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Constructing the Centroid - Problem 2
Recall that the centroid of a triangle is the point of concurrency of the medians of that triangle. Additionally, the centroid divides the median into two pieces of 1/3 and 2/3 the original length, and the piece adjacent to a vertex is the piece is the larger, 2/3 portion. Using this fact, if told that a given point is a centroid, and that the median is a certain length, it is possible to calculate the lengths of the pieces into which the centroid divides the median.
We can use what we know about the centroid to find missing blanks inside a triangle. So let’s talk about what a centroid is first. A centroid is the point of concurrency of the three medians in a triangle. Which means if you're drawing the medians all three of them where they intersect is one point called the point of concurrency and it’s the centroid.
Centroid is also the center of mass of a triangle and what’s key to this problem is that each median is divided into two proportional segments. One piece that’s two thirds of the overall length and one piece that’s one third so let's apply what we know about the way a segment is divided by the median.
In this problem we are told that E is the centroid and we could also infer that just by looking at this diagram. Seeing that all three of these segments inside the triangle go from the vertex to the opposite side is midpoint. Now how do we know which piece is the one third and which piece is the larger two thirds?
Well the piece, the segment that’s in between the centroid and the midpoint of the opposite side that will always be your smaller piece. The piece between the centroid and the vertex will always be the larger two thirds portions. So let’s start off by saying AD equals 12cm. So first I have to locate AD and I see that AD is divided into two pieces, W and Z.
W we said is equal 1/3 of the entire length, the entire length is 12 so a 1/3 of 12 is 4. So our units here are cm because I’m told that AD is 12cm I’m going to write that in W is 4cm. Since I’m already on that segment I’m going to find Z. Z is the larger part because it’s in between the centroid and the vertex, so Z equals 2/3 of the entire length. 2/3 of 12 you could 2 times 12 divide by 3, which will give you 24 divide by 3 which is 8 and our units are cm. So Z is 8cm.
Let’s go back to x where we are told that all we know of this median is that this piece the smaller one is 8. Well 8 is equal to 1/3 of line segment BE, because I know that the 8 part is the smaller piece. So to find out what the total length is I will have to multiply both sides of this equation by 3. So 3 times 8 is 24, 1/3 of 3 is 1, so we find out that BE is 24. X is the larger piece so x is 2/3 of 24. So 2/3 times 24 is equal to x.
Again a couple of different ways you could do this, way I would do this is 3 goes into 24 8 times, 8 times 2 is 16 so we find out that x is 16cm, so let’s go write that in and we’re left with one last problem that’s y. So Y is the smaller piece of this median CF, so 12 is equal to 2/3 of CF. So to find out what the total length of CF is I need to multiply both sides by 2/3 and again the way you could do this is 3 goes into 12 excuse me 4 times, 4 times 2 is 8.
So something is wrong here. Is that CF should be larger than 12, so if 12 oh! I found my mistake. As you remember from algebra if you want to eliminate that fraction you multiply by its reciprocal so that’s why we are coming up with a smaller number. So I see that 2/3 times it reciprocal I’m going to multiply this by 3/2 so 2 goes into 12, 6 times, 6 times 3 is 18.
So what I just did there checking to make sure it made sense is something that you should always do when you are solving a geometry problem. Because if I would end up with 8 and I know that part of that segment is 12 it doesn’t that the whole thing is it so always do an error check. So if the total length is 18 and y is 1/3 of that, so y is 1/3 times 18 we know a 1/3 of 18 is 6. So I’m going to come over here and write in that y is equal to 6cm.
So the key to this problem we said was each median is divided into two proportional segments by the centroid, one piece being 2/3, the other piece being 1/3.