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So these two lines are parallel. These two lines are parallel. They have the exact same slope. And I encourage you to find the equations of both of these lines and graph both of these lines and verify for yourself that they are indeed parallel. Let's do this one. Once again, this is just an exercise in finding slopes. So this first line has those points.
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Parallel & perpendicular lines from graph.mp3
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And I encourage you to find the equations of both of these lines and graph both of these lines and verify for yourself that they are indeed parallel. Let's do this one. Once again, this is just an exercise in finding slopes. So this first line has those points. Let's figure out its slope. The slope of this first line, one line passes through these points. So let's see, 3 minus negative 3, that's our change in y, over 3 minus negative 6.
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Parallel & perpendicular lines from graph.mp3
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So this first line has those points. Let's figure out its slope. The slope of this first line, one line passes through these points. So let's see, 3 minus negative 3, that's our change in y, over 3 minus negative 6. So this is the same thing as 3 plus 3, which is 6, over 3 plus 6, which is 9. So this first line has a slope of 2 thirds. What is the second line's slope?
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Parallel & perpendicular lines from graph.mp3
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So let's see, 3 minus negative 3, that's our change in y, over 3 minus negative 6. So this is the same thing as 3 plus 3, which is 6, over 3 plus 6, which is 9. So this first line has a slope of 2 thirds. What is the second line's slope? This is the second line there. That's the other line passing through these points. So the other line's slope, let's see, we could say negative 8 minus 4, over 2 minus negative 6.
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Parallel & perpendicular lines from graph.mp3
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What is the second line's slope? This is the second line there. That's the other line passing through these points. So the other line's slope, let's see, we could say negative 8 minus 4, over 2 minus negative 6. So what is this equal to? Negative 8 minus 4 is negative 12. 2 minus negative 6, that's the same thing as 2 plus 6.
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Parallel & perpendicular lines from graph.mp3
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So the other line's slope, let's see, we could say negative 8 minus 4, over 2 minus negative 6. So what is this equal to? Negative 8 minus 4 is negative 12. 2 minus negative 6, that's the same thing as 2 plus 6. The negatives cancel out. So it's negative 12 over 8, which is the same thing if we divide the numerator and denominator by 4. That's negative 3 halves.
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Parallel & perpendicular lines from graph.mp3
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2 minus negative 6, that's the same thing as 2 plus 6. The negatives cancel out. So it's negative 12 over 8, which is the same thing if we divide the numerator and denominator by 4. That's negative 3 halves. Notice these guys are the negative inverse of each other. If I take negative 1 over 2 thirds, that is equal to negative 1 times 3 halves, which is equal to negative 3 halves. These guys are the negative inverses of each other.
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Parallel & perpendicular lines from graph.mp3
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That's negative 3 halves. Notice these guys are the negative inverse of each other. If I take negative 1 over 2 thirds, that is equal to negative 1 times 3 halves, which is equal to negative 3 halves. These guys are the negative inverses of each other. You swap the numerator and denominator, make them negative, and they become equal to each other. So these two lines are perpendicular. I encourage you to find the equations.
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Parallel & perpendicular lines from graph.mp3
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Let's think a little bit about the volume of a cone. So a cone would have a circular base, or I guess depends on how you want to draw it. Think of like a conical hat of some kind. It would have a circle as a base, and it would come to some point. So it would look something like that. This you could consider this to be a cone, just like that. Or you can make it upside down if you're thinking of an ice cream cone.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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It would have a circle as a base, and it would come to some point. So it would look something like that. This you could consider this to be a cone, just like that. Or you can make it upside down if you're thinking of an ice cream cone. So it might look something like that. That's the top of it. And then it comes down like this.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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Or you can make it upside down if you're thinking of an ice cream cone. So it might look something like that. That's the top of it. And then it comes down like this. This also is kind of those disposable cups of water. You might see at some water coolers. And the important things that we need to think about when we want to know about the volume of a cone is we definitely want to know the radius of the base.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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And then it comes down like this. This also is kind of those disposable cups of water. You might see at some water coolers. And the important things that we need to think about when we want to know about the volume of a cone is we definitely want to know the radius of the base. So that's the radius of the base. Or here's the radius of the top part. You definitely want to know that radius.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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And the important things that we need to think about when we want to know about the volume of a cone is we definitely want to know the radius of the base. So that's the radius of the base. Or here's the radius of the top part. You definitely want to know that radius. And you want to know the height of the cone. So let's call that h. Or right over here, you could call this distance right over here h. And the formula for the volume of a cone, and it's interesting because it's close to the formula for the volume of a cylinder in a very clean way, which is somewhat surprising. And that's what's neat about a lot of this three dimensional geometry is that it's not as messy as you would think it would be.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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You definitely want to know that radius. And you want to know the height of the cone. So let's call that h. Or right over here, you could call this distance right over here h. And the formula for the volume of a cone, and it's interesting because it's close to the formula for the volume of a cylinder in a very clean way, which is somewhat surprising. And that's what's neat about a lot of this three dimensional geometry is that it's not as messy as you would think it would be. It is the area of the base. Well, what's the area of the base? Well, that's going to be the area of the base is going to be pi r squared times the height.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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And that's what's neat about a lot of this three dimensional geometry is that it's not as messy as you would think it would be. It is the area of the base. Well, what's the area of the base? Well, that's going to be the area of the base is going to be pi r squared times the height. And if you just multiply the height times pi r squared, that would give you the volume of an entire cylinder that looks something like that. So this would give you this entire volume that looks of the figure that looks like this, where its center of the top is the tip right over here. So if I just left it as pi r squared h, or h times pi r squared, it's the volume of this entire can, this entire cylinder.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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Well, that's going to be the area of the base is going to be pi r squared times the height. And if you just multiply the height times pi r squared, that would give you the volume of an entire cylinder that looks something like that. So this would give you this entire volume that looks of the figure that looks like this, where its center of the top is the tip right over here. So if I just left it as pi r squared h, or h times pi r squared, it's the volume of this entire can, this entire cylinder. But if you just want the cone, it's 1 third of that. It is 1 third of that. And that's what I mean when I say it's surprisingly clean, that this cone right over here is 1 third the volume of this cylinder that is essentially, you can kind of think of the cylinder as bounding around it.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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So if I just left it as pi r squared h, or h times pi r squared, it's the volume of this entire can, this entire cylinder. But if you just want the cone, it's 1 third of that. It is 1 third of that. And that's what I mean when I say it's surprisingly clean, that this cone right over here is 1 third the volume of this cylinder that is essentially, you can kind of think of the cylinder as bounding around it. Or if you wanted to rewrite this, you could write this as 1 third times pi, or pi over 3 times h r squared, however you want to view it. The easy way I remember it, for me, the volume of a cylinder is very intuitive. You take the area of the base, and then you multiply that times the height.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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And that's what I mean when I say it's surprisingly clean, that this cone right over here is 1 third the volume of this cylinder that is essentially, you can kind of think of the cylinder as bounding around it. Or if you wanted to rewrite this, you could write this as 1 third times pi, or pi over 3 times h r squared, however you want to view it. The easy way I remember it, for me, the volume of a cylinder is very intuitive. You take the area of the base, and then you multiply that times the height. And so a volume of a cone is just 1 third of that. It's just 1 third the volume of the bounding cylinder is one way to think about it. But let's just apply these numbers just to make sure that it makes sense to us.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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You take the area of the base, and then you multiply that times the height. And so a volume of a cone is just 1 third of that. It's just 1 third the volume of the bounding cylinder is one way to think about it. But let's just apply these numbers just to make sure that it makes sense to us. So let's say that this is some type of a conical glass, the type that you might see at the water cooler. And let's say that we're told that it holds, so this thing holds 131 cubic centimeters of water. And let's say that we're also told that its height right over here, actually I want to do that in a different color.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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But let's just apply these numbers just to make sure that it makes sense to us. So let's say that this is some type of a conical glass, the type that you might see at the water cooler. And let's say that we're told that it holds, so this thing holds 131 cubic centimeters of water. And let's say that we're also told that its height right over here, actually I want to do that in a different color. We're told that the height of this cone is 5 centimeters. And so given that, what is roughly the radius of the top of this glass? Let's just say to the nearest tenth of a centimeter.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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And let's say that we're also told that its height right over here, actually I want to do that in a different color. We're told that the height of this cone is 5 centimeters. And so given that, what is roughly the radius of the top of this glass? Let's just say to the nearest tenth of a centimeter. Well, we just once again have to apply the formula. The volume, which is 131 cubic centimeters, is going to be equal to 1 third times pi times the height, which is 5 centimeters, times the radius squared. Or if we wanted to solve for the radius squared, we could just divide both sides by all of this business.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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Let's just say to the nearest tenth of a centimeter. Well, we just once again have to apply the formula. The volume, which is 131 cubic centimeters, is going to be equal to 1 third times pi times the height, which is 5 centimeters, times the radius squared. Or if we wanted to solve for the radius squared, we could just divide both sides by all of this business. And we would get radius squared is equal to 131 cubic centimeters. You divide by 1 third. That's the same thing as multiplying by 3.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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Or if we wanted to solve for the radius squared, we could just divide both sides by all of this business. And we would get radius squared is equal to 131 cubic centimeters. You divide by 1 third. That's the same thing as multiplying by 3. And then of course, you're going to divide by pi. And you're going to divide by 5 centimeters. Now let's see if we can clean this up.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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That's the same thing as multiplying by 3. And then of course, you're going to divide by pi. And you're going to divide by 5 centimeters. Now let's see if we can clean this up. Centimeters will cancel out with one of these centimeters, so you'll just be left with square centimeters, only in the numerator. And so this is going to be, and then to solve for r, we could take the square root of both sides. So we could say that r is going to be equal to the square root of 3 times 131 is 393 over 5 pi.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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Now let's see if we can clean this up. Centimeters will cancel out with one of these centimeters, so you'll just be left with square centimeters, only in the numerator. And so this is going to be, and then to solve for r, we could take the square root of both sides. So we could say that r is going to be equal to the square root of 3 times 131 is 393 over 5 pi. So that's this part right over here. And the square root, once again, remember, we can treat units just like algebraic quantities. The square root of centimeters squared, well, that's just going to be centimeters, which is nice because we want our units in centimeters.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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So we could say that r is going to be equal to the square root of 3 times 131 is 393 over 5 pi. So that's this part right over here. And the square root, once again, remember, we can treat units just like algebraic quantities. The square root of centimeters squared, well, that's just going to be centimeters, which is nice because we want our units in centimeters. So let's get our calculator out to actually calculate this kind of messy expression. Turn it on. Let's see.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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The square root of centimeters squared, well, that's just going to be centimeters, which is nice because we want our units in centimeters. So let's get our calculator out to actually calculate this kind of messy expression. Turn it on. Let's see. Square root of 393 divided by 5 times pi is equal to 5. Well, that's pretty close. To the nearest, it's pretty much 5 centimeters.
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Volume of a cone Perimeter, area, and volume Geometry Khan Academy.mp3
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And one simple way to think about density is it's a quantity of something, and we're going to think about examples of it, per unit volume, so per volume. So for example, let's say that I have a cubic meter right over here. Let me have two different cubic meters, just to give you an example. So these are both cubic meters. And let's say in the one on the left, I have a quantity of, let's say, six of these dots per cubic meter. And over here, I only have three of these dots per cubic meter. Well, here I have a higher density.
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Volume density.mp3
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So these are both cubic meters. And let's say in the one on the left, I have a quantity of, let's say, six of these dots per cubic meter. And over here, I only have three of these dots per cubic meter. Well, here I have a higher density. And in general, we're gonna take the quantity and divide it by the volume. And the units are going to be some quantity per unit volume. Now, you're typically going to see mass per unit volume, but density, especially in the volume context, can refer to any quantity per unit volume.
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Volume density.mp3
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Well, here I have a higher density. And in general, we're gonna take the quantity and divide it by the volume. And the units are going to be some quantity per unit volume. Now, you're typically going to see mass per unit volume, but density, especially in the volume context, can refer to any quantity per unit volume. Now, with that out of the way, let's give ourselves a little bit of an example. So here we're told that stone spheres thought to be carved by the Deke people, I'm not sure if I'm pronouncing that correctly, more than 1,000 years ago, are a national symbol of Costa Rica. One such sphere has a diameter of about 1.8 meters and masses about 8,300 kilograms.
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Volume density.mp3
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Now, you're typically going to see mass per unit volume, but density, especially in the volume context, can refer to any quantity per unit volume. Now, with that out of the way, let's give ourselves a little bit of an example. So here we're told that stone spheres thought to be carved by the Deke people, I'm not sure if I'm pronouncing that correctly, more than 1,000 years ago, are a national symbol of Costa Rica. One such sphere has a diameter of about 1.8 meters and masses about 8,300 kilograms. Based on these measurements, what is the density of this sphere in kilograms per cubic meter? Round to the nearest 100 kilograms per cubic meter. So pause this video and see if you can figure that out.
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Volume density.mp3
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One such sphere has a diameter of about 1.8 meters and masses about 8,300 kilograms. Based on these measurements, what is the density of this sphere in kilograms per cubic meter? Round to the nearest 100 kilograms per cubic meter. So pause this video and see if you can figure that out. All right, so we're going to wanna get kilograms per cubic meter. So we know the total number of kilograms in one point, in a sphere that has a diameter of 1.8 meters, so that's the total number of kilograms, but we don't know the volume just yet. So we have a sphere like this, this would be a cross section of it, its diameter is 1.8 meters.
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Volume density.mp3
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So pause this video and see if you can figure that out. All right, so we're going to wanna get kilograms per cubic meter. So we know the total number of kilograms in one point, in a sphere that has a diameter of 1.8 meters, so that's the total number of kilograms, but we don't know the volume just yet. So we have a sphere like this, this would be a cross section of it, its diameter is 1.8 meters. Now, you may or may not already know that the volume of a sphere is given by 4 3rds pi r cubed. And so the radius here is 0.9 meters, and so that would be the r right over here. So the volume of one of these spheres is going to be, let me write it over here, the volume is going to be 4 3rds pi times 0.9 to the 3rd power.
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Volume density.mp3
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So we have a sphere like this, this would be a cross section of it, its diameter is 1.8 meters. Now, you may or may not already know that the volume of a sphere is given by 4 3rds pi r cubed. And so the radius here is 0.9 meters, and so that would be the r right over here. So the volume of one of these spheres is going to be, let me write it over here, the volume is going to be 4 3rds pi times 0.9 to the 3rd power. And we know what the mass is. The mass in that volume is 8,300 kilograms. So we would know that the density, the density in this situation is going to be 8,300 kilograms, 8,300 kilograms per this many cubic meters, 4 3rds pi times 0.9 to the 3rd power cubic meters.
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Volume density.mp3
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So the volume of one of these spheres is going to be, let me write it over here, the volume is going to be 4 3rds pi times 0.9 to the 3rd power. And we know what the mass is. The mass in that volume is 8,300 kilograms. So we would know that the density, the density in this situation is going to be 8,300 kilograms, 8,300 kilograms per this many cubic meters, 4 3rds pi times 0.9 to the 3rd power cubic meters. And we're gonna need a calculator for this, and we're gonna round to the nearest 100 kilograms. So we have 8,300 divided by, and let me just open parentheses here, 4 divided by 3 times pi times 0.9 to the 3rd power. And then I'm going to close my parentheses, is equal to this right over here.
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Volume density.mp3
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The measure of two angles of an isosceles triangle are three x plus five degrees, we'll say, and x plus 16 degrees. Find all possible values of x. So let's think about this. Let's draw ourselves an isosceles triangle or two. So it's an isosceles triangle, like that and like that. And actually, let me draw a couple of them, just because we want to think about all of the different possibilities here. All of the different possibilities.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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Let's draw ourselves an isosceles triangle or two. So it's an isosceles triangle, like that and like that. And actually, let me draw a couple of them, just because we want to think about all of the different possibilities here. All of the different possibilities. So we know from what we know about isosceles triangles that the base angles are going to be congruent. So that angle's going to be equal to that angle, that angle's going to be equal to that angle. And so what could the three x plus five degrees and the x plus 16, what could they be measures of?
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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All of the different possibilities. So we know from what we know about isosceles triangles that the base angles are going to be congruent. So that angle's going to be equal to that angle, that angle's going to be equal to that angle. And so what could the three x plus five degrees and the x plus 16, what could they be measures of? Well, maybe this one right over here has a measure of three x plus five degrees, and the vertex is the other one. So maybe this one up here is the x plus 16, x plus 16 degrees. The other possibility, the other possibilities is that this is describing both base angles, in which case they would be equal.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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And so what could the three x plus five degrees and the x plus 16, what could they be measures of? Well, maybe this one right over here has a measure of three x plus five degrees, and the vertex is the other one. So maybe this one up here is the x plus 16, x plus 16 degrees. The other possibility, the other possibilities is that this is describing both base angles, in which case they would be equal. So maybe this one is three x plus five, and maybe this one over here is x plus 16. X plus 16. And that is, and then the final possibility, actually we haven't exhausted all of them, is if we swap these two.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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The other possibility, the other possibilities is that this is describing both base angles, in which case they would be equal. So maybe this one is three x plus five, and maybe this one over here is x plus 16. X plus 16. And that is, and then the final possibility, actually we haven't exhausted all of them, is if we swap these two. If this one is x plus 16 and that one is three x plus five. So let me draw ourselves another triangle. Let me draw ourselves another triangle.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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And that is, and then the final possibility, actually we haven't exhausted all of them, is if we swap these two. If this one is x plus 16 and that one is three x plus five. So let me draw ourselves another triangle. Let me draw ourselves another triangle. And obviously swapping these two aren't going to make a difference because they are equal to each other. And then we can make that one equal to three x plus five, but that's not going to change anything either because they're equal to each other. So the last situation is where this angle down here is x plus 16, and this angle up here is x plus, is three x plus five.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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Let me draw ourselves another triangle. And obviously swapping these two aren't going to make a difference because they are equal to each other. And then we can make that one equal to three x plus five, but that's not going to change anything either because they're equal to each other. So the last situation is where this angle down here is x plus 16, and this angle up here is x plus, is three x plus five. This is three x plus five. So let's just work through each of these. So in this situation, if this base angle is three x plus five, so is this base angle.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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So the last situation is where this angle down here is x plus 16, and this angle up here is x plus, is three x plus five. This is three x plus five. So let's just work through each of these. So in this situation, if this base angle is three x plus five, so is this base angle. And then we know that all three of these are going to have to add up to 180 degrees. So we get three x plus five, plus three x plus five, plus x plus 16, plus x plus 16 is going to be equal to 180 degrees. We have three x, let's just add up the, you have three x plus three x, which gives you six x, plus another x gives you seven x.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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So in this situation, if this base angle is three x plus five, so is this base angle. And then we know that all three of these are going to have to add up to 180 degrees. So we get three x plus five, plus three x plus five, plus x plus 16, plus x plus 16 is going to be equal to 180 degrees. We have three x, let's just add up the, you have three x plus three x, which gives you six x, plus another x gives you seven x. Seven x. And then you have five plus five, which is 10, plus 16 is equal to 26. Is equal to 26, and that is going to be equal to 180.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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We have three x, let's just add up the, you have three x plus three x, which gives you six x, plus another x gives you seven x. Seven x. And then you have five plus five, which is 10, plus 16 is equal to 26. Is equal to 26, and that is going to be equal to 180. And then we have, let's see, 180 minus 26. If we subtract 26 from both sides, we get 180 minus, 180 minus 20 is 160, minus another six is 154. 150, 154.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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Is equal to 26, and that is going to be equal to 180. And then we have, let's see, 180 minus 26. If we subtract 26 from both sides, we get 180 minus, 180 minus 20 is 160, minus another six is 154. 150, 154. You have seven x is equal to 154. Seven x is equal to 154. And let's see how many times this, if we divide both sides by seven, seven will go into 140, 20 times.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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150, 154. You have seven x is equal to 154. Seven x is equal to 154. And let's see how many times this, if we divide both sides by seven, seven will go into 140, 20 times. And then you have another 14, so it looks like it's 22 times. So x is equal to 22. Is that right?
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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And let's see how many times this, if we divide both sides by seven, seven will go into 140, 20 times. And then you have another 14, so it looks like it's 22 times. So x is equal to 22. Is that right? 20 times seven is 140. 140 plus 14 is 154. So we have x is equal to 22 in this first, 22 degrees in this first scenario.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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Is that right? 20 times seven is 140. 140 plus 14 is 154. So we have x is equal to 22 in this first, 22 degrees in this first scenario. Now let's think about the second scenario over here. Now we have, now we have these two characters are going to be equal to each other, because they're both the base angles. So you have three x plus five is equal to x plus 16.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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So we have x is equal to 22 in this first, 22 degrees in this first scenario. Now let's think about the second scenario over here. Now we have, now we have these two characters are going to be equal to each other, because they're both the base angles. So you have three x plus five is equal to x plus 16. Well, you can subtract x from both sides, and so this becomes two x plus five is equal to 16. We can subtract five from both sides, and you get two x is equal to 11. And then you can divide both sides, you can divide both sides by two, and you get x is equal to 11, is equal to 11 halves.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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So you have three x plus five is equal to x plus 16. Well, you can subtract x from both sides, and so this becomes two x plus five is equal to 16. We can subtract five from both sides, and you get two x is equal to 11. And then you can divide both sides, you can divide both sides by two, and you get x is equal to 11, is equal to 11 halves. So that is our second scenario. And then we do our third scenario right over here. If this base angle is x plus 16, then this base angle right over here is also going to be x plus 16.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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And then you can divide both sides, you can divide both sides by two, and you get x is equal to 11, is equal to 11 halves. So that is our second scenario. And then we do our third scenario right over here. If this base angle is x plus 16, then this base angle right over here is also going to be x plus 16. They are congruent. And then we can do the same thing that we did for the first scenario. All of these angles are going to have to add up to 180 degrees.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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If this base angle is x plus 16, then this base angle right over here is also going to be x plus 16. They are congruent. And then we can do the same thing that we did for the first scenario. All of these angles are going to have to add up to 180 degrees. So we have x plus 16 plus x plus 16 plus three x plus five, plus three x plus five. When you add them all together, you're going to get 180 degrees. Now let's add up all the x terms.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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All of these angles are going to have to add up to 180 degrees. So we have x plus 16 plus x plus 16 plus three x plus five, plus three x plus five. When you add them all together, you're going to get 180 degrees. Now let's add up all the x terms. X plus x is two x plus three x is five x. So we get five x, five x, and then you have 16 plus 16 is 32. 32 plus five is 37.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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Now let's add up all the x terms. X plus x is two x plus three x is five x. So we get five x, five x, and then you have 16 plus 16 is 32. 32 plus five is 37. Plus 37 is equal to 180 degrees. Is equal to 180 degrees. Subtract 37 from both sides, and we get five x.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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32 plus five is 37. Plus 37 is equal to 180 degrees. Is equal to 180 degrees. Subtract 37 from both sides, and we get five x. Five x is equal to, 180 minus 30 is 150, so that gets us to 143. So it's not going to divide nicely. Divide both sides by five, and you get x is equal to 143 over five, which we could just leave as an improper fraction.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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Subtract 37 from both sides, and we get five x. Five x is equal to, 180 minus 30 is 150, so that gets us to 143. So it's not going to divide nicely. Divide both sides by five, and you get x is equal to 143 over five, which we could just leave as an improper fraction. You could write it as a mixed number, or however else you might want to write it. And we're done. These are the three possible values, the three possible values of x, given the information that they gave us right up there.
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Another isosceles example problem Congruence Geometry Khan Academy.mp3
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So let's start with an example. Let's say I have the point. I'll do it in a darker color so we can see it on the graph paper, let's say I have the point 3, negative 4. So if I were to graph it, I'd go 1, 2, 3, and then I'd go down 4. 1, 2, 3, 4 right there is 3, negative 4. And let's say I also have the point 6, 0. So 1, 2, 3, 4, 5, 6.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So if I were to graph it, I'd go 1, 2, 3, and then I'd go down 4. 1, 2, 3, 4 right there is 3, negative 4. And let's say I also have the point 6, 0. So 1, 2, 3, 4, 5, 6. And then there's no movement in the y direction. We're just sitting on the x-axis. The y-coordinate is 0, so that's 6, 0.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So 1, 2, 3, 4, 5, 6. And then there's no movement in the y direction. We're just sitting on the x-axis. The y-coordinate is 0, so that's 6, 0. And what I want to figure out is the distance between these two points. How far is this blue point away from this orange point? And at first you're like, gee, Sal, I don't think I've ever seen anything about how to solve for a distance like this, and what are you even talking about, the Pythagorean theorem?
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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The y-coordinate is 0, so that's 6, 0. And what I want to figure out is the distance between these two points. How far is this blue point away from this orange point? And at first you're like, gee, Sal, I don't think I've ever seen anything about how to solve for a distance like this, and what are you even talking about, the Pythagorean theorem? I don't see a triangle there. And if you don't see a triangle, let me draw it for you. Let me draw the triangle.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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And at first you're like, gee, Sal, I don't think I've ever seen anything about how to solve for a distance like this, and what are you even talking about, the Pythagorean theorem? I don't see a triangle there. And if you don't see a triangle, let me draw it for you. Let me draw the triangle. Let me draw this triangle right there, just like that. Let me actually do several colors here just to really hit the point home. So there's our triangle.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Let me draw the triangle. Let me draw this triangle right there, just like that. Let me actually do several colors here just to really hit the point home. So there's our triangle. And you might immediately recognize this is a right triangle. This is a right angle right there. The base goes straight left to right.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So there's our triangle. And you might immediately recognize this is a right triangle. This is a right angle right there. The base goes straight left to right. The right side goes straight up and down. So we're dealing with a right triangle. So if we could just figure out what the base length is and what this height is, we could use the Pythagorean theorem to figure out this long side, the side that is opposite the right angle, the hypotenuse.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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The base goes straight left to right. The right side goes straight up and down. So we're dealing with a right triangle. So if we could just figure out what the base length is and what this height is, we could use the Pythagorean theorem to figure out this long side, the side that is opposite the right angle, the hypotenuse. This right here, the distance is the hypotenuse of this right triangle. Let me write that down. The distance is equal to the hypotenuse of this right triangle.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So if we could just figure out what the base length is and what this height is, we could use the Pythagorean theorem to figure out this long side, the side that is opposite the right angle, the hypotenuse. This right here, the distance is the hypotenuse of this right triangle. Let me write that down. The distance is equal to the hypotenuse of this right triangle. So let me draw it a little bit bigger. So this is the hypotenuse right there. And then we have the side on the right, the side that goes straight up and down.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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The distance is equal to the hypotenuse of this right triangle. So let me draw it a little bit bigger. So this is the hypotenuse right there. And then we have the side on the right, the side that goes straight up and down. And then we have our base. Now, how do we figure out? Let's call this d for distance.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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And then we have the side on the right, the side that goes straight up and down. And then we have our base. Now, how do we figure out? Let's call this d for distance. That's the length of our hypotenuse. How do we figure out the lengths of this up and down side and the base side right here? So let's look at the base first.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Let's call this d for distance. That's the length of our hypotenuse. How do we figure out the lengths of this up and down side and the base side right here? So let's look at the base first. What is this distance? You could even count it on this graph paper. But here, we're at x is equal to 3.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So let's look at the base first. What is this distance? You could even count it on this graph paper. But here, we're at x is equal to 3. And here, we're at x is equal to 6. We're just moving straight right. This is the same distance as that distance right there.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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But here, we're at x is equal to 3. And here, we're at x is equal to 6. We're just moving straight right. This is the same distance as that distance right there. So to figure out that distance, it's literally the end x point. And you can actually go either way, because you're going to square everything. So it doesn't matter if you get negative numbers.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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This is the same distance as that distance right there. So to figure out that distance, it's literally the end x point. And you can actually go either way, because you're going to square everything. So it doesn't matter if you get negative numbers. So it's going to be 6. The distance here is going to be 6 minus 3. That's this distance right here, which is equal to 3.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So it doesn't matter if you get negative numbers. So it's going to be 6. The distance here is going to be 6 minus 3. That's this distance right here, which is equal to 3. So we figured out the base. And just to remind ourselves, that is equal to the change in x. That was equal to your finishing x minus your starting x.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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That's this distance right here, which is equal to 3. So we figured out the base. And just to remind ourselves, that is equal to the change in x. That was equal to your finishing x minus your starting x. 6 minus 3. This is our delta x. Now, by the same exact line of reasoning, this height right here is going to be your change in y.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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That was equal to your finishing x minus your starting x. 6 minus 3. This is our delta x. Now, by the same exact line of reasoning, this height right here is going to be your change in y. Up here, you're at y is equal to 0. That's kind of where you finish. That's your higher y point.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Now, by the same exact line of reasoning, this height right here is going to be your change in y. Up here, you're at y is equal to 0. That's kind of where you finish. That's your higher y point. And over here, you're at y is equal to negative 4. So change in y is equal to 0 minus negative 4. I'm just taking the larger y value minus the smaller y value, the larger x value minus the smaller x value.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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That's your higher y point. And over here, you're at y is equal to negative 4. So change in y is equal to 0 minus negative 4. I'm just taking the larger y value minus the smaller y value, the larger x value minus the smaller x value. But you're going to see, we're going to square it in a second. So even if you did it the other way around, you get a negative number, but you still get the same answer. So this is equal to 4.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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I'm just taking the larger y value minus the smaller y value, the larger x value minus the smaller x value. But you're going to see, we're going to square it in a second. So even if you did it the other way around, you get a negative number, but you still get the same answer. So this is equal to 4. So this side is equal to 4. You could even count it on the graph paper if you like. And this side is equal to 3.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So this is equal to 4. So this side is equal to 4. You could even count it on the graph paper if you like. And this side is equal to 3. And now we can do the Pythagorean theorem. This distance is the distance squared. Be careful.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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And this side is equal to 3. And now we can do the Pythagorean theorem. This distance is the distance squared. Be careful. The distance squared is going to be equal to this delta x squared, the change in x squared plus the change in y squared. This is nothing fancy. Sometimes people will call this the distance formula.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Be careful. The distance squared is going to be equal to this delta x squared, the change in x squared plus the change in y squared. This is nothing fancy. Sometimes people will call this the distance formula. It's just the Pythagorean theorem. This side squared plus that side squared is equal to hypotenuse squared, because this is a right triangle. So let's apply it with these numbers, the numbers that we have at hand.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Sometimes people will call this the distance formula. It's just the Pythagorean theorem. This side squared plus that side squared is equal to hypotenuse squared, because this is a right triangle. So let's apply it with these numbers, the numbers that we have at hand. So the distance squared is going to be equal to delta x squared is 3 squared plus delta y squared plus 4 squared, which is equal to 9 plus 16, which is equal to 25. So the distance is equal to, let me write that, d squared is equal to 25. d, our distance, is equal to, you don't want to take the negative square root, because you can't have a negative distance, so only the principal root, the positive square root of 25, which is equal to 5. So this distance right here is 5.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So let's apply it with these numbers, the numbers that we have at hand. So the distance squared is going to be equal to delta x squared is 3 squared plus delta y squared plus 4 squared, which is equal to 9 plus 16, which is equal to 25. So the distance is equal to, let me write that, d squared is equal to 25. d, our distance, is equal to, you don't want to take the negative square root, because you can't have a negative distance, so only the principal root, the positive square root of 25, which is equal to 5. So this distance right here is 5. Or if we look at this distance right here, that was the original problem. How far is this point from that point? It is 5 units away.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So this distance right here is 5. Or if we look at this distance right here, that was the original problem. How far is this point from that point? It is 5 units away. So what you'll see here, they call it the distance formula, but it's just the Pythagorean theorem. And just so you're exposed to all of the ways that you'll see the distance formula, sometimes people will say, oh, if I have two points, if I have one point, let's call it x1 and y1. So that's just a particular point.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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It is 5 units away. So what you'll see here, they call it the distance formula, but it's just the Pythagorean theorem. And just so you're exposed to all of the ways that you'll see the distance formula, sometimes people will say, oh, if I have two points, if I have one point, let's call it x1 and y1. So that's just a particular point. And let's say I have another point, that is x2, y2. Sometimes you'll see this formula, that the distance, you'll see it in different ways, but you'll see that the distance is equal to, and it looks like this really complicated formula, but I want you to see that this is really just the Pythagorean theorem. You'll see that the distance is equal to x2 minus x1 squared plus y2 minus y1 squared.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So that's just a particular point. And let's say I have another point, that is x2, y2. Sometimes you'll see this formula, that the distance, you'll see it in different ways, but you'll see that the distance is equal to, and it looks like this really complicated formula, but I want you to see that this is really just the Pythagorean theorem. You'll see that the distance is equal to x2 minus x1 squared plus y2 minus y1 squared. You'll see this written in a lot of textbooks as the distance formula. It's a complete waste of your time to memorize it, because it's really just the Pythagorean theorem. This is your change in x.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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You'll see that the distance is equal to x2 minus x1 squared plus y2 minus y1 squared. You'll see this written in a lot of textbooks as the distance formula. It's a complete waste of your time to memorize it, because it's really just the Pythagorean theorem. This is your change in x. And it really doesn't matter which x you pick to be first or second, because even if you get the negative of this value, when you square it, the negative disappears. This right here is your change in y. So it's just saying that the distance squared, remember if you square both sides of this equation, the radical will disappear, and this will be the distance squared is equal to this expression squared, delta x squared, change in x. Delta means change in.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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This is your change in x. And it really doesn't matter which x you pick to be first or second, because even if you get the negative of this value, when you square it, the negative disappears. This right here is your change in y. So it's just saying that the distance squared, remember if you square both sides of this equation, the radical will disappear, and this will be the distance squared is equal to this expression squared, delta x squared, change in x. Delta means change in. Delta x squared plus delta y squared. Don't want to confuse you. Delta y just means change in y. I should have probably said that earlier in the video.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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So it's just saying that the distance squared, remember if you square both sides of this equation, the radical will disappear, and this will be the distance squared is equal to this expression squared, delta x squared, change in x. Delta means change in. Delta x squared plus delta y squared. Don't want to confuse you. Delta y just means change in y. I should have probably said that earlier in the video. But let's apply it to a couple of more. And I'll just pick some points at random. Let's say I have the point 1, 2, 3, 4, 5, 6.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Delta y just means change in y. I should have probably said that earlier in the video. But let's apply it to a couple of more. And I'll just pick some points at random. Let's say I have the point 1, 2, 3, 4, 5, 6. Negative 6 comma negative 4. And let's say I want to find the distance between that and 1 comma 1, 2, 3, 4, 5, 6, 7. And the point 1 comma 7.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Let's say I have the point 1, 2, 3, 4, 5, 6. Negative 6 comma negative 4. And let's say I want to find the distance between that and 1 comma 1, 2, 3, 4, 5, 6, 7. And the point 1 comma 7. So I want to find this distance right here. So it's the exact same idea. We just use the Pythagorean theorem.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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And the point 1 comma 7. So I want to find this distance right here. So it's the exact same idea. We just use the Pythagorean theorem. You figure out this distance, which is our change in x. This distance, which is our change in y. This distance squared plus this distance squared is going to equal that distance squared.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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We just use the Pythagorean theorem. You figure out this distance, which is our change in x. This distance, which is our change in y. This distance squared plus this distance squared is going to equal that distance squared. So let's do it. So our change in x, you just take, it doesn't matter. I mean, in general, you want to take the larger x value minus the smaller x value, but you could do it either way.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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This distance squared plus this distance squared is going to equal that distance squared. So let's do it. So our change in x, you just take, it doesn't matter. I mean, in general, you want to take the larger x value minus the smaller x value, but you could do it either way. So we could write the distance squared is equal to, what's our change in x? So let's take the larger x minus the smaller x. 1 minus negative 6 squared plus the change in y.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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I mean, in general, you want to take the larger x value minus the smaller x value, but you could do it either way. So we could write the distance squared is equal to, what's our change in x? So let's take the larger x minus the smaller x. 1 minus negative 6 squared plus the change in y. The larger y is here. It's 7 minus negative 4 squared. And I just picked these numbers at random, so they're probably not going to come out too cleanly.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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1 minus negative 6 squared plus the change in y. The larger y is here. It's 7 minus negative 4 squared. And I just picked these numbers at random, so they're probably not going to come out too cleanly. So we get that the distance squared is equal to 1 minus negative 6, that is 7 squared. And you'll even see it over here if you count it. You go 1, 2, 3, 4, 5, 6, 7.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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And I just picked these numbers at random, so they're probably not going to come out too cleanly. So we get that the distance squared is equal to 1 minus negative 6, that is 7 squared. And you'll even see it over here if you count it. You go 1, 2, 3, 4, 5, 6, 7. That's that number right here. That's what your change in x is. Plus 7 minus negative 4.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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You go 1, 2, 3, 4, 5, 6, 7. That's that number right here. That's what your change in x is. Plus 7 minus negative 4. That's 11. This is this distance right here. And you can count it on the blocks.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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Plus 7 minus negative 4. That's 11. This is this distance right here. And you can count it on the blocks. We're going up 11. We're just taking 7 minus negative 4 to get a distance of 11. So plus 11 squared is equal to d squared.
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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And you can count it on the blocks. We're going up 11. We're just taking 7 minus negative 4 to get a distance of 11. So plus 11 squared is equal to d squared. So let me just take the calculator out. So the distance, let's just take, if we get 7 squared plus 11 squared is equal to 170. That distance is going to be the square root of that, right?
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Distance formula Analytic geometry Geometry Khan Academy.mp3
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