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And I'm trying to draw it freehand. So I would get a shape that looks like this. And actually, let me draw it in. I don't want to make you think it's only the points where that there's white. It's all of these points right over here. I want to draw a dashed line over there. Maybe I should just clear out all of these.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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I don't want to make you think it's only the points where that there's white. It's all of these points right over here. I want to draw a dashed line over there. Maybe I should just clear out all of these. And I'll just draw a solid line. So this could look something like that, my best attempt. And this set of all of the points that are exactly 2 centimeters away from A, this is a circle, as I'm sure you are already familiar with.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Maybe I should just clear out all of these. And I'll just draw a solid line. So this could look something like that, my best attempt. And this set of all of the points that are exactly 2 centimeters away from A, this is a circle, as I'm sure you are already familiar with. But that is the formal definition, the set of all points that have a fixed distance from A, that are a given distance from A. If I said the set of all points that are 3 centimeters from A, it might look something like this. It might look something like that.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And this set of all of the points that are exactly 2 centimeters away from A, this is a circle, as I'm sure you are already familiar with. But that is the formal definition, the set of all points that have a fixed distance from A, that are a given distance from A. If I said the set of all points that are 3 centimeters from A, it might look something like this. It might look something like that. That would give us another circle. It would give us another circle. I think you get the general idea.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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It might look something like that. That would give us another circle. It would give us another circle. I think you get the general idea. Now, what I want to do in this video is introduce ourselves into some of the concepts and words that we use when dealing with circles. So let me get rid of that 3 centimeter circle. So first of all, let's think about this distance.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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I think you get the general idea. Now, what I want to do in this video is introduce ourselves into some of the concepts and words that we use when dealing with circles. So let me get rid of that 3 centimeter circle. So first of all, let's think about this distance. This distance or one of these line segments that join A, which we would call the center of the circle. So we'll call A the center of the circle. And it makes sense just from the way we use the word center in everyday life.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So first of all, let's think about this distance. This distance or one of these line segments that join A, which we would call the center of the circle. So we'll call A the center of the circle. And it makes sense just from the way we use the word center in everyday life. What I want to do is think about what line segment AB is. So AB. AB connects the center, and it connects a point on the circle itself.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And it makes sense just from the way we use the word center in everyday life. What I want to do is think about what line segment AB is. So AB. AB connects the center, and it connects a point on the circle itself. Remember, the circle itself is all the points that are equal distance from the center. So AB, any point, any line segment, I should say, that connects the center to a point on the circle, we would call a radius. Is a radius.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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AB connects the center, and it connects a point on the circle itself. Remember, the circle itself is all the points that are equal distance from the center. So AB, any point, any line segment, I should say, that connects the center to a point on the circle, we would call a radius. Is a radius. And so the length of the radius, AB, over here is equal to 2 centimeters. And you're probably already familiar with the word radius, but I'm just being a little bit more formal. And what's interesting about geometry, at least when you start learning it at the high school level, is it's probably the first class where you're introduced into a slightly more formal mathematics, where we're a little bit more careful.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Is a radius. And so the length of the radius, AB, over here is equal to 2 centimeters. And you're probably already familiar with the word radius, but I'm just being a little bit more formal. And what's interesting about geometry, at least when you start learning it at the high school level, is it's probably the first class where you're introduced into a slightly more formal mathematics, where we're a little bit more careful. About giving our definitions, and then building on those definitions to come up with interesting results, and proving to ourselves that we definitely know what we think we know. And so that's why we're being a little bit more careful with our language over here. So AB is a radius, line segment AB.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And what's interesting about geometry, at least when you start learning it at the high school level, is it's probably the first class where you're introduced into a slightly more formal mathematics, where we're a little bit more careful. About giving our definitions, and then building on those definitions to come up with interesting results, and proving to ourselves that we definitely know what we think we know. And so that's why we're being a little bit more careful with our language over here. So AB is a radius, line segment AB. And so is line segment, let me draw another. So let me put another point on here. Let's say this is x.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So AB is a radius, line segment AB. And so is line segment, let me draw another. So let me put another point on here. Let's say this is x. So line segment AX is also a radius. Now you can also have other forms of lines and line segments that interact in interesting ways with the circle. So you could have a line that just intersects that circle at exactly one point.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Let's say this is x. So line segment AX is also a radius. Now you can also have other forms of lines and line segments that interact in interesting ways with the circle. So you could have a line that just intersects that circle at exactly one point. So let's call that point right over there. And let's call that D. And let's say you have a line. And the only point on the circle that the only point in the set of all the points that are equal distance from A, the only point on that circle that is also on that line is point D. And we could call that line L. So sometimes you'll see lines specified by some of the points on them.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So you could have a line that just intersects that circle at exactly one point. So let's call that point right over there. And let's call that D. And let's say you have a line. And the only point on the circle that the only point in the set of all the points that are equal distance from A, the only point on that circle that is also on that line is point D. And we could call that line L. So sometimes you'll see lines specified by some of the points on them. So for example, I've got another point right over here called E. We could call this line DE. Or we could just put a little script letter here with an L and say this is line L. But this line that only has one point in common with our circle, we call this a tangent line. So line L is a tangent.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And the only point on the circle that the only point in the set of all the points that are equal distance from A, the only point on that circle that is also on that line is point D. And we could call that line L. So sometimes you'll see lines specified by some of the points on them. So for example, I've got another point right over here called E. We could call this line DE. Or we could just put a little script letter here with an L and say this is line L. But this line that only has one point in common with our circle, we call this a tangent line. So line L is a tangent. It is tangent to the circle. So let me write it this way. Line L is tangent.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So line L is a tangent. It is tangent to the circle. So let me write it this way. Line L is tangent. And you normally wouldn't write it in caps like this. I'm just doing that for emphasis. Is tangent to, instead of writing the circle centered at A, you'll sometimes see this notation, to the circle centered at A.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Line L is tangent. And you normally wouldn't write it in caps like this. I'm just doing that for emphasis. Is tangent to, instead of writing the circle centered at A, you'll sometimes see this notation, to the circle centered at A. So this tells us that this is the circle we're talking about. Because who knows? Maybe we had another circle over here that is centered at M. Another circle.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Is tangent to, instead of writing the circle centered at A, you'll sometimes see this notation, to the circle centered at A. So this tells us that this is the circle we're talking about. Because who knows? Maybe we had another circle over here that is centered at M. Another circle. So we have to specify it's not tangent to that one. It's tangent to this one. So this circle with a dot in the middle tells we're talking about a circle.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Maybe we had another circle over here that is centered at M. Another circle. So we have to specify it's not tangent to that one. It's tangent to this one. So this circle with a dot in the middle tells we're talking about a circle. And this is a circle centered at point A. I want to be very clear. Point A is not on the circle. Point A is the center of the circle.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So this circle with a dot in the middle tells we're talking about a circle. And this is a circle centered at point A. I want to be very clear. Point A is not on the circle. Point A is the center of the circle. The points on the circle are the points equal distant from point A. Now, L is tangent because it only intersects the circle on one point. You could just as easily imagine a line that intersects the circle at two points.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Point A is the center of the circle. The points on the circle are the points equal distant from point A. Now, L is tangent because it only intersects the circle on one point. You could just as easily imagine a line that intersects the circle at two points. So we could call, maybe this is F and this is G. You could call that line FG. So we could write it like this. Line FG.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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You could just as easily imagine a line that intersects the circle at two points. So we could call, maybe this is F and this is G. You could call that line FG. So we could write it like this. Line FG. And this line that intersects at two points, we call this a secant of circle A. Is a secant. It is a secant line to this circle right over here.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Line FG. And this line that intersects at two points, we call this a secant of circle A. Is a secant. It is a secant line to this circle right over here. Because it intersects it in two points. Now, if FG was just a segment, if it didn't keep on going forever like lines like to do, if we only spoke about this line segment, let me do this in a new color. If we only spoke about this line segment between F and G and not thinking about going on forever, then all of a sudden we have a line segment, which we would specify there.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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It is a secant line to this circle right over here. Because it intersects it in two points. Now, if FG was just a segment, if it didn't keep on going forever like lines like to do, if we only spoke about this line segment, let me do this in a new color. If we only spoke about this line segment between F and G and not thinking about going on forever, then all of a sudden we have a line segment, which we would specify there. And we would call this a chord of the circle. Is a chord of circle A. It starts at a point on the circle, a point that is exactly, in this case, 2 centimeters away.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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If we only spoke about this line segment between F and G and not thinking about going on forever, then all of a sudden we have a line segment, which we would specify there. And we would call this a chord of the circle. Is a chord of circle A. It starts at a point on the circle, a point that is exactly, in this case, 2 centimeters away. And then it finishes at a point on the circle. So it connects two points on the circle. Now, you can have chords like this.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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It starts at a point on the circle, a point that is exactly, in this case, 2 centimeters away. And then it finishes at a point on the circle. So it connects two points on the circle. Now, you can have chords like this. And you can also have a chord, as you can imagine, a chord that actually goes through the center of the circle. So let's call this point H. And you have a straight line connecting F to H through A. So that's about as straight as I could draw it.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now, you can have chords like this. And you can also have a chord, as you can imagine, a chord that actually goes through the center of the circle. So let's call this point H. And you have a straight line connecting F to H through A. So that's about as straight as I could draw it. So if you have a chord like that, that contains the actual center of the circle, a chord that goes from one point on the circle to another point on the circle, and it goes through the center of the circle, we call that a diameter of A. Is a diameter of circle A. And you've probably seen this in tons of problems before when we were not talking about geometry as formally.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So that's about as straight as I could draw it. So if you have a chord like that, that contains the actual center of the circle, a chord that goes from one point on the circle to another point on the circle, and it goes through the center of the circle, we call that a diameter of A. Is a diameter of circle A. And you've probably seen this in tons of problems before when we were not talking about geometry as formally. But a diameter is made up of exactly two radiuses. We already know that a radius connects a point to the center. So you have one radius right over here that connects F and A.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And you've probably seen this in tons of problems before when we were not talking about geometry as formally. But a diameter is made up of exactly two radiuses. We already know that a radius connects a point to the center. So you have one radius right over here that connects F and A. That's one radius. And then you have another radius connecting A and H, the center to a point on the circle. So a diameter is made up of these two radiuses, or radii I should call.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So you have one radius right over here that connects F and A. That's one radius. And then you have another radius connecting A and H, the center to a point on the circle. So a diameter is made up of these two radiuses, or radii I should call. I think that's the plural for radius. And so the length of a diameter is going to be twice the length of a radius. So we could say the length of the diameter, so the length of FH.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So a diameter is made up of these two radiuses, or radii I should call. I think that's the plural for radius. And so the length of a diameter is going to be twice the length of a radius. So we could say the length of the diameter, so the length of FH. And once again, I don't put the line on top of it when I'm just talking about the length, is going to be equal to FA, the length of segment FA, plus the length of segment AH. Now there's one last thing I want to talk about when we're dealing with circles. And that's the idea of an arc.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So we could say the length of the diameter, so the length of FH. And once again, I don't put the line on top of it when I'm just talking about the length, is going to be equal to FA, the length of segment FA, plus the length of segment AH. Now there's one last thing I want to talk about when we're dealing with circles. And that's the idea of an arc. So we also have the parts of the circle itself. So let me draw another circle over here. Let's center this circle right over here at B.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And that's the idea of an arc. So we also have the parts of the circle itself. So let me draw another circle over here. Let's center this circle right over here at B. And I'm going to find all the points that are a given distance from B. So it has some radius. I'm not going to specify it right over here.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Let's center this circle right over here at B. And I'm going to find all the points that are a given distance from B. So it has some radius. I'm not going to specify it right over here. And let me pick some random points on this circle. So let's call this J, let's call that K, let's call that S, let's call this T, let's call this U right over here. And I know it doesn't look that centered.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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I'm not going to specify it right over here. And let me pick some random points on this circle. So let's call this J, let's call that K, let's call that S, let's call this T, let's call this U right over here. And I know it doesn't look that centered. Let me try to center B a little bit better. Let me erase that and put B a little bit closer to the center of the circle. So that's my best shot.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And I know it doesn't look that centered. Let me try to center B a little bit better. Let me erase that and put B a little bit closer to the center of the circle. So that's my best shot. So let's put B right over there. Now, one interesting thing is, what do you call the length of the circle that goes between two points? So what would you call this?
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So that's my best shot. So let's put B right over there. Now, one interesting thing is, what do you call the length of the circle that goes between two points? So what would you call this? Well, you could imagine in everyday language we would call something that looks like that an arc. And we would also call that an arc in geometry. And to specify this arc, we would call this, we would say JK, the two endpoints of the arc, the two points on the circle that are the endpoints of the arc.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So what would you call this? Well, you could imagine in everyday language we would call something that looks like that an arc. And we would also call that an arc in geometry. And to specify this arc, we would call this, we would say JK, the two endpoints of the arc, the two points on the circle that are the endpoints of the arc. And you use a little notation like that. So you put a little curve on top instead of a straight line. Now, you can also have another arc that connects J and K. This is the minor arc.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And to specify this arc, we would call this, we would say JK, the two endpoints of the arc, the two points on the circle that are the endpoints of the arc. And you use a little notation like that. So you put a little curve on top instead of a straight line. Now, you can also have another arc that connects J and K. This is the minor arc. It is the shortest way along the circle to connect J and K. But you could also go the other way around. You could also have this thing that goes all the way around the circle. And we would call that the major arc.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now, you can also have another arc that connects J and K. This is the minor arc. It is the shortest way along the circle to connect J and K. But you could also go the other way around. You could also have this thing that goes all the way around the circle. And we would call that the major arc. And normally when you specify a major arc, just to show that you're going kind of the long way around, the way that you, it's not the shortest way to get between J and K, you'll often specify another point that you're going through. So for example, this major arc we could specify. We started J.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And we would call that the major arc. And normally when you specify a major arc, just to show that you're going kind of the long way around, the way that you, it's not the shortest way to get between J and K, you'll often specify another point that you're going through. So for example, this major arc we could specify. We started J. We went through. We could have said U, T, or S. But I'll put T right over there. We went through T. And then we went all the way to K. And so this specifies the major arc.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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We started J. We went through. We could have said U, T, or S. But I'll put T right over there. We went through T. And then we went all the way to K. And so this specifies the major arc. And this thing could have been the same thing as if I wrote JUK. These are specifying the same thing. Or JSK.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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We went through T. And then we went all the way to K. And so this specifies the major arc. And this thing could have been the same thing as if I wrote JUK. These are specifying the same thing. Or JSK. So there's multiple ways to specify this major arc right over here. But the one thing I want to make clear is that the minor arc is the shortest distance. So this is the minor arc.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Or JSK. So there's multiple ways to specify this major arc right over here. But the one thing I want to make clear is that the minor arc is the shortest distance. So this is the minor arc. And the longer distance around is the major arc. And I'll leave you there. And maybe in the next few videos, we'll start playing with some of this notation a little bit more.
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Language and notation of the circle Introduction to Euclidean geometry Geometry Khan Academy.mp3
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It's isosceles, which means it has two equal sides. And we also know from isosceles triangles that the base angles must be equal. So these two base angles are going to be equal, and this side right over here is going to be equal in length to this side over here. We can say AC is going to be equal to CE. So we get all of that from this first statement right over there. Then they give us some more clues or some more information. They say CG is equal to 24.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We can say AC is going to be equal to CE. So we get all of that from this first statement right over there. Then they give us some more clues or some more information. They say CG is equal to 24. So this is CG right over here. It has length 24. They tell us BH is equal to DF.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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They say CG is equal to 24. So this is CG right over here. It has length 24. They tell us BH is equal to DF. BH is equal to DF, so those two things are going to be congruent. They're going to be the same length. Then they tell us that GF is equal to 12.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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They tell us BH is equal to DF. BH is equal to DF, so those two things are going to be congruent. They're going to be the same length. Then they tell us that GF is equal to 12. So this is GF right over here. So GF is equal to 12. That's that distance right over there.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Then they tell us that GF is equal to 12. So this is GF right over here. So GF is equal to 12. That's that distance right over there. Then finally they tell us that FE is equal to 6. So this is FE. Then finally they ask us what is the area of CBHFD.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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That's that distance right over there. Then finally they tell us that FE is equal to 6. So this is FE. Then finally they ask us what is the area of CBHFD. So CBHFD. So they're asking us for the area of this part right over here. That part and that part right over there.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Then finally they ask us what is the area of CBHFD. So CBHFD. So they're asking us for the area of this part right over here. That part and that part right over there. That is CBHFD. So let's think about how we can do this. We can figure out the area of the larger triangle.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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That part and that part right over there. That is CBHFD. So let's think about how we can do this. We can figure out the area of the larger triangle. Then from that we can subtract the areas of these little pieces at the end. Then we'll be able to figure out this middle area, this area that I've shaded. We don't have all the information yet to solve that.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We can figure out the area of the larger triangle. Then from that we can subtract the areas of these little pieces at the end. Then we'll be able to figure out this middle area, this area that I've shaded. We don't have all the information yet to solve that. We know what the height or the altitude of this triangle is, but we don't know its base. If we knew its base, we'd say 1 half base times height, we'd get the area of this triangle. Then we'd have to subtract out these areas.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We don't have all the information yet to solve that. We know what the height or the altitude of this triangle is, but we don't know its base. If we knew its base, we'd say 1 half base times height, we'd get the area of this triangle. Then we'd have to subtract out these areas. We don't have full information there either. We don't know this height. Once we know that height, then we can figure out this height, but we also don't quite yet know what this length right over here is.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Then we'd have to subtract out these areas. We don't have full information there either. We don't know this height. Once we know that height, then we can figure out this height, but we also don't quite yet know what this length right over here is. Let's just take it piece by piece. The first thing we might want to do, and you might guess because we've been talking a lot about similarity, is making some type of argument about similarity here because there's a bunch of similar triangles. For example, triangle CGE shares this angle with triangle DFE.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Once we know that height, then we can figure out this height, but we also don't quite yet know what this length right over here is. Let's just take it piece by piece. The first thing we might want to do, and you might guess because we've been talking a lot about similarity, is making some type of argument about similarity here because there's a bunch of similar triangles. For example, triangle CGE shares this angle with triangle DFE. They both share this orange angle right here. They both have this right angle right over here. They have two angles in common.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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For example, triangle CGE shares this angle with triangle DFE. They both share this orange angle right here. They both have this right angle right over here. They have two angles in common. They are going to be similar by angle-angle. You can actually show that there's going to be a third angle in common because these two are parallel lines. We can write that triangle CGE is similar to triangle DFE.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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They have two angles in common. They are going to be similar by angle-angle. You can actually show that there's going to be a third angle in common because these two are parallel lines. We can write that triangle CGE is similar to triangle DFE. We know that by angle-angle. We have one set of corresponding angles congruent, and then this angle is in both triangles, so it is a set of corresponding congruent angles right over there. Then once we know that they are similar, we can set up the ratio between sides because we have some information about some of the sides.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We can write that triangle CGE is similar to triangle DFE. We know that by angle-angle. We have one set of corresponding angles congruent, and then this angle is in both triangles, so it is a set of corresponding congruent angles right over there. Then once we know that they are similar, we can set up the ratio between sides because we have some information about some of the sides. We know that the ratio between DF and this side right over here, which is a corresponding side, the ratio between DF and CG, which is 24, is going to be the same thing as the ratio between FE, which is 6, and GE, which is not 12. It's 12 plus 6. It is 18.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Then once we know that they are similar, we can set up the ratio between sides because we have some information about some of the sides. We know that the ratio between DF and this side right over here, which is a corresponding side, the ratio between DF and CG, which is 24, is going to be the same thing as the ratio between FE, which is 6, and GE, which is not 12. It's 12 plus 6. It is 18. Then let's see. 6 over 18, this is just 1 over 3. You get 3DF.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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It is 18. Then let's see. 6 over 18, this is just 1 over 3. You get 3DF. You get 3DF is equal to 24. I just cross-multiplied, or you could multiply both sides by 24, multiply both sides by 3. You would get this.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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You get 3DF. You get 3DF is equal to 24. I just cross-multiplied, or you could multiply both sides by 24, multiply both sides by 3. You would get this. Actually, you could just multiply both sides times 24, and you'll get 24 times 1 third, but we'll just do it this way. Divide both sides by 3. You get DF is equal to 8.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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You would get this. Actually, you could just multiply both sides times 24, and you'll get 24 times 1 third, but we'll just do it this way. Divide both sides by 3. You get DF is equal to 8. We found out that DF is equal to 8, that length right over there. That's useful for us because we know that this length right over here is also equal to 8. Now what can we do?
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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You get DF is equal to 8. We found out that DF is equal to 8, that length right over there. That's useful for us because we know that this length right over here is also equal to 8. Now what can we do? It seems like we can make another similarity argument because we have this angle right over here. It is congruent to that angle right over there. We also have this angle, which is going to be 90 degrees.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Now what can we do? It seems like we can make another similarity argument because we have this angle right over here. It is congruent to that angle right over there. We also have this angle, which is going to be 90 degrees. We have a 90-degree angle there. That by itself is actually enough to say that we have two similar triangles. We don't even have to show that they have a congruent side here.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We also have this angle, which is going to be 90 degrees. We have a 90-degree angle there. That by itself is actually enough to say that we have two similar triangles. We don't even have to show that they have a congruent side here. Actually, we're going to show that these are actually congruent triangles that we're dealing with right over here. We have two angles. Actually, we could just go straight to that because when we talk about congruency, if you have an angle that's congruent to another angle, another angle that's congruent to another angle, and then a side that's congruent to another side, you're dealing with two congruent triangles.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We don't even have to show that they have a congruent side here. Actually, we're going to show that these are actually congruent triangles that we're dealing with right over here. We have two angles. Actually, we could just go straight to that because when we talk about congruency, if you have an angle that's congruent to another angle, another angle that's congruent to another angle, and then a side that's congruent to another side, you're dealing with two congruent triangles. We can write triangle A, H, B is congruent to triangle E, F, D. We know that because we have angle, angle, side. Angle, angle, side postulate for congruency. If the two triangles are congruent, that makes things convenient.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Actually, we could just go straight to that because when we talk about congruency, if you have an angle that's congruent to another angle, another angle that's congruent to another angle, and then a side that's congruent to another side, you're dealing with two congruent triangles. We can write triangle A, H, B is congruent to triangle E, F, D. We know that because we have angle, angle, side. Angle, angle, side postulate for congruency. If the two triangles are congruent, that makes things convenient. That means if this side is 8, that side is 8. We already knew that. That's how we established our congruency.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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If the two triangles are congruent, that makes things convenient. That means if this side is 8, that side is 8. We already knew that. That's how we established our congruency. That means if this side has length 6, then the corresponding side on this triangle is also going to have length 6. We can write this length right over here is also going to be 6. I can imagine you can imagine where all of this is going to go, but we want to prove to ourselves.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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That's how we established our congruency. That means if this side has length 6, then the corresponding side on this triangle is also going to have length 6. We can write this length right over here is also going to be 6. I can imagine you can imagine where all of this is going to go, but we want to prove to ourselves. We want to know for sure what the area is. We don't want to say, hey, maybe this is the same thing as that. Let's just actually prove it to ourselves.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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I can imagine you can imagine where all of this is going to go, but we want to prove to ourselves. We want to know for sure what the area is. We don't want to say, hey, maybe this is the same thing as that. Let's just actually prove it to ourselves. How do we figure out? We've almost figured out the entire base of this triangle, but we still haven't figured out the length of HG. Now we can use a similarity argument again because we can see that triangle ABH is actually similar to triangle ACG.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Let's just actually prove it to ourselves. How do we figure out? We've almost figured out the entire base of this triangle, but we still haven't figured out the length of HG. Now we can use a similarity argument again because we can see that triangle ABH is actually similar to triangle ACG. They both have this angle here, and then they both have a right angle. They have one. ABH has a right angle there.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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Now we can use a similarity argument again because we can see that triangle ABH is actually similar to triangle ACG. They both have this angle here, and then they both have a right angle. They have one. ABH has a right angle there. ACG has a right angle right over there. So you have two angles, two corresponding angles are equal to each other. You're now dealing with similar triangles.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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ABH has a right angle there. ACG has a right angle right over there. So you have two angles, two corresponding angles are equal to each other. You're now dealing with similar triangles. We know that triangle ABH, I'll just write it as AHB since I already wrote it this way, AHB is similar to triangle AGC. You want to make sure you get the vertices in the right order. A is the orange angle, G is the right angle, and C is the unlabeled angle.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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You're now dealing with similar triangles. We know that triangle ABH, I'll just write it as AHB since I already wrote it this way, AHB is similar to triangle AGC. You want to make sure you get the vertices in the right order. A is the orange angle, G is the right angle, and C is the unlabeled angle. This is similar to triangle AGC. What that does for us is now we can use the ratios to figure out what HG is equal to. What can we say over here?
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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A is the orange angle, G is the right angle, and C is the unlabeled angle. This is similar to triangle AGC. What that does for us is now we can use the ratios to figure out what HG is equal to. What can we say over here? We can say that 8 over 24, BH over its corresponding side of the larger triangle, so we say 8 over 24 is equal to 6 over not HG but over AG. I think you can see where this is going. You have 1 third is equal to 6 over AG, or we can cross multiply here and we can get AG is equal to 18.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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What can we say over here? We can say that 8 over 24, BH over its corresponding side of the larger triangle, so we say 8 over 24 is equal to 6 over not HG but over AG. I think you can see where this is going. You have 1 third is equal to 6 over AG, or we can cross multiply here and we can get AG is equal to 18. This entire length right over here is 18. If AG is 18 and AH is 6, then HG is 12. This is what you might have guessed if you were just trying to guess the answer right over here.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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You have 1 third is equal to 6 over AG, or we can cross multiply here and we can get AG is equal to 18. This entire length right over here is 18. If AG is 18 and AH is 6, then HG is 12. This is what you might have guessed if you were just trying to guess the answer right over here. Now we have proven to ourselves that this base has length of 18 here and then we have another 18 here. It has a length of 36. The entire base here is 36.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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This is what you might have guessed if you were just trying to guess the answer right over here. Now we have proven to ourselves that this base has length of 18 here and then we have another 18 here. It has a length of 36. The entire base here is 36. Now we can figure out the area of this larger, of the entire isosceles triangle. The area of ACE is going to be equal to 1 half times the base, which is 36, times 24. This is going to be the same thing as 1 half times 36 is 18 times 24.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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The entire base here is 36. Now we can figure out the area of this larger, of the entire isosceles triangle. The area of ACE is going to be equal to 1 half times the base, which is 36, times 24. This is going to be the same thing as 1 half times 36 is 18 times 24. I'll just do that over here on the top. 18 times 24, 8 times 4 is 32, 1 times 4 is 4 plus 3 is 7. We put a zero here because we're not dealing with 2 but 20.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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This is going to be the same thing as 1 half times 36 is 18 times 24. I'll just do that over here on the top. 18 times 24, 8 times 4 is 32, 1 times 4 is 4 plus 3 is 7. We put a zero here because we're not dealing with 2 but 20. You have 2 times 8 is 16, 2 times 1 is 2 plus 1, so it's 360. Then you have the 2, 7 plus 6 is 13, 1 plus 3 is 4. The area of ACE is equal to 432, but we're not done yet.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We put a zero here because we're not dealing with 2 but 20. You have 2 times 8 is 16, 2 times 1 is 2 plus 1, so it's 360. Then you have the 2, 7 plus 6 is 13, 1 plus 3 is 4. The area of ACE is equal to 432, but we're not done yet. This area that we care about is the area of the entire triangle minus this area and minus this area right over here. What is the area of each of these little wedges right over here? It's going to be 1 half times 8 times 6.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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The area of ACE is equal to 432, but we're not done yet. This area that we care about is the area of the entire triangle minus this area and minus this area right over here. What is the area of each of these little wedges right over here? It's going to be 1 half times 8 times 6. 1 half times 8 is 4 times 6, so this is going to be 24 right over there. This is going to be another 24 right over there. This is going to be equal to 432 minus 24 minus 24 or minus 48, which is equal to, and we could try this to do this in our head, if we subtract 32, we're going to get to 400.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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It's going to be 1 half times 8 times 6. 1 half times 8 is 4 times 6, so this is going to be 24 right over there. This is going to be another 24 right over there. This is going to be equal to 432 minus 24 minus 24 or minus 48, which is equal to, and we could try this to do this in our head, if we subtract 32, we're going to get to 400. Then we're going to have to subtract another 16. If you subtract 10 from 400, you get 390, so you get to 384, whatever the units were. If these were in meters, then this would be meters squared.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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This is going to be equal to 432 minus 24 minus 24 or minus 48, which is equal to, and we could try this to do this in our head, if we subtract 32, we're going to get to 400. Then we're going to have to subtract another 16. If you subtract 10 from 400, you get 390, so you get to 384, whatever the units were. If these were in meters, then this would be meters squared. If this was centimeters, this would be centimeters squared. Did I do that right? Let me go the other way.
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Finding area using similarity and congruence Similarity Geometry Khan Academy.mp3
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We're told the triangle PIN is rotated negative 270 degrees about the origin. So this is the triangle PIN. We're gonna rotate it negative 270 degrees about the origin. Draw the image of this rotation using the interactive graph. The direction of rotation by a positive angle is counterclockwise. So positive is counterclockwise, which is the standard convention. And this is negative, so negative degree would be clockwise.
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Points after rotation Transformations Geometry Khan Academy.mp3
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Draw the image of this rotation using the interactive graph. The direction of rotation by a positive angle is counterclockwise. So positive is counterclockwise, which is the standard convention. And this is negative, so negative degree would be clockwise. And we wanna use this tool here. And this tool, I can put points in, or I could delete points, I can draw a point by clicking on it. So what we wanna do is think about, well look, if we rotate the points of this triangle around the origin by negative 270 degrees, where is it gonna put these points?
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Points after rotation Transformations Geometry Khan Academy.mp3
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And this is negative, so negative degree would be clockwise. And we wanna use this tool here. And this tool, I can put points in, or I could delete points, I can draw a point by clicking on it. So what we wanna do is think about, well look, if we rotate the points of this triangle around the origin by negative 270 degrees, where is it gonna put these points? And to help us think about that, I've copied and pasted this on our scratch pad. So actually, let me go over here so I can actually draw on it. So let's just first think about what a negative 270 degree rotation actually is.
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Points after rotation Transformations Geometry Khan Academy.mp3
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So what we wanna do is think about, well look, if we rotate the points of this triangle around the origin by negative 270 degrees, where is it gonna put these points? And to help us think about that, I've copied and pasted this on our scratch pad. So actually, let me go over here so I can actually draw on it. So let's just first think about what a negative 270 degree rotation actually is. So if I were to start, if I were to, let me draw some coordinate axes here. So that's a x-axis and that's a y-axis. If you were to start right over here, and you were to rotate around the origin by negative, so this is the origin here, by negative 270 degrees, what would that be?
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Points after rotation Transformations Geometry Khan Academy.mp3
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So let's just first think about what a negative 270 degree rotation actually is. So if I were to start, if I were to, let me draw some coordinate axes here. So that's a x-axis and that's a y-axis. If you were to start right over here, and you were to rotate around the origin by negative, so this is the origin here, by negative 270 degrees, what would that be? Well let's see, this would be rotating negative 90. This would be rotating negative, another negative 90, which together would be negative 180. And then this would be another negative 90, which would give you in total negative 270 degrees.
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Points after rotation Transformations Geometry Khan Academy.mp3
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If you were to start right over here, and you were to rotate around the origin by negative, so this is the origin here, by negative 270 degrees, what would that be? Well let's see, this would be rotating negative 90. This would be rotating negative, another negative 90, which together would be negative 180. And then this would be another negative 90, which would give you in total negative 270 degrees. That's negative 270 degrees. Now notice, that would get that point here, which we could have also gotten there by just rotating it by positive 90 degrees. We could have just said that this is equivalent to a positive 90 degree rotation.
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Points after rotation Transformations Geometry Khan Academy.mp3
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And then this would be another negative 90, which would give you in total negative 270 degrees. That's negative 270 degrees. Now notice, that would get that point here, which we could have also gotten there by just rotating it by positive 90 degrees. We could have just said that this is equivalent to a positive 90 degree rotation. So if they want us to rotate the points here around the origin by negative 270 degrees, that's equivalent to just rotating all of the points, and I'll just focus on the vertices, because those are the easiest ones to think about, to visualize. I can just rotate each of those around the origin by positive 90 degrees. But how do we do that?
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Points after rotation Transformations Geometry Khan Academy.mp3
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We could have just said that this is equivalent to a positive 90 degree rotation. So if they want us to rotate the points here around the origin by negative 270 degrees, that's equivalent to just rotating all of the points, and I'll just focus on the vertices, because those are the easiest ones to think about, to visualize. I can just rotate each of those around the origin by positive 90 degrees. But how do we do that? And to do that, what I am going to do, to do that, what I'm gonna do is I'm gonna draw a series of right triangles. So let's first focus on, actually let's first focus on point I right over here. And if I were to, let me draw a right triangle.
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Points after rotation Transformations Geometry Khan Academy.mp3
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But how do we do that? And to do that, what I am going to do, to do that, what I'm gonna do is I'm gonna draw a series of right triangles. So let's first focus on, actually let's first focus on point I right over here. And if I were to, let me draw a right triangle. And I could draw it several ways, but let me draw it like this. So it's a right triangle, where the line between the origin and I is its hypotenuse. So let me see if I can, it's, I could probably draw, I could use a line tool for that.
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Points after rotation Transformations Geometry Khan Academy.mp3
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And if I were to, let me draw a right triangle. And I could draw it several ways, but let me draw it like this. So it's a right triangle, where the line between the origin and I is its hypotenuse. So let me see if I can, it's, I could probably draw, I could use a line tool for that. So let me, so that's the hypotenuse of the line. Now if I'm gonna rotate I 90 degrees about the origin, that's equivalent to rotating this right triangle 90 degrees. So what's going to happen there?
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Points after rotation Transformations Geometry Khan Academy.mp3
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So let me see if I can, it's, I could probably draw, I could use a line tool for that. So let me, so that's the hypotenuse of the line. Now if I'm gonna rotate I 90 degrees about the origin, that's equivalent to rotating this right triangle 90 degrees. So what's going to happen there? Well, this side, right over here, if I rotate this 90 degrees, where is that gonna go? Well, instead of going seven along the x-axis, it's gonna go seven along the y-axis. So it's going to, if you rotate it positive 90 degrees, that side is going to look like this.
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Points after rotation Transformations Geometry Khan Academy.mp3
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So what's going to happen there? Well, this side, right over here, if I rotate this 90 degrees, where is that gonna go? Well, instead of going seven along the x-axis, it's gonna go seven along the y-axis. So it's going to, if you rotate it positive 90 degrees, that side is going to look like this. So that's rotating it 90 degrees, just like that. Now what about this side over here? What about this side?
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Points after rotation Transformations Geometry Khan Academy.mp3
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So it's going to, if you rotate it positive 90 degrees, that side is going to look like this. So that's rotating it 90 degrees, just like that. Now what about this side over here? What about this side? Let me do this in a different color. What about, what about this side right over here? Well, this side over here, notice we've gone down from the origin, we've gone down seven.
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Points after rotation Transformations Geometry Khan Academy.mp3
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What about this side? Let me do this in a different color. What about, what about this side right over here? Well, this side over here, notice we've gone down from the origin, we've gone down seven. But if you were to rotate it up, notice this forms a right angle between this magenta side and this blue side. So you're gonna form a right angle again. And so from this point, you're gonna go straight as a right angle, and instead of going down seven, you're gonna go to the right seven.
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Points after rotation Transformations Geometry Khan Academy.mp3
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Well, this side over here, notice we've gone down from the origin, we've gone down seven. But if you were to rotate it up, notice this forms a right angle between this magenta side and this blue side. So you're gonna form a right angle again. And so from this point, you're gonna go straight as a right angle, and instead of going down seven, you're gonna go to the right seven. So you're gonna go to the right seven, just like this. And so the point I, or the corresponding point in the image after the rotation, is going to be right over here. So that green line, let me draw the hypotenuse now, it's gonna look, it's gonna look like, whoops, I wanted to do that in a different color, I wanted to do that in the green, I have trouble changing colors.
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Points after rotation Transformations Geometry Khan Academy.mp3
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