In the figure, tan B = 3/4. If BC = 15 and DA = 4, what is the length of DE?

Explanation

The correct answer is 6. Since tan B = 3/4 , ΔABC and ΔDBE are both 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since BC = 15, which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of ΔABC to the 3-4-5 triangle is 3:1. Therefore, the length of _ AC (the side opposite to B) is 3 × 3 = 9, and the length of AB (the side adjacent to angle B) is 4 × 3 = 12. It is also given that DA = 4. Since AB = DA + DB and AB = 12, it follows that DB = 8, which means that the similarity ratio of ΔDBE to the 3-4-5 triangle is 2:1 (DB is the side adjacent to angle B). Therefore, the length of DE, which is the side opposite to angle B, is 3 × 2 = 6.