How do you solve this triangle problem?

Question

http://mdtp.ucsd.edu/crtest/showquestion.asp?qu=28

The correct answer is B I just cant find out how to get it?

Answer 1

The triangles ABC and BCD are the same shape.  Both of them have a right angle (at C for ABC and D for BCD) and share the angle at B.  So all dimensions will be in ratio.

So:   BD/CB = CB/AB  

Or  BD= CB*CB/AB

But CB=12 and AB = 14.

So BD = 12*12/14   =  144/14    = 72/7

Answer 2

There is a theorem that states: when the altitude is drawn to  the hypotenuse of a right triangle, then the altitude divides the triangle into two triangles similar to the original triangle and similar to each other.  The proportion would be:  12:14::BD:12

Use product of means =  product of extremes to solve for BD

BD = 72/7