Median of a Right Triangle

[00:00:00]In a right triangle, the median drawn from the right angle to the hypotenuse equals half of the hypotenuse.

[00:00:07]Take the right triangle ABC and drop a segment [music] from C to a point O on the hypotenuse, chosen so that CO equals OA. Triangle AOC is isosceles with base [music] AC, so its base angles are equal. Call them alpha. The acute angles of a right triangle add up to 90°. So, the angle at vertex B equals 90 minus alpha. The full angle at C is 90°.

[00:00:31]We already used alpha for angle OCA, so the remaining [music] angle BCO is also 90 minus alpha. Angles CBO and BCO are equal, [music] so triangle BCO is isosceles, which means BO equals CO.

[00:00:43]Combining the two [music] equalities, CO equals OA and BO equals CO. So, all three are equal, [music] and the median CO is exactly half of the hypotenuse.

[00:00:55]Quod erat demonstrandum.