Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Medians of a Triangle


Date: 02/16/99 at 22:06:24
From: Jin Park
Subject: Medians of a Triangle

I learned in school that the 3 medians of a triangle divide themselves 
up into a ratio of 1:2. I clearly understand what this means but could 
you give me a proof of it (that the medians of triangle divide 
themselves up into a ratio of 1:2)?  I guess I'm not stuck on anything 
right now because I do not know where to start!

Thanks,
Jin Park


Date: 02/17/99 at 02:54:03
From: Doctor Floor
Subject: Re: Medians of a Triangle

Let us consider a triangle ABC. The midpoints of the sides form a 
triangle A'B'C'. AA', BB' and CC' meet in a point G (known as the 
centroid):

  

First we notice that triangle A'B'C' is similar to ABC; the sides of 
A'B'C' have half the length of the the sidelengths of ABC. We can see 
that, because for instance B'C' is an image of BC from multiplication 
over A with factor 1/2. So indeed B'C' must be half the length of BC.

Also we notice that AA' passes through the midpoint of B'C', since in 
the multiplication mentioned above, A' moves to the midpoint of B'C'. 
Similar results are found for BB' and CC'. But that means that the 
medians of ABC are also medians of A'B'C'.

We now can conclude that the distance AG in ABC corresponds to A'G in 
A'B'C'. From the observation that A'B'C' has lengths half of those in 
ABC, we can conclude that AG:A'G = 2:1. And that is what we wanted to 
prove.

I hope this helps!

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/