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### Course: Geometry (all content)>Unit 4

Lesson 6: Medians & centroids

# Triangle medians and centroids (2D proof)

Showing that the centroid is 2/3 of the way along a median. Created by Sal Khan.

## Want to join the conversation?

• I wonder if someone can prove a theorem that the median of any triangle (not just isosceles)divides the triangle into two triangles with equal areas. Thanks
• Since the median, by definition, divides a base in half, then you will have two triangle with the same base, as well as the same height. Therefore, the triangles will have the same area.
• I wish could have more practice or at least some mini tests here to see how much we have learned from these videos.
• Is it possible for you to give us the equation for the median line in an equation without using 0 as one of the coordinate points?
• I don't think x=(a-b)/3. Cause b is positive.
Say a=6, b=3, then the length of line AB should equal to a+b, which means 9.
So the centroid should be (a-(-b)/3 and c/3.
Does anyone agree?
• It's already correct that the x-coordinate of the centroid is (a-b)/3.
The false thing is to say that it is 1/3 of the entire base length ;)
(1 vote)
• Will this be used in further math progression? Say, Algebra II for example?
• Well, maybe not. But this comes in helpful with much more difficult problems.
(1 vote)
• Is there a way to prove this without graphing it on a coordinate grid?
• I still don't understand the theorem. Can anyone help me understand? Thank you, in advance. :)
• If you connect a line from the midpoint of one side to the vertex opposite to that side (which is a median), then the centroid is where all 3 medians intersect.

The theorem basically says that:
The length of the centroid to the midpoint of the opposite side is 2 times the length of the centroid to the vertex.

Hope this helps!
• How does this relate to the centroid formula? (x1+x2+x3)/3 , (y1+y2+y3)/3
I recently found some information on this formula but I'm not exactly sure why it works.
• At x-coordinate is (a-b)/3. I worked a problem with a triangle with vertices (7,0), (-2,0), and (0,9) (a=7, b=-2, and c=9). Then x-coordinate centroid (7-(-2))/3 = 3. This, however, is incorrect (a simple drawing shows that the x-coordinate is less than 2). Can you elaborate?
• At Sal draws a green segment from the vertex at the top of the triangle through the centroid to the side of the triangle at the bottom (on the x axis). He calls it a median but am I wrong in thinking that he does not prove that it is a median? He proves that d is 2/3 of l, but doesn't quite prove that l is the median. I don't think he quite proves that that green segment meets the triangle side at the bottom at its midpoint. He comes close; he just needs to show, using those similar triangles at that (a-b)/3 is 2/3 of the distance from the origin to the midline which is (a-b)/2.
(1 vote)