Question

in the figure below, triangle rpq is similar to triangle rts.
what is the distance between p and q?
24
42
50
54

Explanation:

Step1: Recall Similar Triangles Property

For similar triangles \( \triangle RPQ \) and \( \triangle RTS \), the corresponding sides are proportional. So, \( \frac{PQ}{TS}=\frac{RP}{RT} \).

Step2: Identify Corresponding Sides

We know \( RP = 42 \), \( RT = 28 \), and \( TS = 36 \). Let \( PQ = x \). Substitute into the proportion: \( \frac{x}{36}=\frac{42}{28} \).

Step3: Solve for \( x \)

Simplify \( \frac{42}{28}=\frac{3}{2} \). Then the equation becomes \( \frac{x}{36}=\frac{3}{2} \). Cross - multiply: \( 2x = 36\times3 \). Calculate \( 36\times3 = 108 \), so \( 2x = 108 \). Divide both sides by 2: \( x=\frac{108}{2}=54 \).

Answer:

The distance between \( P \) and \( Q \) is \( 54 \), so the correct option is the one with \( 54 \) (e.g., if the options are labeled as in the problem, the option with \( 54 \)).