Problem 4.
In triangle ABC pictured below, DA, EB, and FC are altitudes (perpendicular to the sides of the triangle with which they intersect). Side AB is 21 units in length,
altitude FC is 12 units long, and altitude DA is units long.
Find the length of altitude EB.


Solution:

The key to this problem is that we can choose any side to be the base (and the corresponding altitude as its height), and the area of the triangle does not change. Therefore, the areas of the 3 triangles with their respective bases and heights are expressed in the following equation:


From the last pair of equality,


Hence the only missing information is side AC. Side AC is found as follows:



In the triangle BCF,


From the graph,


In the triangle ACF,


Therefore,