In triangle ABC, AB = 12, AC= 9, D is a point on AC, AD = 3. E is a point on AB, so that triangle ADE is similar to ABC, what is the length of AE? (two different answers)

Answer 1

Using the standard notation of listing similar triangles by listing the order of matching vertices,
triangle ADE is similar to ABC
(i.e. angle A matches with angle A, angle D matches with angle B, angle E matches with angle C, that way all ratios can be easily stated)

AE/AD = AC/AB
AE/3 = 9/12
AE = 27/12 = 9/4

or
AED is similar to ABC
AE/AB = AD/AC
AE/12 = 3/9
AE = 36/9 = 4