Chapter 6.4, Problem 23WE

To determine

To discover, state, and prove a theorem that compares the length of the longest side of a quadrilateral with the sum of the lengths of other three sides.

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,

  

Apply the above theorem,

  $\begin{array}{l}AB+BC>AC\mathrm{........}(1)\\ CD+DA>AC\mathrm{........}(2)\end{array}$

Add equation (1) and (2) to get,

  $AB+BC+CD+DA>2AC$

Therefore,

  $AB+BC+CD+DA>2AC$