What is the length of DC?

Solution:

The diagram shown above comprises two triangles △ADE and △ACD. 

These triangles △ADE and △ACD can be proved to be similar as shown below.

The angles ∠CAB and ∠DAE are the same angles as they are common angles. Sides DE and BC are parallel and are in proportion. Side AE is also proportional to AB, it can be concluded that the two triangles △ADE and △ACD are similar.

Hence we can write the expression using the basic proportionality theorem as: 

AD / AC  =  AE  / AB =  DE / CB

9/AC  =  12/16

AC =  (9 × 16 ) / 12 

AC   =  12

AC = AD + DC

12      =  9   + DC

DC  = 12 - 9 = 3

What is the length of DC?

Summary:

 The length of DC is 3.