Chapter 6.4, Problem 24WE

To determine

To prove: that $P$ is any point inside $\Delta XYZ$ , then $ZX+ZY>PX+PY$ .

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 a triangle $XYZ$ and $P$ is any point inside triangle.

So the figure can be drawn as shown below:

  

From the above diagram that it can be observed that $\angle ZXY$ is greater than $\angle PXY$ . Hence, side $ZY$ will be greater than side $PY$ . Similarly, $\angle ZYX$ is greater than $\angle PYX$ . Therefore, side $ZX$ will be greater than side $PX$ .

Hence, from above observation find that $ZX+ZY>PX+PY$ proved.