What is the length of ab in the accompanying figure ?

Since the line from top angle creates a right angle at the perpendicular, you can do the Pythagorean Theorem to get the length of the line on either side of it, then add the two numbers together.
a²+b²= c²
12²+b²= 13²
144+b²= 169
b²= 25
b= 5
a²+b²= c²
12²+b²= 20²
144+b²= 400
b²= 256
b= 16
16+5= 21feet