Question

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

diagram of points p, q, r, s, t collinear
find the length of ( qr ).
( \bigcirc ) a. 20
( \bigcirc ) b. 3
( \bigcirc ) c. 9
( \bigcirc ) d. 17

Explanation:

Step1: Analyze segment relations

Given \(PQ = QR = RS\), let \(PQ = QR = RS = x\). Then \(QS=QR + RS=2x\) (since \(Q\) to \(R\) is \(x\) and \(R\) to \(S\) is \(x\)). We know \(QS = 6\), so \(2x=6\)? Wait, no, wait. Wait, actually, let's look at \(PT\). Wait, maybe I made a mistake. Wait, \(PT\) is the total length from \(P\) to \(T\). Let's re - express.

Wait, \(QS = QR+RS\), and \(QR = RS=x\), so \(QS = 2x\). Given \(QS = 6\), so \(2x = 6\)? No, that can't be, because then \(x = 3\), but let's check with \(PT\). Wait, maybe the diagram is \(P - Q - R - S - T\)? Wait, the problem says all points are collinear: \(P\), \(Q\), \(R\), \(S\), \(T\) in order. So \(PQ=QR = RS=x\), so \(PS=PQ + QR+RS = 3x\). Then \(PT = PS+ST=20\). But we also know \(QS=QR + RS = 2x=6\)? Wait, no, \(QS\) is from \(Q\) to \(S\), which is \(QR + RS\), so \(QS=2x\). Given \(QS = 6\), so \(2x=6\) implies \(x = 3\)? But let's check again. Wait, maybe the total length: Wait, if \(PQ = QR=RS=x\), then \(QS = 2x\), so \(x=\frac{QS}{2}=\frac{6}{2}=3\). Wait, but let's confirm with \(PT\). Wait, maybe the diagram has \(P - Q - R - S - T\), so \(PT=PQ + QR+RS+ST\). But we know \(PQ = QR = RS=x\), so \(PT = 3x+ST = 20\). But we don't know \(ST\), but the question is about \(QR\), which is \(x\). Since \(QS = QR + RS=2x = 6\), then \(x = 3\). So \(QR=x = 3\).

Step2: Solve for \(x\)

We have \(QS=QR + RS\). Let \(QR = x\), then \(RS=x\) (since \(QR = RS\)). So \(QS=x + x=2x\). Given \(QS = 6\), we set up the equation \(2x=6\). Solving for \(x\), we divide both sides by 2: \(x=\frac{6}{2}=3\). Since \(QR=x\), the length of \(QR\) is 3.

Answer:

B. 3