
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,


