Question

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Explanation:

Step1: Identify Similar Triangles

Since \(\overline{RS} \parallel \overline{QT}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle RSU \sim \triangle QTU\) (similar triangles).

Step2: Set Up Proportion

Let \(QU = x\), then \(RU = RQ + QU = 30 + x\). The ratio of corresponding sides in similar triangles is equal. So, \(\frac{QU}{RU}=\frac{TU}{SU}\). We know \(TU = 40\), \(SU = 64\), \(RQ = 30\), \(QU = x\), \(RU = 30 + x\). Substituting values: \(\frac{x}{30 + x}=\frac{40}{64}\).

Step3: Solve the Proportion

Simplify \(\frac{40}{64}=\frac{5}{8}\). So, \(\frac{x}{30 + x}=\frac{5}{8}\). Cross - multiply: \(8x = 5(30 + x)\). Expand: \(8x = 150 + 5x\). Subtract \(5x\) from both sides: \(3x = 150\). Divide by 3: \(x = 50\).

Answer:

\(50\)