SOLUTION: Suppose that ABC is a right triangle with right angle at B. If AC = 25 and the altitude BD = 20, what is AD? Answer and explain please.

Here is a triangle drawn to scale with AC = 25, and BD = 20.  
But as you see, it cannot have a right angle at B.  The
altitude BD = 20 is just too long to have a right angle at B.



Here is a triangle also drawn with he same AC = 25 as the
first that has a right angle at B.  However the altitude
is only about 12.2, not 20.



This makes us believe that it is impossible to have such a
triangle.

To show that it is impossible, let's suppose that the altitude
BD is actually 20 and angle B is a right angle. Let DC = x, then
AD will = 25-x.




Then triangle ADB is similar to triangle BDC, so





Cross-multiplying:







However this quadratic has no real solutions,
because its discriminant is negative:



Therefore we have proved that there is no such triangle.

Edwin