Unizor - Geometry2D - Right Triangles
Channel: Zor Shekhtman on UNIZOR Education
Date: 2011-10-14

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Right triangles are those that have one of their angles measured at 90 degrees.
Since exterior angle supplemental to this 90 degrees interior angle is also equal to 90 degrees and, according to a previously proven theorem, this exterior angle is greater than any interior not supplemental with it, two other interior angles must have measure of less than 90 degrees, i.e. they are acute.

A side opposite to an angle of 90 degrees is called hypotenuse. Since it is opposite to a 90 degrees interior angle, while two other sides (called legs or catheti) lie opposite to acute angles measured less than 90 degrees, a hypotenuse of any right triangle is longer than any of its cathetus (leg), as a side in a triangle that lies opposite to a bigger angle. 

Consequently, if you have a straight line and a point outside it, and you draw a perpendicular to a line from this point and another, non-perpendicular, segment to connect this point with a line, a perpendicular will be shorter, as a cathetus in a right triangle with the other segment as a hypotenuse. In a word, a perpendicular is a shortest segment connecting a point to a line.

Incidentally, let's prove another simple theorem.
If two different segments connect a point to a line, the one closer to a perpendicular will be shorter.