geometry

[missa0312]

Prove theorem 4.2.18: The median to the hypotenuse of a right triangle is one-half the length of the hypotenuse.

 

[TchrWill]

Prove theorem 4.2.18: The median to the hypotenuse of a right triangle is one-
half the length of the hypotenuse.

Clearlly not the only way:

1--Consider right triangle ABC wih C the right angle
2--Draw BC' equal and parallel to AC and AC' wqual to and parallel to BC.
3--The figure thus created is rectangle ABCC' with the diagonals (the two hypotenuses of triangles ABC and AC'B) crossing at D
4--Since the diagonals of a rectangle bisect one another, CD, a median of triangle ABC is equal to AB/2.
4--Similar statement s can be made about AD, BD and DC'.

 

[soroban]

Hello, missa0312!

Prove theorem 4.2.18: The median to the hypotenuse of a right triangle is one-half the length of the hypotenuse.

Here's an "eyeball" approach . . .

A right triangle can be inscribed in a semicircle.

                  C
              * * *
           *    /  \ *
         *    /     \  *
        *   /        \  *
          /           \  *
       */              \ *
       *- - - - * - - - -*
       A        O        B

The hypotenuse \(\displaystyle AB\) is a diameter of the circle.

The median \(\displaystyle CO\) is a radius of the circle. \(\displaystyle \;\;\) . . . ta-DAA!