Question

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

Explanation:

Step1: Find the scale factor

The ratio of corresponding sides of similar triangles is equal. So, we take the ratio of side \( AB \) (length 40) and side \( DE \) (length 5). The scale factor \( k = \frac{40}{5}=8 \).

Step2: Use the scale factor to find the missing side

We know side \( BC = 64 \) and side \( EF \) is the missing side. Since the triangles are similar, \( \frac{BC}{EF}=k \). We can also use another pair to verify. Let's use \( AC = 56 \) and \( DF = 7 \), \( \frac{56}{7}=8 \), which matches the scale factor. Now, for side \( BC = 64 \) and \( EF \), we have \( \frac{64}{EF}=8 \), so \( EF=\frac{64}{8}=8 \). Alternatively, we can set up a proportion with the corresponding sides. Let the missing side be \( x \). The proportion is \( \frac{40}{5}=\frac{64}{x} \) (cross - multiplying: \( 40x = 5\times64 \), \( 40x = 320 \), \( x=\frac{320}{40}=8 \)).

Answer:

8