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AMC 10/12 Daily Bonus #29

 
 
LiMinkai的头像
AMC 10/12 Daily Bonus #29
LiMinkai 发表于 2019年03月2日 Saturday 11:53
 

I was working on the problem #29, the solution mentions that if a line splits the triangle in two it must pass through the centroid. Why is that?

 
ReynosoDavid的头像
Re: AMC 10/12 Daily Bonus #29
ReynosoDavid 发表于 2019年03月4日 Monday 10:42
 

The line $y = mx$ goes through $(0,0)$, which is one of the vertices of the triangle.

If a line goes through a vertex, then it splits the triangle in two triangles that have the same height from that vertex. Thus, the triangles have equal area if and only if the line is a median, in particular, the line goes through the centroid of the triangle.

Note we would have reached the same answer if we had used the midpoint of side $BC$ instead of the centroid of the triangle.