Question

1. in the diagram below, all points are collinear. ( pt = 20 ), ( qs = 6 ), and ( pq = qr = rs ). diagram find the length of ( qr ).

a. 20
b. 3
c. 9
d. 17

1. in the diagram below, all points are collinear. ( pt = 20 ), ( qs = 6 ), and ( pq = qr = rs ). diagram find the length of ( pq ).

a. 20
b. 3
c. 9
d. 17

Explanation:

Question 8

Step1: Analyze QS segment

Given \( QS = 6 \) and \( QS=QR + RS \), also \( QR = RS \). Let \( QR = x \), then \( RS=x \). So \( QS=x + x=2x \).
\( 2x = 6 \)

Step2: Solve for x

Divide both sides by 2: \( x=\frac{6}{2}=3 \)? Wait, no, wait. Wait, the total length \( PT = 20 \), and \( PQ=QR = RS \). Let \( PQ = QR=RS = x \). Then the length from \( P \) to \( S \) is \( PQ+QR + RS=3x \). Then \( ST=PT - PS=20 - 3x \). But for QS, \( QS = QR+RS=2x \), and \( QS = 6 \), so \( 2x=6 \)? No, that can't be, because then \( 3x = 9 \), and \( ST=20 - 9 = 11 \), but maybe I misread. Wait, the problem says \( PQ = QR = RS \), so each of those is equal. Let \( PQ=QR=RS = x \). Then \( QS=QR + RS=x + x = 2x \), and \( QS = 6 \), so \( 2x=6 \), so \( x = 3 \)? But wait, then \( PT=PQ + QR+RS + ST=3x+ST = 20 \), but maybe the diagram is \( P---Q---R---S---T \), so \( PQ=QR=RS=x \), so \( PS = 3x \), and \( QS=QR + RS=2x = 6 \), so \( x = 3 \)? But then option B is 3. Wait, but let's check again. Wait, maybe I made a mistake. Wait, the problem says "Find the length of QR". If \( QS = 6 \) and \( QS = QR + RS \), and \( QR = RS \), then \( QR=\frac{QS}{2}=\frac{6}{2}=3 \). So that's correct.

Step1: Use given equal segments

Given \( PQ = QR = RS \), let each be \( x \). So \( PQ=x \), \( QR=x \), \( RS=x \). The segment \( QS=QR + RS=x + x = 2x \), and \( QS = 6 \), so \( 2x=6 \), so \( x = 3 \)? Wait, no, wait. Wait, \( PT = 20 \), and \( PT=PQ + QR+RS + ST \). But since \( PQ=QR=RS=x \), then \( PS = 3x \), and \( QS=2x = 6 \), so \( x = 3 \). Therefore, \( PQ=x = 3 \)? But wait, that would mean \( PS = 9 \), and \( ST=20 - 9 = 11 \), but the question is about \( PQ \). Since \( PQ = QR = RS \), and \( QS = 6 = QR + RS = 2x \), so \( x = 3 \). So \( PQ = 3 \).

Answer:

B. 3

Question 9