Question

find rt.
rt =

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that if a line segment connects the midpoints of two sides of a triangle, then it is parallel to the third side and half its length. Here, \(PS\) is a mid - segment. So, \(\frac{QP}{PT}=\frac{RS}{ST}\). Since \(PS\) is a mid - segment (marked by the red arrows indicating mid - points), we know that \(QS = SR\) (not directly used in the ratio for length calculation here, but the ratio of the divided sides of \(\triangle QRT\) by the mid - segment \(PS\) gives a proportion). Also, we can use the property of similar triangles (by the basic proportionality theorem or Thales' theorem as \(PS\parallel QR\)). The ratio of the sides of \(\triangle TPS\) and \(\triangle TQR\) is \(\frac{PT}{QT}=\frac{TS}{RT}\). Given \(QP = 30\), \(PT=20\), so \(QT=QP + PT=30 + 20=50\). Let \(RT=x\), \(TS = 45\).

Step2: Set up the proportion

Since \(\triangle TPS\sim\triangle TQR\) (by AA similarity as \(\angle T=\angle T\) (common angle) and \(\angle TPS=\angle TQR\) (corresponding angles as \(PS\parallel QR\))), we have \(\frac{PT}{QT}=\frac{TS}{RT}\). Substituting the values: \(\frac{20}{50}=\frac{45}{RT}\).

Step3: Cross - multiply to solve for \(RT\)

Cross - multiplying gives \(20\times RT=50\times45\). Then \(RT=\frac{50\times45}{20}\). Simplify \(\frac{50\times45}{20}=\frac{5\times45}{2}=\frac{225}{2}=112.5\).

Answer:

\(RT = 112.5\)