![]() So once again, its slope is change and y over change and x, so the change and y,Ĭ over two minus zero, so the slope here is equal This line right over here, this median right over here. If the line just kept going, or the median is a line segment This business over here, so this is essentially the equation of this median right here And obviously, this would just simplify to y is equal to all of Of this line in purple, let me stay in the purple, is going to be y minus zero, is equal to x minus negative b, or I could say x plus b, times our slope, times c over a plus 2b. We know the point negativeī comma zero is on the line, so we know that the equation We could just use the point slope formula for the equation of a line. This line right over here, and then we know some points here. I could add them, or I could just multiply the numerator and theĭenominator both by two and I'll get the slope is equal to, c over a plus 2b, so that's the slope of Thing as adding a 2b over two, and I did that so these I just subtracted a negative b, which is the same thing as adding a b, and adding a b is the same Write a plus b over here, but just to makes the To be change and y, so c over two minus zero, so that's just c over two, over change and x, so a over two minus minus b, so minus negative b, I should say, so it's plus b, I could So what would be the equationįor this line right over here? Well, our slope is going Now that we know all the coordinates for the vertices and the midpoints, at least of these two sides, we can find the equations for the lines that the medians are part of, so we can find the equation for this line, we could find the equationįor this line in purple, then we can find theĮquation for the other line that connects to these two dots, find their intersection, and we'll essentially have theĬoordinates of our centroid, of the intersection of the medians, and we can do it for this one, too, but that would be redundant, because it's going to The midpoint of this side right over here is going to be negativeī plus zero over two, so it's negative b over two, and then zero plus c over two, so it's just going to be c over two. Would just be a over two, and then c plus zero over two, which would just be c over two. I'm only going to do itįor two of the medians, because we know that the third median will also intersect at the same centroid, so we have a median on this, we have a midpoint, I should say, on this side of the triangle, and its coordinates are just going to be the midpoint of those two points, or zero plus a over two, which Negative b comma zero, and then let's make this distance, this is the point y is equal to c, so this is a point zero comma c, so it's an arbitrary triangle, any triangle can be represented this way. Let's make it equal to b, so this is the point Let's make this distance right here be equal to, I don't know, ![]() Is an arbitrary triangle, so let's just make this distance equal to a, so this point right here, this vertex would be the point a comma zero, so that distance is a. So that's one side of the triangle, that's the other side of the triangle, and let's put the height of the triangle, let's put it on the y-axis, so this triangle right here, this is the y-axis that is the y-axis, and then this right here, is our x-axis. Math a little bit easier, maybe there's easier ways of doing it. Version of the proof, so why not do that? So let's just draw an arbitrary triangle, and I'll make one side of the triangle right along the x-axis just to make, I think it'll make the That they'd be interested in seeing the two-dimensional I did it using a two-dimensional triangle in three dimensions, and I mentioned that I thought, at least it made the math a little simpler, but someone mentioned Proof that the centroid is 2/3 along the way of a median. According to the theorem: ‘the sum of the squares of any two sides of a triangle equals twice the square on half the third side and twice the square on the median bisecting the third side’.Triangle medians and centroids I did, essentially, the ![]() Thus, the 3 medians divide the main triangle into 6 smaller triangles having equal area median AD forms triangles △ABD and △ACD, median BF into △ABF and △CBF, and median CE into △CAE and △CBEįormulas How to Find the Median of a TriangleĪ theorem called Apollonius’s Theorem gives the length of the median of a triangle. Each median divides the main triangle into 2 smaller triangles having equal area.The centroid is the center of gravity or center of mass of the triangle the three medians, m a, m b, & m c meet at point O, which is the centroid of the given triangle The 3 medians meet at a common point, called the centroid of the triangle.Each triangle has 3 medians, one coming from each vertex in △ABC, AD, BF, and CE are the 3 medians.
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