Chapter 5.3, Problem 48HP

To determine

To Prove: The reason why the hypotenuse of the right angle triangle is always the longest.

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider a triangle that has one side longer than the other side only when the angle that is only that is opposite to the longer side greater measure than the angle opposite to the shorter side.

Since, $\angle X$ is the right angle and triangle XYZ is right angled triangle, the other angle are then acute angle. Then, by the definition of the acute angle triangle,

  $m\angle Y<90=m\angle X$

As, the angle across from segment YZ is larger than the angle across the angle across from $XZ$ line. Then, by the angle side relationship in the triangles theorem shows that YZ segment $\overline{YZ}=\overline{XZ}$ .

Then, by the angle side relationship of the triangles. Likewise, one has $\overline{YZ}$ larger then $\overline{XY}$ .

Thus, it is proved that the hypotenuse of the right angle triangle is always the longest.