Chapter 6.4, Problem 22WE

To determine

To prove: that sum of the lengths of the altitudes is greater than half the perimeter of the triangle.

Explanation of Solution

Consider the theorem,

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

  $\begin{array}{l}AB+BC>AC\\ AB+AC>BC\\ AC+BC>AB\end{array}$

  

Consider the following figure,

  

In case of medians, as median divides the side into two equal segments but in case of altitudes it is not always true. If a triangle is equilateral triangle than the median and altitude are at same point. So it can not prove that sum of the lengths of the altitudes is greater than half the perimeter of the triangle. But in case of equilateral triangle this is true.

Therefore, it can not prove that sum of the lengths of the altitudes is greater than half the perimeter of the triangle.