Question

qr || ps. find pt.
pt =

Explanation:

Step1: Apply the basic proportionality theorem (Thales' theorem)

Since \( \overline{QR}\parallel\overline{PS}\), we have \(\frac{RT}{ST}=\frac{QT}{PT}\). Given \(RT = 14\), \(ST=42 - 14=28\), and \(QT = PT + 75\).

Step2: Substitute values into the proportion

Substitute into \(\frac{RT}{ST}=\frac{QT}{PT}\), we get \(\frac{14}{28}=\frac{PT + 75}{PT}\).

Step3: Simplify the proportion

Simplify \(\frac{14}{28}=\frac{1}{2}\), so \(\frac{1}{2}=\frac{PT + 75}{PT}\). Cross - multiply: \(PT=2(PT + 75)\).

Step4: Solve the equation

Expand: \(PT = 2PT+150\). Subtract \(2PT\) from both sides: \(PT-2PT=150\), \(-PT = 150\), so \(PT = 150\).

Answer:

\(150\)