In triangle ABC, the point D lies on side AB and the point E lies on side AC. The measurements are AB = 12 cm, AC = 9 cm, AD = 3 cm and AE = 4 cm, and triangles ADE and ABC share the angle at A. If BC = 15 cm, what is the length of DE?

Look for two sides in proportion around the shared angle. Here AD/AC = 3/9 = 1/3 and AE/AB = 4/12 = 1/3, so by SAS similarity triangle ADE is similar to triangle ACB, with D matching C and E matching B. Hence DE/CB = 1/3 and DE = 15/3 = 5 cm. The obvious pairing of D with B fails, since AD/AB = 1/4 while AE/AC = 4/9; forcing it gives the wrong value 15/4 = 3.75 cm.