Verify that the length from any point in a right triangle to the endpoints of hypotenuse is less than the length of the hypotenuse.

#### Question

#### Answer

In the figure is a right triangle \triangle ABC. Point D resides inside the triangle, which forms \triangle ACD along with the endpoints of the hypotenuse.

\because \angle CAD <\angle CAB

\angle ACD <\angle ACB

Addition of the two inequalities

\angle CAD+\angle ACD < \angle CAB+\angle ACB =90\degree

Therefore, the subtended angle of the hypotenuse \angle ADC is an obtuse angle, of which the corresponding side is the longest among the three sides of \triangle ACD.

Now we have verified that the length from any point in a right triangle to the endpoints of hypotenuse is less than the length of the hypotenuse.