Question

$\overline{pq} \parallel \overline{rt}$. find $qs$.

(there is a diagram with points p, t, s on a horizontal line where pt = 39, ts = 26, sr = 20, and triangle-like figure with pq parallel to rt, and we need to find qs.)

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{PQ} \parallel \overline{RT}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle SRT \sim \triangle SQP\) (similar triangles).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{ST}{SP}=\frac{SR}{SQ}\). Let \(QS = x\), then \(SR = x - 20\), \(ST = 26\), and \(SP=39 + 26=65\). Substituting these values, we get \(\frac{26}{65}=\frac{20}{x - 20}\)? Wait, no, correct proportion: \(\frac{ST}{SP}=\frac{SR}{SQ}\) should be \(\frac{ST}{SP}=\frac{SR}{SQ}\)? Wait, actually, corresponding sides: \(ST\) corresponds to \(SP\), \(SR\) corresponds to \(SQ\), and \(RT\) corresponds to \(PQ\). Wait, maybe better: \(\frac{ST}{SP}=\frac{SR}{SQ}\) is incorrect. Let's re - establish: Since \(\triangle SRT \sim \triangle SQP\), \(\frac{ST}{SP}=\frac{SR}{SQ}\). Wait, \(SP=PT + TS=39 + 26 = 65\), \(ST = 26\), \(SR = 20\), \(SQ=SR + RQ\)? No, wait, \(SQ\) is the side we need to find, and \(SR = 20\), so let \(SQ=x\), then the ratio of sides: \(\frac{ST}{SP}=\frac{SR}{SQ}\) is wrong. The correct ratio is \(\frac{ST}{SP}=\frac{SR}{SQ}\)? Wait, no, the similarity ratio: \(\triangle SRT\) and \(\triangle SQP\), so \(\frac{ST}{SP}=\frac{SR}{SQ}=\frac{RT}{PQ}\). Let's use \(\frac{ST}{SP}=\frac{SR}{SQ}\). So \(ST = 26\), \(SP=39 + 26=65\), \(SR = 20\), \(SQ=x\). Then \(\frac{26}{65}=\frac{20}{x}\)? Wait, no, that's not right. Wait, maybe the other way: \(\frac{PT}{ST}=\frac{QR}{SR}\)? No, let's think again. Since \(PQ\parallel RT\), \(\angle SRT=\angle SQP\) and \(\angle STR=\angle SPQ\) (corresponding angles), so \(\triangle SRT\sim\triangle SQP\) by AA similarity. Therefore, \(\frac{ST}{SP}=\frac{SR}{SQ}\). Wait, \(SP = PT+TS = 39 + 26=65\), \(ST = 26\), \(SR = 20\), \(SQ=x\). So \(\frac{26}{65}=\frac{20}{x}\)? Solving for \(x\): Cross - multiply, \(26x=65\times20\), \(26x = 1300\), \(x=\frac{1300}{26}=50\). Wait, but let's check the ratio again. Alternatively, \(\frac{PT}{ST}=\frac{QR}{SR}\), but we don't know \(QR\). Wait, maybe the ratio of \(PT\) to \(ST\) is equal to the ratio of \(QR\) to \(SR\), but we need \(SQ=SR + RQ\). Wait, \(PT = 39\), \(ST = 26\), so \(\frac{PT}{ST}=\frac{39}{26}=\frac{3}{2}\). So the ratio of similarity is \(\frac{3}{2}\) (since \(\triangle SQP\) is larger than \(\triangle SRT\)). So \(SQ=SR\times\frac{3 + 2}{2}\)? No, wait, if \(\triangle SRT\sim\triangle SQP\) with ratio \(\frac{ST}{SP}=\frac{26}{65}=\frac{2}{5}\)? No, I think I made a mistake earlier. Let's start over.

Since \(PQ\parallel RT\), by the Basic Proportionality Theorem (Thales' theorem) in \(\triangle SQP\), the line \(RT\) is parallel to \(PQ\) and intersects \(SP\) at \(T\) and \(SQ\) at \(R\). So \(\frac{ST}{PT}=\frac{SR}{RQ}\)? No, Thales' theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So in \(\triangle SQP\), line \(RT\parallel PQ\), so it intersects \(SP\) at \(T\) and \(SQ\) at \(R\). Therefore, \(\frac{ST}{PT}=\frac{SR}{RQ}\). We know \(ST = 26\), \(PT = 39\), \(SR = 20\). So \(\frac{26}{39}=\frac{20}{RQ}\). Simplify \(\frac{26}{39}=\frac{2}{3}\), so \(\frac{2}{3}=\frac{20}{RQ}\), cross - multiply: \(2RQ=60\), \(RQ = 30\). Then \(SQ=SR + RQ=20 + 30 = 50\).

Step1: Apply Thales' Theorem

In \(\triangle SQP\), since \(RT\parallel PQ\), by Thales' theorem (Basic Proportionality Theorem), \(\frac{ST}{PT}=\frac{SR}{RQ}\).
We have \(ST = 26\), \(PT = 39\), \(SR = 20\). Substitute into the proportion: \(\frac{26}{39}=\frac{20}{RQ}\).
Simplify \(\frac{26}{39}\) to \(\frac{2}{3}\), so \(\frac{2}{3}=\frac{20}{RQ}\).
Cross - multiply: \(2\times RQ=3\times20\), \(2RQ = 60\), so \(RQ = 30\).

Step2: Calculate \(QS\)

We know that \(QS=SR + RQ\). Given \(SR = 20\) and \(RQ = 30\), then \(QS=20 + 30=50\).

Answer:

\(50\)