back to Geometry

Proving The Median Theorem

Question

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.

Proof

Extend CD to E, to make CD = DE.

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.