Expert: Paul Klarreich - 12/15/2008

Question
Hello Mr.Klarreich, thank you so much for helping me with my last calculus question.  Well, i have another one for you:

prove using vectors that the median on the hypotenuse of a right triangle is half of the hypotenuse.

My thinking so far: ok so i draw a triangle with a and b being the vectors forming the sides it.  So then a+b is the hypotenuse of my triangle.  I don't know what to do next or if i m on the right track or not.
Thank you ahead of time for any assistance you might be able to offer.

Answer
Questioner:   sinclair
Category:  Calculus
 
Subject:  vector proof
Question:  Hello Mr.Klarreich, thank you so much for helping me with my last calculus question.  Well, i have another one for you:

prove using vectors that the median on the hypotenuse of a right triangle is half of the hypotenuse.

My thinking so far: ok so i draw a triangle with a and b being the vectors forming the sides it.  So then a+b is the hypotenuse of my triangle.  I don't know what to do next or if i m on the right track or not.
Thank you ahead of time for any assistance you might be able to offer.
....................................
Hi, Sinclair,

Yes, A+B is the hypotenuse.  Then (see pic) the median goes from the end of B to the midpoint of the hypotenuse.

Now you want to prove that

|X| = 1/2|A + B|

Somehow you must use the fact that this is a right triangle.  That means  A * B = 0,[the * is the dot product]

So you have this vector equation:

B + X = (A + B)/2

   X = A/2 - B/2         << solve

   X = 1/2(A - B)        << factor

 |X| = 1/2|(A - B)|      << lengths must be equal.

|X|^2 = 1/2|(A - B)|^2       << square both sides.

>>>>> Now recall that  V * V = |V|^2, for any vector V.

|X|^2 = 1/2(A - B)*(A - B)

|X|^2 = 1/2[A*A - 2A*B +  B*B]       << multiply out

Aha!  A*B = 0, since this is a right triangle,
So  -2A*B is the same as +2A*B

|X|^2 = 1/4[A*A PLUS 2A*B + B*B]

|X|^2 = 1/4[(A + B) * (A + B)]  << factor

|X|^2 = 1/4|A + B|^2       << use the V*V property again.

and

|X| = 1/2|A + B|

which is what you wanted to prove.