Question

△abc and △pqr are similar. find the missing side length.
(the triangles are not drawn to scale.)

Explanation:

Step1: Find the scale factor

Since the triangles are similar, the ratio of corresponding sides is equal. First, find the scale factor by comparing the sides \( AC = 5 \) and \( PR = 40 \). The scale factor \( k=\frac{PR}{AC}=\frac{40}{5} = 8 \). Alternatively, we can check with \( AB = 2 \) and \( PQ = 16 \), \( \frac{PQ}{AB}=\frac{16}{2}=8 \), so the scale factor is 8.

Step2: Find the missing side

Let the missing side in \( \triangle ABC \) (corresponding to \( QR = 48 \) in \( \triangle PQR \)) be \( x \). Using the scale factor, \( \frac{QR}{x}=8 \), so \( x=\frac{QR}{8}=\frac{48}{8}=6 \). We can also set up a proportion: \( \frac{AB}{PQ}=\frac{BC}{QR} \), substituting values \( \frac{2}{16}=\frac{x}{48} \). Cross - multiplying gives \( 16x = 2\times48 \), \( 16x = 96 \), and \( x=\frac{96}{16}=6 \).

Answer:

6