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Show that for any triangle, the medians intersect in a point which is $2 / 3$ of the way from each vertex to the opposite side. Hint: Choose the vertices of the triangle to be $(0. 0).$ ( $b, c$ ) and ( $a$. 0).

$$(16 / 5,36 / 5)$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 9

Elements of Geometry

Derivatives

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:35

Prove that the centroid of…

04:28

02:39

01:39

12:27

Triangle Show that the cen…

01:06

Recall that a median of a …

52:39

a. Using coordinates, prov…

Okay, so let's have a triangle. Here. We have coordinates 00, we have B, C. And a zero. So we want to see you, let's say these are the big points are the mid segments. We want to say that this point right here is 2/3 away from any source vertex of the triangle. So first what we want to do is find each of these mid points. So to do that we want to go around the triangle and take the up the mid points of each of these segments. So let's do right here right here, Henry here, so to do that we take the average of the ordered pairs or the midpoint of the ordered pairs. So here we have for the blue we have a B plus zero over to C. Plus your over to which is Be over two and see over two. I'm just gonna raise it for the sake of space. Be over two and see over to. Yeah. And then for this green section right here we have B plus A over to and C plus zero Over to that one. I'll keep. And then over here we have a plus 0/2 which is a over to and zero plus 0/2 which is zero. So going from here, we want to see how far this the vertex the intersection point is from all these. So We're going to we're going to see if it's actually 2/3 or one third away now. So if we apply the two thirds, one third rule to each of these, say to each of these points on the order pair to their corresponding vertex then they should all intersect at the same point. So we'll start with the 1st 1. Let's try 2/3 of the way from A plus B over two right here. So that's actually the green one plus one third zero. That's this guy right here and then two thirds. See over to plus one third zero and two thirds. See over to notice it. Sure I'm sorry that was it actually. Mhm. It's right here if you saw solve this you get A plus B. Over to sorry three. Mhm. Yeah come on see over three. So that's where that for texas. So let's try that again with the next one. Bill to. It must be to that right here over to see over to. Yes I'm sorry we actually just said that one. It's confusing if you don't keep it in order so make sure you do that, we'll do two thirds times be over. Yeah so two thirds be over to Plus 1 3rd. A. I'm on 2/3 of the way from See over to there was 1/30 and that two gives us a plus B. Or 3 & C. or three. So you can try that for the last one and then I'll show you that right here the intersecting coordinates A plus B over three comma. See over three and that will be the vertex at which all of them. And keep in mind this drawing is not to scale, But where they all intersect, which is 2/3 of the way from anyone.

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