Question

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

Explanation:

Step1: Identify corresponding sides

In similar triangles, corresponding sides are proportional. Let the missing side (AB) be \( x \). The sides of \( \triangle XYZ \) are \( XY = 4 \), \( YZ = 8 \), \( XZ = 7 \). The sides of \( \triangle ABC \) are \( AB = x \), \( BC = 72 \), \( AC = 63 \). We match the sides: \( XY \) corresponds to \( AB \), \( YZ \) corresponds to \( BC \), \( XZ \) corresponds to \( AC \)? Wait, no, let's check the ratios. Wait, \( YZ = 8 \), \( BC = 72 \); \( XZ = 7 \), \( AC = 63 \); \( XY = 4 \), \( AB = x \). Let's check the ratio of \( YZ \) to \( BC \): \( \frac{8}{72}=\frac{1}{9} \)? No, wait \( 72\div8 = 9 \), \( 63\div7 = 9 \). Ah, so the scale factor from \( \triangle XYZ \) to \( \triangle ABC \) is 9, because \( 7\times9 = 63 \), \( 8\times9 = 72 \). So then \( XY \times 9 = AB \). \( XY = 4 \), so \( 4\times9 = 36 \)? Wait, no, wait \( AC = 63 \), \( XZ = 7 \), so \( 63\div7 = 9 \). \( BC = 72 \), \( YZ = 8 \), \( 72\div8 = 9 \). So the ratio of \( \triangle ABC \) to \( \triangle XYZ \) is 9. Therefore, \( AB = XY \times 9 \). \( XY = 4 \), so \( AB = 4\times9 = 36 \). Wait, but let's set up the proportion correctly. Let's use the sides: \( \frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ} \). So \( \frac{x}{4}=\frac{72}{8}=\frac{63}{7} \). Let's check \( \frac{72}{8}=9 \), \( \frac{63}{7}=9 \). So \( \frac{x}{4}=9 \), so \( x = 4\times9 = 36 \).

Step2: Solve for \( x \)

From the proportion \( \frac{x}{4}=9 \) (since \( \frac{72}{8}=9 \) and \( \frac{63}{7}=9 \)), multiply both sides by 4: \( x = 4\times9 = 36 \).

Answer:

36