Question

in the diagram below, △abc ~ △def. find y.

(diagram: two similar triangles. triangle abc has sides: ab = y, bc = 7, ac = 9. triangle def has sides: de = 15, ef = 35, df = 45.)

Explanation:

Step1: Identify Corresponding Sides

Since \( \triangle ABC \sim \triangle DEF \), corresponding sides are proportional. Let's match the sides: \( AB \) corresponds to \( DE \), \( BC \) corresponds to \( EF \), and \( AC \) corresponds to \( DF \). So, \( \frac{AB}{DE} = \frac{AC}{DF} \).

Step2: Substitute Known Values

We know \( AB = y \), \( DE = 15 \), \( AC = 9 \), and \( DF = 45 \). Substituting into the proportion: \( \frac{y}{15} = \frac{9}{45} \).

Step3: Solve for \( y \)

Simplify \( \frac{9}{45} = \frac{1}{5} \). Then the equation becomes \( \frac{y}{15} = \frac{1}{5} \). Cross - multiply: \( 5y = 15\times1 \), so \( 5y = 15 \). Divide both sides by 5: \( y=\frac{15}{5}=3 \).

Answer:

\( y = 3 \)