Question

qr || su. find rt.
rt =
submit

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{QR} \parallel \overline{SU} \), triangles \( QRT \) and \( SUT \) are similar by the Basic Proportionality Theorem (Thales' theorem). So, the ratios of corresponding sides are equal. Let \( RT = x \), then \( ST = x - 36 \). The ratio of \( QT \) to \( UT \) is \( \frac{33}{11}=3 \).

Step2: Set Up Proportion

Using the similarity of triangles, \( \frac{RT}{ST}=\frac{QT}{UT} \). Substituting the known values: \( \frac{x}{x - 36}=3 \).

Step3: Solve for \( x \)

Multiply both sides by \( x - 36 \): \( x = 3(x - 36) \). Expand the right side: \( x = 3x - 108 \). Subtract \( x \) from both sides: \( 0 = 2x - 108 \). Add 108 to both sides: \( 2x = 108 \). Divide by 2: \( x = 54 \).

Answer:

\( 54 \)