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Proving The Median Theorem

median theorem example figure
In the figure above, ABC is a right triangle, angle C is 90 degrees. D is the midpoint of the hypotenuse AB. Prove CD = AB/2.
Extend CD to E, to make CD = DE.
median theorem example solution figure
Since D is the midpoint of AB (given), so AD = DB
Since CD = DE, so quadrilateral AEBC is a parallelogram.
(Theorem: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram).
Since angle C is 90 degrees (given), so parallelogram AEBC is a rectangle.
(A rectangle can be defined as a parallelogram having a right angle). The properties of a parallelogram can then be used to prove that each of the remaining angles of a rectangle must be a right angle.
Since AEBC is a rectangle, so AB = CE. (Theorem: The diagonals of a rectangle are congruent).
So CD = CE/2 = AB/2
So we proof the theorem that the length of the median drawn to the hypotenuse of a right triangle is one-half the length of the hypotenuse.