Question

verify that \\( \triangle a b c \sim \triangle d e f \\). find the scale factor of \\( \triangle a b c \\) to \\( \triangle d e f \\).
\\( \triangle a b c: b c=18, a b=15, a c=12 \\)
\\( \triangle d e f: e f=12, d e=10, d f=8 \\)
\\( k= \\)

Explanation:

Step1: Calculate the ratios of corresponding sides

For \( \frac{BC}{EF} \), substitute \( BC = 18 \) and \( EF = 12 \): \( \frac{18}{12}=\frac{3}{2} \).
For \( \frac{AB}{DE} \), substitute \( AB = 15 \) and \( DE = 10 \): \( \frac{15}{10}=\frac{3}{2} \).
For \( \frac{AC}{DF} \), substitute \( AC = 12 \) and \( DF = 8 \): \( \frac{12}{8}=\frac{3}{2} \).

Step2: Verify similarity and find scale factor

Since \( \frac{BC}{EF}=\frac{AB}{DE}=\frac{AC}{DF}=\frac{3}{2} \), by the Side - Side - Side (SSS) similarity criterion, \( \triangle ABC\sim\triangle DEF \). The scale factor \( k \) of \( \triangle ABC \) to \( \triangle DEF \) is the ratio of corresponding sides.

Answer:

\( \frac{3}{2} \)