \) and \( QR \) is \( 11x - 3 \)? Wait, no, maybe I misread. Wait, the segment \( PR \) has \( PQ = 5x - 3 \) and \( QR = 11x - 3 \)? Wait, no, midpoint means \( PQ = QR \), so \( 5x - 3=11x - 3 \)? Wait, that would give \( 5x=11x \), \( - 6x = 0 \), \( x = 0 \). But then \( QR=11(0)-3=-3 \), which doesn't make sense. Wait, maybe the labels are wrong. Wait, maybe \( PQ \) is \( 5x - 3 \) and \( QR \) is \( 11x - 3 \), but actually, the length from \( P \) to \( Q \) and \( Q \) to \( R \) should be equal. Wait, maybe I made a mistake. Wait, let's re - express. If \( Q \) is the midpoint, then \( PQ=QR \). So \( 5x - 3=11x - 3 \)? Wait, subtract \( 5x \) from both sides: \( - 3 = 6x-3 \), then add 3 to both sides: \( 0 = 6x \), so \( x = 0 \). Then \( QR=11(0)-3=-3 \), which is impossible. Wait, maybe the expression for \( PQ \) is \( 5x + 3 \) or \( QR \) is \( 11x + 3 \)? Wait, the original problem says \( 5x - 3 \) and \( 11x - 3 \). Wait, maybe there is a typo, but assuming the problem is correct as given. Wait, no, maybe I misread the segments. Wait, the line is \( P \)---\( Q \)---\( R \), with \( PQ = 5x - 3 \) and \( QR = 11x - 3 \), and \( Q \) is the midpoint, so \( PQ = QR \). So:

\( 5x-3 = 11x - 3 \)

Step2: Solve for \( x \)

Subtract \( 5x \) from both sides:

\( - 3=6x - 3 \)

Add 3 to both sides:

\( 0 = 6x \)

So \( x = 0 \)

Step3: Find \( QR \)

Substitute \( x = 0 \) into \( QR=11x - 3 \):

\( QR=11(0)-3=-3 \)

Wait, this is a negative length, which is impossible. So there must be a mistake in the problem statement. But assuming the problem is as given, following the math:

If we ignore the negative length (maybe a typo, like \( PQ = 5x + 3 \) and \( QR = 11x - 3 \)), but with the given problem:

From \( 5x-3=11x - 3 \), we get \( x = 0 \), then \( QR=-3 \). But since length can't be negative, maybe the problem has a typo. But according to the given equations:

Answer:

\( QR=-3 \) (Note: This result suggests a possible error in the problem's segment length expressions as length cannot be negative in reality.)