Problem: In triangle is the altitude to AB and AE is the altitude to BC. If the lengths of , and are known, the length of is:
Answer Choices:
A. not determined by the information given
B. determined only if A is an acute angle
C. determined only if B is an acute angle
D. determined only if is an acute triangle.
E. none of these is correct
Solution:
When angles and are acute and angle is either acute, obtuse, or right, triangles and are similar. Since , and are known, can be found. Now, by applying the Pythagorean theorem to triangle CBD, where and are known, we can find .
When angle is obtuse the same analysis holds.
When angle is obtuse, triangles and are similar. From this fact and the given lengths, can be found. Next can be found with the aid of the Pythagorean theorem. Finally. .
When either angle or angle is right, the problem is trivial. In the former case ; in the latter, .