Suppose that ABC is a right triangle with right angle at B. If AC = 25 and the altitude BD = 20, what is AD?
Draw right triangle(ABD) with angle(B) the right angle. Therefore, AC will be the hypotenuse. AC = 25.
Draw altitude BD. BD = 20
Since B is on AC, let AD = x and DC = 25 - x.
Triangle(ADB) is a right triangle with angle(D) the right angle.
Using the Pythagorean Theorem: AD2 + DB2 = AB2 ---> AB2 = x2 + 202.
Triangle(BDC) is a right triangle with angle(D) the right angle.
Using the Pythagorean Theorem: CD2 + DB2 = CB2 ---> CB2 = (25 - x)2 + 202.
Using the Pythagorean Theorem on right triangle(ABC): AB2 + CB2 = AC2,
Substituting: [x2 + 202] + [(25 - x)2 + 202] = 252
Simplifying: x2 + 400 + 625 -50x + x2 + 400 = 625
---> 2x2 - 50x +1425 = 625
---> 2x2 - 50x + 800 = 0
---> x2 - 15x + 400 = 0
This equation has no real roots, so the original problem is impossible to occur.