Questions: In the figure, PQ is parallel to RS. The length of PQ is 18 cm; the length of RS is 72 cm. What is the length of SO? A. 9 cm B. 3 cm C. 54 cm

Transcript text: In the figure, PQ is parallel to RS. The length of PQ is 18 cm; the length of RS is 72 cm. What is the length of SO? A. 9 cm B. 3 cm C. 54 cm

Solution

Solution Steps

Step 1: Identify Given Information

\( PQ \parallel RS \)
\( TP = 2 \) cm
\( PR = 18 \) cm
\( QT = 27 \) cm

Step 2: Understand the Relationship

Since \( PQ \parallel RS \), triangles \( TPQ \) and \( TRS \) are similar by the Basic Proportionality Theorem (Thales' theorem).

Step 3: Set Up the Proportion

Using the similarity of triangles \( TPQ \) and \( TRS \): \[ \frac{TP}{TR} = \frac{PQ}{RS} = \frac{TQ}{TS} \]

Step 4: Calculate \( TR \)

Given \( TP = 2 \) cm and \( PR = 18 \) cm: \[ TR = TP + PR = 2 + 18 = 20 \) cm

Step 5: Calculate \( TS \)

Given \( TQ = 27 \) cm: \[ TS = TQ + QS \] Since \( PQ \parallel RS \), \( \frac{TP}{TR} = \frac{TQ}{TS} \): \[ \frac{2}{20} = \frac{27}{TS} \] \[ \frac{1}{10} = \frac{27}{TS} \] \[ TS = 27 \times 10 = 270 \) cm

Step 6: Calculate \( QS \)

\[ QS = TS - TQ = 270 - 27 = 243 \) cm