Let us consider the right triangle ABC with the right angle and let AD be the median drawn from the vertex A to the hypotenuse BC. We need to prove that the length of the median AD is half the length of the hypotenuse BC.
Draw the straight line DE passing through the midpoint D parallel to the leg AC till the intersection with the other leg AB at the point E. The angle BAC is the right angle by the condition. The angles BED and BAC are congruent as they are corresponding angles at the parallel lines AC and ED and the transverse. AB Therefore, the angle BED is the right angle.
Now, since the straight line DE passes through the midpoint D and is parallel to AC, it cuts the side AB in two congruent segments of equal length: AE = EB So, the triangles AED and BED are right triangles that have congruent AE and EB and the common DE. Hence, these triangles are congruent in accordance to the postulate (SAS) It implies that the segments AD and DB are congruent as corresponding sides of these triangles. Since DB has the length half the length of the hypotenuse BC, we have proved that the median AD has the length half the length of the hypotenuse.