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Problem 39 Hard Difficulty

Prove that the centroid of any triangle is located at the point of intersection of the medians. [Hints: Place the axes so that the vertices are $ (a, 0) $, $ (0, b) $ , and $ (c, 0) $. Recall that a median is a line segment from a vertex to the midpoint of the opposite side. Recall also that the medians intersect at a point two-thirds of the way from each vertex (along the median) to the opposite side.]

Answer

$=\frac{b}{3}$

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Video Transcript

in this problem. Worse to show that the central right off any triangle is the intersection off its medians. No. Now let's draw the meanings. I said I'd This is the 1st 1 from a two line B C. This is from vortex See toe like a bee and that is for cortex. Be to line a C. Now we know that the median do you guys lend to equal length. So it means that this language is equal to this one. So if the cord until point B is zero to be assuming that this is our X and this is our waxes And if the coordinate off point C c come zero than a cornet off, this very point will then be c over to and be over, too. Now we're given a hint and we are set not the ratio between the vortex and intersection of medians. And the line that connects that works to the end of its median is to over three. So meaning that if this link is to kay, this length will then be K. Now we know that then in X direction so less that that were only interested in next to the business between point A to this word point in extraction is done equal to a minus. C over to. I want to say that this is a good three K unless said, we are interested in finding what to K is sort of coordinates off intersection off the medians. So from this we see that merciful K physical to 1/3 off a one C or two. And let's say that the distance from point A to into sexual comedians is equal to explore, and that will be equal to coordinates of point against. Since we're an extraction that will be equal to a minus. It is two cases too. Times 1/3 off a minus C or two. Question number one is Why don't have a minus here Business a point days here that we're moving in this direction. So the coordinate off this point will be, ah, value once will be less than develop the off this lower case K. That is why we have a month's year. So doing that, we find this to be a plus three plus c. Sorry, they were three plus security that is equal to a plus C over three and that will be the X coordinate off the central and now a Senate. This we're interested in. Why direction and we're interested in why coordinate off the central oId? It's same idea, You know, the total length This is from point A to this point is three k, Um, and the difference in why corded it off those points iss be or two minus zero. And unless that this one to be equal to three k from this we see that kay in white direction, let's make it a case of wine case a wide and is equal to 1/3 off your to in order to find why co ordinate, since we're moving from point a two in positive wind direction, why bar will be located that then Ah, why cordoning off point A, which is zero plus to K So that is two times won over three be over to these will cancel out and why I will be allocated that be over three sort of center of mess off this system will be that at the intersection off its medians and accordance off. That will be April, see over three and be over three