Triangle's Medians Make Smaller Triangles with Equal Area
Date: 04/15/99 at 00:33:18 From: Courtney Peurasaari Subject: Area of triangles I am studying to be a teacher at MSU and I have to take a geometry course, which is quite difficult for me. In class recently my teacher posed the problem: given a triangle in which you construct the three medians, resulting in six triangles, why is the area of each of these six triangles the same? Now, I know that if you draw only one median, it spilts the triangle in half. I was playing around with this on Geometers Sketchpad, by doing things such as getting the triangle to form a parallelogram, but that just got complicated. I think there must be a simpler answer. So, if you can help me in explaining that no matter what kind of triangle it is, when you find the medians six new triangles form and the area to these six triangles is the same, it would be very helpful. Thank You, Courtney Peurasaari
Date: 04/15/99 at 11:49:05 From: Doctor Rob Subject: Re: Area of triangles Thanks for writing to Ask Dr. Math! I don't know what the simplest way to see this is, but here is how I figured this out. Let the triangle be ABC, and let the midpoints of the sides be P (on BC), Q (on AC), and R (on AB). Let the intersection of the medians be O. First, the areas of AOR and BOR are the same because OR is the median from O to AB. The same is true of the areas of AOQ and COQ, and BOP and COP. Next, you know that the areas of APB and APC are equal because AP is the median from A to BC. That means that the areas of AOB and AOC are equal. Dividing by 2, that means that the areas of AOR and AOQ are equal. Do the same with BQA and BQC, concluding that BOR and BOP are equal. Do the same with CRA and CRB, concluding that COP and COQ are equal. Putting this all together, the six small triangles all have the same area. Another approach would be to show that the perpendicular distance from Q to AP equals half the distance from C to AP, which equals half the distance from B to AP, which equals the distance from R to AP, so that AOQ and AOR have the same base AO and equal altitudes, so equal areas. Then do the same for BOP and BOR, and for COP and COQ. Another approach would be to use the fact that OP = AP/3, OQ = BQ/3, and OR = CR/3, provided you are aware of that. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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