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Determine the equation of the perpendicular bisector of the line joining the points $(1,-3)$ and $(7,13).$

$$y=-\frac{3}{8} x+\frac{13}{2}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 9

Elements of Geometry

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Baylor University

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.

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for this problem, we need to find the perpendicular by sector of the line segment joined by the two points that we've been given. Now, we do not need a picture to solve this, but I'm going to graph it first. Just help us visualize what we're talking about. So I have two points. I have one negative three and I have 7. 13. 1234567 13. And so I'm going to do my best to draw a straight line connecting those two points. Okay, so what does a perpendicular by sector look like? Well, first it bisects my line, which means it's going to go through the mid point. And I'm just going to kind of approximate this year. Roughly there is going to be our midpoint and is perpendicular, so it's going to look about like this. Okay, So because it's a by sector, I know I need the midpoint. That will be a point on that blue line that we're trying to find. Okay, so I want the midpoint of the red line. Okay. If I have a point and a slope, I can write the equation. So what's the slope? Well, it's perpendicular to the red line, so I'm going to find the red slope. Perpendicular lines have negative reciprocal for slopes. So if I find the red slope, I can then take the negative reciprocal to find the slope of my blue line. Okay, so those are the two things we need. We need this point and a slope. So let's do these one at a time. First, let's find the midpoint. Remember, the midpoint of a line segment means I'm going to average the excess, and then I'm going to average the wise. So what is my midpoint? Well, first, I'm going to average the exes, so I'm going to add and divide by two. Same thing for the wise. I add and divide by two. And when I do, that gives me a midpoint of four five. Okay, so that's a point on my line. That's the point right here. And actually, 45 is right that greenpoint. So just eyeballing it, we got pretty close to having the correct midpoint, which is which is pretty good. Hey, now, what about our slope? Member Slope is rise over run. So I'm going to do this in red because This is my red slope. Okay, rise over. Run. That's the change. And why? Over the change in X. So let's see what this is changing. White. And these are bigger numbers. So I'm going to take that as my first point. So 13 minus negative. Three or 13 plus three over my change in x seven minus one. That gives me 16/6, which can simplify to eight thirds. Okay, so that's my red slope for my blue slope. I need the negative. Reciprocal. So that will be negative. 3/8. Okay, I have a point. I have a slope we can write. The equation of this line will use slope intercept use point slope form. So why minus y one equals m times X minus X one Our points Why minus five equals negative 3/8 X minus four And that's just pretty this up. We often put these into slope intercept form just little bit easier to read sometimes. So we'll get rid of our parenthesis. That gives me negative 3/8 x plus. Well, it's going to be 12 8th, which I can simplify to three halves. And now we're gonna add five So negative 3/8 x Adding five via common denominators was plus 10 halves, which is going to equal plus 13 halves, so this is the equation of that perpendicular by sector.

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