Question

△abc and △def are similar. find the missing side length.

b e
/\\ /\\
32 \\ ? 4 \\ 8
/ \\ / \\
a-----c d-----f
56 7

(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. Let's take sides \( AB = 32 \) and \( DE = 4 \). The scale factor \( k \) is \( \frac{AB}{DE}=\frac{32}{4} = 8 \).

Step2: Find the missing side

The side \( EF = 8 \) in \( \triangle DEF \), so the corresponding side in \( \triangle ABC \) (let's call it \( BC \)) is \( EF\times k \). Substituting the values, we get \( 8\times8 = 64 \). We can also verify with the other pair of sides: \( AC = 56 \), \( DF = 7 \), and \( \frac{56}{7}=8 \), which confirms the scale factor is 8. So the missing side length is \( 8\times8 = 64 \).

Answer:

64