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Given the triangle with vertices $A(1,1), B(3,7),$ and $C(9,3)$. (a) Determine the equation of the three lines drawn from each vertex to the midpoint on the opposite side. These lines are called the medians of the triangle. (b) Determine the length of each median. (c) show these medians intersect in a point which is $2 / 3$ of the way from each vertex to the opposite side.

(a) $y=\frac{4}{5} x+\frac{1}{5}, y=-\frac{5}{2} x+\frac{29}{2}, y=-\frac{1}{7} x+\frac{30}{7}$(b) $\sqrt{41}, \sqrt{29}, 5 \sqrt{2}$$(c)(13 / 3,11 / 3)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 9

Elements of Geometry

Derivatives

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University of Michigan - Ann Arbor

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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

A median of a triangle is …

02:13

01:31

Geometry The medians of a …

for this problem. We're going to be examining the medians of a triangle. Now, as a reminder, if I have a triangle here, the medians connect that advertises with the corresponding midpoint on the other side of the triangle. And if you do it with straight lines and actually hit the medians or the mid points, the medians will intersect in the center of the triangle. So we're going to examine the media is we're gonna do this step by step. First, we're going to find the equation of these lines of these median lines. Okay, so our first one connects point A to the midpoint of line BC. So what is the midpoint of line BC? Well, to find the midpoint, we take the average of the exes and the average of the wise. So for the X is three plus nine is 12 average is six for the Y. Seven plus three is 10, the average is five. And now I need the slope of this line going from a to that 0.65 member slope is why one minus y two over X one, minus x two. So, my wise are five minus one. My exes are six minus one. So my slope is 4/5. So now I have a point in the slope so I can use the point slope form, which is why minus Why one and I'm going to use point is my point here. Why? Minus why one equals my slope. M times x minus X one and I'm going to solve this it put it in the slope intercept form. I'm gonna get rid of my parenthesis and now we add one. So that gives me why equals 4 50 x plus 1/5. So there's my first media. Second one. I'm gonna do this in different colours just to keep them separate here. My second median goes from Vertex B to the mid point of the line A C So we're gonna do just what we did last time. We're going to find the midpoint, and we're going to find the slope. Okay, so the midpoint we take points A and C and we average them. One plus nine is 10. So the average is five for the wise. One plus three is four and the average is choose. Now for my slope. I'm going to compare the wise so three minus one over the X is nine minus one. So that's going to give me 2/8 or 1/4. I apologize. I just realized I pulled the wrong number. I'm going to raise that slope. The midpoint is correct, but I used the wrong number for the slope. So let me try that again. I'm looking for the slope from B to the midpoint. I used a I apologize for that. So, using B and the midpoint, the difference in my wise over the difference in my exes that becomes negative. Five halves. Much better. Okay, so now we have a point in a slope. We can use the point slope form. Why? Minus white. One equals m times X minus X one. Okay. And I just used the midpoint this time and I could have put be in. They both are on the line. They both work. Um, so let's put this into the same form we did for the other one. First we get rid of our parentheses, so I have negative Five have X plus 25 halves. And now we add to negative five halves X plus 29 halves. Okay, so there's the equation for our second medium. And now for our third, we'll do this one in red. I'm going from the Vertex at sea to the midpoint of the line A B. So we've had gotten all three here again, just like last time. We're going to find the midpoint and we're going to find the slope. So let's look at a B and find the mid point. One plus three is four. The average is 21 plus seven is eight and the average is four. So there's our midpoint and our slope. I'm going from sea to my mid point. So four minus three over two minus nine. That gives me a slope of negative 1/7. So again I can use either point C or the midpoint. How many is the midpoint? Because the numbers are a little smaller. Y minus four equals negative. 1/7 times, X minus two. Let's get rid of our parentheses. Negative 1/7 X plus 2/7 and I'm going to add four to both sides. That gives me negative 17 x Common denominator means I'm adding 28 7th or 37th. Okay, so those are my three, uh equations for, uh, those three mediums. Now, our next step, we want to find the length of each medium. So the length is the distance. How far is it between these two points? Okay, so once again, I'm gonna keep my colors consistent. Um, I'm gonna write the equations here, and then I'm going to scroll down, so I've got a little bit more room. Remember our distance formula. This is based on the Pythagorean theorem. We take the square root of two different numbers squared. I'm looking at the difference of my exes squared, plus the difference of my wife square. So those would be the legs of a right triangle, the distance that corresponds to the hypotenuse. So, inside the parentheses, the difference between my exes and again I'm going from a to that midpoint. So the difference of my exes is six minus one. The difference of my wise is five minus one. Okay, I'm just gonna go across and write all of these in and then solve them again. Square root of two numbers, the difference of my exes. And that's the midpoint and point B. And the difference of my wise again, mid point and point B and for my third median. Same idea. I'm looking at the midpoint. Look at my ex's first midpoint and see Midpoint and see. Okay, so let me scroll lots of room here and I'll solve these one at a time. Well, six minus one is five. So that's gonna be 25 four square to 16. So this first medium has a length of the square root of 41. Okay, Second media, five minutes. Three. That's two squared, which is four negative. Five squared is 25. So this distance is the square root of 29. And for the third one, negative seven square, just 49. I have one which is the square root of 50 and I can factor out of five and say this is five square root of two. Okay, Next we have we have two more pieces here. Next, we want to show that the medians intersect in a point which is two thirds of the way from each Vertex to the opposite side. So one question, but a two part answer. We need to figure out where they intersect. And then we'll see how far that is from each Vertex well compared to those distances we just found. So where do they intersect? And I'll do black on this one because it's not really tied to any one particular medium. Where they intersect is where the X's and wise that solves these equations the same X and Y need to go into each of the three equations we found. So I'm going to take the first two equations and set them equal to each other. Okay, they're both equal to y. So I can say that 4 50 x plus 1/5 equals my second equation. Negative five halves X plus 29 halves. I could've picked any two of these as they all meet in the same place. Any pairing will give me the same answer for X and Y. I just picked the first two because they were first. Okay, so let's put our life terms together. I have 4 50 x plus five halves X and then I'm going to have 29 halves minus 1/5. So the X is on one side, Constance on the other. Hey, I'm going to have to find a common denominator. And across the board it's going to be 10 So first I'm multiplying by two Dana multiplying by five. Same thing over here. I'm going to have common denominator of 10. I multiplied by five. That gives me 143. I'm sorry. 145 and I multiply by two. So putting these together I have 33 10th times X equals 143 10th. Multiply both sides by that reciprocal. So that's gone. My tents cancel. And I have a common thing. 11 goes into 33 3 times and 11 goes into 33 or 143 13 times. So X is 13 3rd. Okay. How is this value? What's why? Well, again, I can take any one of these. I want to, um I'm not sure I take the third equation. Why not let this one have a little bit of a say in here again? They all intersect the same place. I could pick any one of the three. I have negative 1/7 times X, which is 13 3rd plus 37th. So I have negative 13. 20 firsts. Common denominator. I'm going to multiply by three. That's 90 20 firsts. Okay, 90 minus 13. That's going to give me 77/21 and seven will go into both of these, so I'll have 11 over. I'm sorry. That is not a very nice neat number. I would have 11/3, so that's my Y value. So the point of intersection is 13 3rd and 11 3rd. Okay, so we are almost done going to come up just a little bit more. Okay? So you can see all three of our distances. This is our last step. I want to know the distance from the Vertex to that point we just found so for a again, I'm going to rewrite this because I've scrolled up a is 11 and I want to find the distance from that point to that point, we just found that point of intersection. So we'll set up that distance that is going to be the difference in my X is squared, plus the difference in my wise squared. And when we do that, we actually find that it turns into two square root of 41/3. Now, the whole distance of that median was square root of 41. We found that two steps ago. Now the distance from that vertex to the intersection point is two thirds of that distance. And if we set up the same thing the distance from B to the intersection, same thing with our. If we set up that distance formula, you'll find that it's two thirds of square root of 29. And if we do see distance from sea to that point of intersection, it's going to be two thirds of that five halves or that 55 squared of 2 10 square root of 2/3. So each one of these really, truly, is two thirds of the distance.

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