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AD is the same thing as CD. And so we know the ratio of AB to AD is equal to CF over CD. But we just proved to ourselves, because this is an isosceles triangle, that CF is the same thing as BC right over here. So CF is the same thing as BC. And we're done. We've just proven AB over AD is equal to BC over CD. So there's kind of two things we had to do here.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So CF is the same thing as BC. And we're done. We've just proven AB over AD is equal to BC over CD. So there's kind of two things we had to do here. One, construct this other triangle that allowed us, assuming this was parallel, that gave us two things. That gave us another angle to show that they're similar, and also allowed us to establish, sorry, I have something stuck in my throat. Just coughed off camera.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So there's kind of two things we had to do here. One, construct this other triangle that allowed us, assuming this was parallel, that gave us two things. That gave us another angle to show that they're similar, and also allowed us to establish, sorry, I have something stuck in my throat. Just coughed off camera. So I should go get a drink of water after this. So we were able to, using constructing this triangle here, we were able to both show it's similar and to construct this larger isosceles triangle to show, look, if we can find a ratio between two sides of this triangle and this one, then that's going to be the ratio of this. If we could find the ratio of this side to this side, it's the same as the ratio of this side to this side.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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What I want to talk about in this video is the notion of arc measure when we're dealing with circles. And as we'll see, sometimes when you see something like arc measure, you might think it's the length of an arc, but arc length is actually a different idea, so we will compare these two things. Arc length to arc measure. So arc measure, all that is, it's just a fancy way of saying, if I have a circle right over here, this is my best attempt at drawing a circle. I have a circle here. The center of the circle, let's call that point O. And let me put some other points over here.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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So arc measure, all that is, it's just a fancy way of saying, if I have a circle right over here, this is my best attempt at drawing a circle. I have a circle here. The center of the circle, let's call that point O. And let me put some other points over here. So let's say that this is point A. Let's say this is point B. And let's say this is point C right over here.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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And let me put some other points over here. So let's say that this is point A. Let's say this is point B. And let's say this is point C right over here. And let's say that I have, let's say the central angle right over here, because it includes the center of the circle. So the central angle, angle AOB. Let's say it has a measure of, let's say it has a measure of 120 degrees.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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And let's say this is point C right over here. And let's say that I have, let's say the central angle right over here, because it includes the center of the circle. So the central angle, angle AOB. Let's say it has a measure of, let's say it has a measure of 120 degrees. And if someone were to say, what is the measure of arc AB? So let me write that down. The measure, so if someone were to say, what is the measure of arc AB?
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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Let's say it has a measure of, let's say it has a measure of 120 degrees. And if someone were to say, what is the measure of arc AB? So let me write that down. The measure, so if someone were to say, what is the measure of arc AB? And they'd write it like this. So that's referring to arc AB right over here. It's the minor arc.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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The measure, so if someone were to say, what is the measure of arc AB? And they'd write it like this. So that's referring to arc AB right over here. It's the minor arc. So there's two ways to connect AB. You could connect it right over here. This is the shorter distance.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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It's the minor arc. So there's two ways to connect AB. You could connect it right over here. This is the shorter distance. Or you could go the other way around, which would be, which is considered the major arc. Now if someone's referring to the major arc, they would say mark ACB. So when you're given just two letters, you assume it's the shortest distance between the two.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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This is the shorter distance. Or you could go the other way around, which would be, which is considered the major arc. Now if someone's referring to the major arc, they would say mark ACB. So when you're given just two letters, you assume it's the shortest distance between the two. You assume that it is the minor arc. In order to specify the major arc, you would give the third letter to go the long way around. So the measure of arc AB, and sometimes you'll see it with the parentheses right over here, all this is, this is the same thing as the measure of the central angle that intercepts that arc.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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So when you're given just two letters, you assume it's the shortest distance between the two. You assume that it is the minor arc. In order to specify the major arc, you would give the third letter to go the long way around. So the measure of arc AB, and sometimes you'll see it with the parentheses right over here, all this is, this is the same thing as the measure of the central angle that intercepts that arc. Well, the central angle that intercepts that arc has a measure of 120 degrees. So this is just going to be 120 degrees. Now some of y'all might be saying, well, what about the major arc?
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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So the measure of arc AB, and sometimes you'll see it with the parentheses right over here, all this is, this is the same thing as the measure of the central angle that intercepts that arc. Well, the central angle that intercepts that arc has a measure of 120 degrees. So this is just going to be 120 degrees. Now some of y'all might be saying, well, what about the major arc? Well, let's write that. So if we're talking about arc ACB, so we're going the other way around. So this is the major arc.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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Now some of y'all might be saying, well, what about the major arc? Well, let's write that. So if we're talking about arc ACB, so we're going the other way around. So this is the major arc. So what is the measure of arc ACB? Once again, we're using three letters so that we're specifying the major arc. Well, this angle, this central angle right over here, to go all the way around the circle is 360 degrees.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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So this is the major arc. So what is the measure of arc ACB? Once again, we're using three letters so that we're specifying the major arc. Well, this angle, this central angle right over here, to go all the way around the circle is 360 degrees. So this is going to be the 360 minus the 120 that we're not including. So 360 degrees minus 120 is going to be 240 degrees. So the measure of this angle right over here is 240 degrees.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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Well, this angle, this central angle right over here, to go all the way around the circle is 360 degrees. So this is going to be the 360 minus the 120 that we're not including. So 360 degrees minus 120 is going to be 240 degrees. So the measure of this angle right over here is 240 degrees. So the arc, the measure of this arc, have to be careful not to say length of that arc, the measure of this arc is going to be the same as the measure of the central angle. So it's going to be 240 degrees. Now, this is going to be, these arc measures are going to be the case regardless of the size of the circle.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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So the measure of this angle right over here is 240 degrees. So the arc, the measure of this arc, have to be careful not to say length of that arc, the measure of this arc is going to be the same as the measure of the central angle. So it's going to be 240 degrees. Now, this is going to be, these arc measures are going to be the case regardless of the size of the circle. And that's where the difference starts to be from arc measure to arc length. So I could have two circles. So this circle right over here and that circle right over here.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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Now, this is going to be, these arc measures are going to be the case regardless of the size of the circle. And that's where the difference starts to be from arc measure to arc length. So I could have two circles. So this circle right over here and that circle right over here. And as long as the central angle that intercepts the arc has the same degree measure, so let's say that that degree measure is the same as, these are central angles, so we're assuming the vertex of the angle is the center of the circle. As long as these two are the same, these two central angles have the same degree measure, then the arc measures, then the corresponding arc measures are going to be the same. But clearly these two arc lengths are different.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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So this circle right over here and that circle right over here. And as long as the central angle that intercepts the arc has the same degree measure, so let's say that that degree measure is the same as, these are central angles, so we're assuming the vertex of the angle is the center of the circle. As long as these two are the same, these two central angles have the same degree measure, then the arc measures, then the corresponding arc measures are going to be the same. But clearly these two arc lengths are different. The arc length is not going to depend only on the measure of the central angle, the arc length is going to depend on the size of the actual circle. Arc measure is only dependent on the measure of the central angle that intercepts that arc. So your maximum arc measure is going to be 360 degrees.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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But clearly these two arc lengths are different. The arc length is not going to depend only on the measure of the central angle, the arc length is going to depend on the size of the actual circle. Arc measure is only dependent on the measure of the central angle that intercepts that arc. So your maximum arc measure is going to be 360 degrees. Your minimum arc measure is going to be zero degrees. It's measured in degrees, not in units of length that arc length would be measured in. So let me write this down.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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So your maximum arc measure is going to be 360 degrees. Your minimum arc measure is going to be zero degrees. It's measured in degrees, not in units of length that arc length would be measured in. So let me write this down. This only depends, so this is, what's going to drive this is the measure, measure of central angle, central angle that intercepts the arc, that intercepts, intercepts the arc. When you talk about arc length, yes, it's going to be dependent on the angle, but it's also dependent, it's going to be dependent on the measure of that central angle plus the size of the circle. Size of the circle.
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Intro to arc measure Mathematics II High School Math Khan Academy.mp3
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What I want to do in this video, see if we can find the measure of angle D. If we could find the measure of angle D. And like always, pause this video and see if you can figure it out. And I'll give you a little bit of a hint. It'll involve thinking about how an inscribed angle relates to the corresponding, to the measure of the arc that it intercepts. So think about it like that. All right, so let's work on this a little bit. So what do we know? What do we know?
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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So think about it like that. All right, so let's work on this a little bit. So what do we know? What do we know? Well, angle D, angle D intercepts an arc. It intercepts this fairly large arc that I'm going to highlight right now in this purple color. So it intercepts that arc.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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What do we know? Well, angle D, angle D intercepts an arc. It intercepts this fairly large arc that I'm going to highlight right now in this purple color. So it intercepts that arc. We don't know the measure of that arc, or at least we don't know the measure of that arc yet. If we did know the measure of this arc that I'm highlighting, then we know that the measure of angle D would just be half that, because the measure of an inscribed angle is half the measure of the arc that it intercepts. We've seen that multiple times.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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So it intercepts that arc. We don't know the measure of that arc, or at least we don't know the measure of that arc yet. If we did know the measure of this arc that I'm highlighting, then we know that the measure of angle D would just be half that, because the measure of an inscribed angle is half the measure of the arc that it intercepts. We've seen that multiple times. So if we knew the measure of this arc, we would be able to figure out what the measure of angle D is. But we do know, we don't know the measure of that arc, but we do know the measure of another arc. We do know the measure of the arc that completes the circle.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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We've seen that multiple times. So if we knew the measure of this arc, we would be able to figure out what the measure of angle D is. But we do know, we don't know the measure of that arc, but we do know the measure of another arc. We do know the measure of the arc that completes the circle. So we do know the measure of this arc. You might be saying, hey, wait, how do we know that measure? It's not labeled.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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We do know the measure of the arc that completes the circle. So we do know the measure of this arc. You might be saying, hey, wait, how do we know that measure? It's not labeled. Well, the reason why we know the measure of this arc that I've just highlighted in this teal color is because the inscribed angle that intercepts it, they gave us the information. They said this is a 45 degree angle. So if this is a 45 degree angle, then this over here is a 90 degree arc.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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It's not labeled. Well, the reason why we know the measure of this arc that I've just highlighted in this teal color is because the inscribed angle that intercepts it, they gave us the information. They said this is a 45 degree angle. So if this is a 45 degree angle, then this over here is a 90 degree arc. The measure of this arc is 90 degrees. The measure of arc, I guess you could say this is the measure of arc, let me write it this way. The measure of arc WL, WL is equal to 90 degrees.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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So if this is a 45 degree angle, then this over here is a 90 degree arc. The measure of this arc is 90 degrees. The measure of arc, I guess you could say this is the measure of arc, let me write it this way. The measure of arc WL, WL is equal to 90 degrees. It's twice that, the inscribed angle that intercepts it. Now why is that helpful? Well, if you go all the way around the circle, you're 360 degrees.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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The measure of arc WL, WL is equal to 90 degrees. It's twice that, the inscribed angle that intercepts it. Now why is that helpful? Well, if you go all the way around the circle, you're 360 degrees. So this purple arc that we cared about, that we said, hey, if we could figure out the measure of that, we're gonna be able to figure out the measure of angle D, that plus arc WL, they are going to add up to 360 degrees. Let me write that down. So the measure of arc, let's see, and this is going to be, this is going to be a major arc right over here.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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Well, if you go all the way around the circle, you're 360 degrees. So this purple arc that we cared about, that we said, hey, if we could figure out the measure of that, we're gonna be able to figure out the measure of angle D, that plus arc WL, they are going to add up to 360 degrees. Let me write that down. So the measure of arc, let's see, and this is going to be, this is going to be a major arc right over here. This is so LIW, the measure of arc LIW plus the measure of arc WL, plus the measure of arc WL, WL plus this right over here, that's going to be equal to 360 degrees. This is going to be equal to 360 degrees. Now we already know that this is 90 degrees.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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So the measure of arc, let's see, and this is going to be, this is going to be a major arc right over here. This is so LIW, the measure of arc LIW plus the measure of arc WL, plus the measure of arc WL, WL plus this right over here, that's going to be equal to 360 degrees. This is going to be equal to 360 degrees. Now we already know that this is 90 degrees. We already know WL is 90 degrees. So if you subtract 90 degrees from both sides, you get that the measure of this large arc right over here, measure of arc LIW is going to be equal to 270 degrees. 270 degrees.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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Now we already know that this is 90 degrees. We already know WL is 90 degrees. So if you subtract 90 degrees from both sides, you get that the measure of this large arc right over here, measure of arc LIW is going to be equal to 270 degrees. 270 degrees. I just took 300, I went all the way around the circle, I subtracted out this 90 degrees, and I'm left with 270 degrees. So let me write that down. This is the measure of this arc in purple is 270 degrees.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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270 degrees. I just took 300, I went all the way around the circle, I subtracted out this 90 degrees, and I'm left with 270 degrees. So let me write that down. This is the measure of this arc in purple is 270 degrees. And now we can figure out the measure of angle D. It's an inscribed angle that intercepts that arc, so it's going to have half the measure. The angle's going to have half the measure. So half of 270 is 135 degrees.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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This is the measure of this arc in purple is 270 degrees. And now we can figure out the measure of angle D. It's an inscribed angle that intercepts that arc, so it's going to have half the measure. The angle's going to have half the measure. So half of 270 is 135 degrees. And we're done. You might notice something interesting, that if you add 135 degrees plus 45 degrees, that they add up to 180 degrees. So it looks like at least for this case, that these angles, these opposite angles of this inscribed quadrilateral, it looks like they are supplementary.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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So half of 270 is 135 degrees. And we're done. You might notice something interesting, that if you add 135 degrees plus 45 degrees, that they add up to 180 degrees. So it looks like at least for this case, that these angles, these opposite angles of this inscribed quadrilateral, it looks like they are supplementary. So an interesting question is, are they always going to be supplementary? If you have a quadrilateral, an arbitrary quadrilateral, inscribed in a circle, so each of the vertices of the quadrilateral sit on the circle, if you have that, are opposite angles of that quadrilateral, are they always supplementary? Do they always add up to 180 degrees?
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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So it looks like at least for this case, that these angles, these opposite angles of this inscribed quadrilateral, it looks like they are supplementary. So an interesting question is, are they always going to be supplementary? If you have a quadrilateral, an arbitrary quadrilateral, inscribed in a circle, so each of the vertices of the quadrilateral sit on the circle, if you have that, are opposite angles of that quadrilateral, are they always supplementary? Do they always add up to 180 degrees? So I encourage you to think about that, and even prove it if you get a chance. And the proof is very close to what we just did here. In order to prove it, you would just have to do it with more general numbers, like instead of saying 45 degrees, you could call this x.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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Do they always add up to 180 degrees? So I encourage you to think about that, and even prove it if you get a chance. And the proof is very close to what we just did here. In order to prove it, you would just have to do it with more general numbers, like instead of saying 45 degrees, you could call this x. And then you would want to prove that this right over here would have to be 180 minus x. So I encourage you to do that on your own. But I'm going to do it in a video as well, so you can check if our reasoning is similar.
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Solving inscribed quadrilaterals Mathematics II High School Math Khan Academy.mp3
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There are 10 lights per meter of the string. How many total lights are on the string? So pause this video and see if you can work this out. All right, now let's work through this together. And I think this one warrants some type of a diagram. So let me draw this doorframe that looks like this. And this doorframe is 2.5 meters tall.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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All right, now let's work through this together. And I think this one warrants some type of a diagram. So let me draw this doorframe that looks like this. And this doorframe is 2.5 meters tall. So that's its height right over there. And what they're going to do, what Laney's doing is she is stringing up these lights. So that's this yellow right over here.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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And this doorframe is 2.5 meters tall. So that's its height right over there. And what they're going to do, what Laney's doing is she is stringing up these lights. So that's this yellow right over here. So it goes up to the top of that doorframe. And then they run the rest of the light in a straight line to a point on the ground that is six meters from the base of the doorframe. So let me show a point that is six meters from the base of the doorframe.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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So that's this yellow right over here. So it goes up to the top of that doorframe. And then they run the rest of the light in a straight line to a point on the ground that is six meters from the base of the doorframe. So let me show a point that is six meters from the base of the doorframe. So it would look something like, maybe like that. So this distance right over here is six meters. So they run the rest of the light from the top of the doorframe to that point that's six meters away.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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So let me show a point that is six meters from the base of the doorframe. So it would look something like, maybe like that. So this distance right over here is six meters. So they run the rest of the light from the top of the doorframe to that point that's six meters away. So the yellow right over here, that is the light. And so we need to figure out how many total lights are on the string. So the way I would tackle it is, first of all, I wanna figure out how long is the total string.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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So they run the rest of the light from the top of the doorframe to that point that's six meters away. So the yellow right over here, that is the light. And so we need to figure out how many total lights are on the string. So the way I would tackle it is, first of all, I wanna figure out how long is the total string. And to figure that out, I just need to figure out, okay, it's going to be 2.5 plus whatever the hypotenuse is of this right triangle. I think it's safe to assume that this is a standard house where doorframes are at a right angle to the floor. And so we have to figure out the length of this hypotenuse.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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So the way I would tackle it is, first of all, I wanna figure out how long is the total string. And to figure that out, I just need to figure out, okay, it's going to be 2.5 plus whatever the hypotenuse is of this right triangle. I think it's safe to assume that this is a standard house where doorframes are at a right angle to the floor. And so we have to figure out the length of this hypotenuse. And if we know that plus this 2.5 meters, then we know how long the entire string of lights are. And then we just have to really multiply it by 10 because there's 10 lights per meter of the string. So let's do that.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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And so we have to figure out the length of this hypotenuse. And if we know that plus this 2.5 meters, then we know how long the entire string of lights are. And then we just have to really multiply it by 10 because there's 10 lights per meter of the string. So let's do that. So how do we figure out the hypotenuse here? Well, of course we would use Pythagorean theorem. So let's call this, let's call this H for hypotenuse.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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So let's do that. So how do we figure out the hypotenuse here? Well, of course we would use Pythagorean theorem. So let's call this, let's call this H for hypotenuse. We know that the hypotenuse squared is equal to 2.5 squared plus six squared. So this is going to be equal to 6.25 plus 36, which is equal to 42.25. Or we could say that the hypotenuse is going to be equal to the square root of 42.25.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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So let's call this, let's call this H for hypotenuse. We know that the hypotenuse squared is equal to 2.5 squared plus six squared. So this is going to be equal to 6.25 plus 36, which is equal to 42.25. Or we could say that the hypotenuse is going to be equal to the square root of 42.25. And I could get my calculator out at this point, but I'll actually just keep using this expression to figure out the total number of lights. So what's the total length of the string? We have to be careful here.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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Or we could say that the hypotenuse is going to be equal to the square root of 42.25. And I could get my calculator out at this point, but I'll actually just keep using this expression to figure out the total number of lights. So what's the total length of the string? We have to be careful here. A lot of folks would say, oh, I figured out the hypotenuse. Let me just multiply that by 10. In fact, my brain almost did that just now.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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We have to be careful here. A lot of folks would say, oh, I figured out the hypotenuse. Let me just multiply that by 10. In fact, my brain almost did that just now. But we gotta realize that the entire string is the hypotenuse plus this 2.5. So the whole string length, let me write it this way, string length is equal to 2.5 plus the square root of 42.25. And then we would just multiply that times 10 to get the total number of lights.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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In fact, my brain almost did that just now. But we gotta realize that the entire string is the hypotenuse plus this 2.5. So the whole string length, let me write it this way, string length is equal to 2.5 plus the square root of 42.25. And then we would just multiply that times 10 to get the total number of lights. So now let's actually get the calculator out. So we have 42.25, and then we were to take the square root of that, gets us to 6.5, and then we add the other 2.5 plus 2.5 equals that. That's the total length of string.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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And then we would just multiply that times 10 to get the total number of lights. So now let's actually get the calculator out. So we have 42.25, and then we were to take the square root of that, gets us to 6.5, and then we add the other 2.5 plus 2.5 equals that. That's the total length of string. So the total length of string is nine meters, and there are 10 lights per meter. So the number of lights, number of lights, is equal to nine total meters of string times 10 lights per meter, which would give us 90 lights. Now, some of you might debate, if you think really deeply about it, is if you have a light right at the beginning, if these were kind of set up like fence posts, that maybe you could argue that there's one extra light in there.
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Multi-step word problem with Pythagorean theorem Geometry Khan Academy.mp3
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We are told that pentagon A prime, B prime, C prime, D prime, E prime, which is in red right over here, is the image of pentagon ABCDE under a dilation. So that's ABCDE. What is the scale factor of the dilation? So they don't even tell us the center of the dilation, but in order to figure out the scale factor, you just have to realize when you do a dilation, the distance between corresponding points will change according to the scale factor. So for example, we could look at the distance between point A and point B right over here. What is our change in y? Our change in, or even what is our distance?
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Dilation scale factor examples.mp3
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So they don't even tell us the center of the dilation, but in order to figure out the scale factor, you just have to realize when you do a dilation, the distance between corresponding points will change according to the scale factor. So for example, we could look at the distance between point A and point B right over here. What is our change in y? Our change in, or even what is our distance? Our change in y is our distance because we don't have a change in x. Well, this is one, two, three, four, five, six. So this length right over here is equal to six.
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Dilation scale factor examples.mp3
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Our change in, or even what is our distance? Our change in y is our distance because we don't have a change in x. Well, this is one, two, three, four, five, six. So this length right over here is equal to six. Now what about the corresponding side from A prime to B prime? Well, this length right over here is equal to two. And so you could see we went from having a length of six to a length of two, so you would have to multiply by 1 3rd.
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Dilation scale factor examples.mp3
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So this length right over here is equal to six. Now what about the corresponding side from A prime to B prime? Well, this length right over here is equal to two. And so you could see we went from having a length of six to a length of two, so you would have to multiply by 1 3rd. So our scale factor right over here is 1 3rd. Now you might be saying, okay, that was pretty straightforward because we had a very clear, you could just see the distance between A and B. How would you do it if you didn't have a vertical or a horizontal line?
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Dilation scale factor examples.mp3
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And so you could see we went from having a length of six to a length of two, so you would have to multiply by 1 3rd. So our scale factor right over here is 1 3rd. Now you might be saying, okay, that was pretty straightforward because we had a very clear, you could just see the distance between A and B. How would you do it if you didn't have a vertical or a horizontal line? Well, one way to think about it is the changes in y and the changes in x would scale accordingly. So if you looked at the distance between point A and point E, our change in y is negative three right over here and our change in x is positive three right over here. And you could see over here between A prime and E prime, our change in y is negative one, which is 1 3rd of negative three, and our change in x is one, which is 1 3rd of three.
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Dilation scale factor examples.mp3
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How would you do it if you didn't have a vertical or a horizontal line? Well, one way to think about it is the changes in y and the changes in x would scale accordingly. So if you looked at the distance between point A and point E, our change in y is negative three right over here and our change in x is positive three right over here. And you could see over here between A prime and E prime, our change in y is negative one, which is 1 3rd of negative three, and our change in x is one, which is 1 3rd of three. So once again, you see our scale factor being 1 3rd. Let's do another example. So we are told that pentagon A prime, B prime, C prime, D prime, E prime is the image, and they haven't drawn that here, is the image of pentagon A, B, C, D, E under a dilation with a scale factor of 5 1st.
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Dilation scale factor examples.mp3
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And you could see over here between A prime and E prime, our change in y is negative one, which is 1 3rd of negative three, and our change in x is one, which is 1 3rd of three. So once again, you see our scale factor being 1 3rd. Let's do another example. So we are told that pentagon A prime, B prime, C prime, D prime, E prime is the image, and they haven't drawn that here, is the image of pentagon A, B, C, D, E under a dilation with a scale factor of 5 1st. So they're giving us our scale factor. What is the length of segment A prime, E prime? So as I was mentioning while I read it, they didn't actually draw this one out.
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Dilation scale factor examples.mp3
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So we are told that pentagon A prime, B prime, C prime, D prime, E prime is the image, and they haven't drawn that here, is the image of pentagon A, B, C, D, E under a dilation with a scale factor of 5 1st. So they're giving us our scale factor. What is the length of segment A prime, E prime? So as I was mentioning while I read it, they didn't actually draw this one out. So how do we figure out the length of a segment? Well, I encourage you to pause the video and try to think about it. Well, they give us the scale factor, and so what it tells us, if the scale factor is 5 1st, that means that the corresponding lengths will change by a factor of 5 1st.
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Dilation scale factor examples.mp3
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So as I was mentioning while I read it, they didn't actually draw this one out. So how do we figure out the length of a segment? Well, I encourage you to pause the video and try to think about it. Well, they give us the scale factor, and so what it tells us, if the scale factor is 5 1st, that means that the corresponding lengths will change by a factor of 5 1st. So to figure out the length of segment A prime, E prime, this is going to be, you could think of it as the image of segment AE, and so you could see that the length of AE is equal to two, and so the length of A prime, E prime is going to be equal to AE, which is two, times the scale factor, times 5 1st. This is our scale factor right over here, and of course, what's two times 5 1st? Well, it is going to be equal to five, five of these units right over here.
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Dilation scale factor examples.mp3
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Well, they give us the scale factor, and so what it tells us, if the scale factor is 5 1st, that means that the corresponding lengths will change by a factor of 5 1st. So to figure out the length of segment A prime, E prime, this is going to be, you could think of it as the image of segment AE, and so you could see that the length of AE is equal to two, and so the length of A prime, E prime is going to be equal to AE, which is two, times the scale factor, times 5 1st. This is our scale factor right over here, and of course, what's two times 5 1st? Well, it is going to be equal to five, five of these units right over here. So in this case, we didn't even have to draw A prime, B prime, C prime, D prime, E prime. In fact, they haven't even given us enough information. I could draw the scale of that, but I actually don't know where to put it because they didn't even give us our center of dilation, but we know that corresponding sides or the lengths between corresponding points are going to be scaled by the scale factor.
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Dilation scale factor examples.mp3
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Well, it is going to be equal to five, five of these units right over here. So in this case, we didn't even have to draw A prime, B prime, C prime, D prime, E prime. In fact, they haven't even given us enough information. I could draw the scale of that, but I actually don't know where to put it because they didn't even give us our center of dilation, but we know that corresponding sides or the lengths between corresponding points are going to be scaled by the scale factor. Now, with that in mind, let's do another example. So we are told that triangle A prime, B prime, C prime, which they depicted right over here, is the image of triangle ABC, which they did not depict under a dilation with a scale factor of two. What is the length of segment AB?
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Dilation scale factor examples.mp3
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I could draw the scale of that, but I actually don't know where to put it because they didn't even give us our center of dilation, but we know that corresponding sides or the lengths between corresponding points are going to be scaled by the scale factor. Now, with that in mind, let's do another example. So we are told that triangle A prime, B prime, C prime, which they depicted right over here, is the image of triangle ABC, which they did not depict under a dilation with a scale factor of two. What is the length of segment AB? Once again, they haven't drawn AB here. How do we figure it out? Well, it's gonna be a similar way as the last example, but here, they've given us the image and they didn't give us the original, so how do we do it?
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Dilation scale factor examples.mp3
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What is the length of segment AB? Once again, they haven't drawn AB here. How do we figure it out? Well, it's gonna be a similar way as the last example, but here, they've given us the image and they didn't give us the original, so how do we do it? Well, the key, and pause the video again and try to do it on your own. Well, the key realization here is that if you take the length of segment AB and you were to multiply by the scale factor, so you multiply it by two, then you're going to get the length of segment A prime, B prime. The image's length is equal to the scale factor times the corresponding length on our original triangle.
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Dilation scale factor examples.mp3
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Well, it's gonna be a similar way as the last example, but here, they've given us the image and they didn't give us the original, so how do we do it? Well, the key, and pause the video again and try to do it on your own. Well, the key realization here is that if you take the length of segment AB and you were to multiply by the scale factor, so you multiply it by two, then you're going to get the length of segment A prime, B prime. The image's length is equal to the scale factor times the corresponding length on our original triangle. So what is the length of A prime, B prime? Well, this is straightforward to figure out. It is one, two, three, four, five, six, seven, eight.
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Dilation scale factor examples.mp3
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So the fact that it's perpendicular means that this line will make a 90 degree angle where it intersects with AB. And it's going to bisect it, so it's going to go halfway in between. And I have at my disposal some tools. I can put out, I can draw things with a compass, and I can add a straight edge. So let's try this out. So let me add a compass. And so this is kind of a virtual compass.
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Constructing a perpendicular bisector using a compass and straightedge Geometry Khan Academy.mp3
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I can put out, I can draw things with a compass, and I can add a straight edge. So let's try this out. So let me add a compass. And so this is kind of a virtual compass. So in a real compass, it's one of those little metal things where you can pivot it on one point, and you can draw a circle of any radius. And so here I'm going to center it at A. And I'm going to make the radius equal to the length of AB.
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Constructing a perpendicular bisector using a compass and straightedge Geometry Khan Academy.mp3
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And so this is kind of a virtual compass. So in a real compass, it's one of those little metal things where you can pivot it on one point, and you can draw a circle of any radius. And so here I'm going to center it at A. And I'm going to make the radius equal to the length of AB. Now I'm going to add another circle with my compass. And now I'm going to center it at B and make the radius equal to AB. And now this gives me two points that I can actually use to draw my perpendicular bisector.
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Constructing a perpendicular bisector using a compass and straightedge Geometry Khan Academy.mp3
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And I'm going to make the radius equal to the length of AB. Now I'm going to add another circle with my compass. And now I'm going to center it at B and make the radius equal to AB. And now this gives me two points that I can actually use to draw my perpendicular bisector. If I connect this point and this point, it is going to bisect AB, and it's also going to be perpendicular. So let's add a straight edge here. So this is to draw a line.
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Constructing a perpendicular bisector using a compass and straightedge Geometry Khan Academy.mp3
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And now this gives me two points that I can actually use to draw my perpendicular bisector. If I connect this point and this point, it is going to bisect AB, and it's also going to be perpendicular. So let's add a straight edge here. So this is to draw a line. So I'm going to draw a line between that point and that point right over there. Let me scroll down so you can look at it a little bit clearer. So there you go.
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Constructing a perpendicular bisector using a compass and straightedge Geometry Khan Academy.mp3
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So this is to draw a line. So I'm going to draw a line between that point and that point right over there. Let me scroll down so you can look at it a little bit clearer. So there you go. That's my construction. I've made a perpendicular bisector for segment AB. Check my answer.
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Constructing a perpendicular bisector using a compass and straightedge Geometry Khan Academy.mp3
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And my goal is to draw a new line that goes through this point and is perpendicular to my original line. How do I do that? Well, you might imagine that our compass will come in handy. It's been handy before. And so what I will do is, I'll pick an arbitrary point on our original line, let's say this point right over here, and then I'll adjust my compass. So the distance between the pivot point and my pencil tip is the same as the distance between those two points. And then I can now use my compass to trace out an arc of that radius.
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Geometric constructions perpendicular line through a point off the line Geometry Khan Academy.mp3
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It's been handy before. And so what I will do is, I'll pick an arbitrary point on our original line, let's say this point right over here, and then I'll adjust my compass. So the distance between the pivot point and my pencil tip is the same as the distance between those two points. And then I can now use my compass to trace out an arc of that radius. So there you go. Now my next step is to find another point on my original line that has the same distance from that point that is off the line. And I can do that by centering my compass on that offline point, and then drawing another arc.
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Geometric constructions perpendicular line through a point off the line Geometry Khan Academy.mp3
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And then I can now use my compass to trace out an arc of that radius. So there you go. Now my next step is to find another point on my original line that has the same distance from that point that is off the line. And I can do that by centering my compass on that offline point, and then drawing another arc. And I can see very clearly that this point also has the same distance from this point up here. And then I can center my compass on that point. And notice I haven't changed the radius of my compass to draw another arc like this, to draw another arc like this.
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Geometric constructions perpendicular line through a point off the line Geometry Khan Academy.mp3
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And I can do that by centering my compass on that offline point, and then drawing another arc. And I can see very clearly that this point also has the same distance from this point up here. And then I can center my compass on that point. And notice I haven't changed the radius of my compass to draw another arc like this, to draw another arc like this. And then what I can do is connect this point and that point, and it at least looks perpendicular, but we're going to prove to ourselves that it is indeed perpendicular to our original line. So let me just draw it so you have that like that. So how do we feel good that this new line that I just drew is perpendicular to our original one?
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Geometric constructions perpendicular line through a point off the line Geometry Khan Academy.mp3
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And notice I haven't changed the radius of my compass to draw another arc like this, to draw another arc like this. And then what I can do is connect this point and that point, and it at least looks perpendicular, but we're going to prove to ourselves that it is indeed perpendicular to our original line. So let me just draw it so you have that like that. So how do we feel good that this new line that I just drew is perpendicular to our original one? Well, let's connect the dots that we made. So if we connect all the dots, we're going to get a rhombus. We know that this distance, this distance is the same as this distance, the same as this one right over here, which is the same as this distance.
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Geometric constructions perpendicular line through a point off the line Geometry Khan Academy.mp3
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So how do we feel good that this new line that I just drew is perpendicular to our original one? Well, let's connect the dots that we made. So if we connect all the dots, we're going to get a rhombus. We know that this distance, this distance is the same as this distance, the same as this one right over here, which is the same as this distance. Let me make sure I get my straight edge right. Same as that distance, which is the same as this distance, same as that distance. And then, so this is a rhombus, and we know that the diagonals of a rhombus intersect at right angles.
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Geometric constructions perpendicular line through a point off the line Geometry Khan Academy.mp3
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We tend to be told in algebra class that if we have a line, our line will have a constant rate of change of y with respect to x. Or another way of thinking about it, that a line will have a constant inclination, or that our line will have a constant slope. And our slope is literally defined as your change in y. This triangle is the Greek letter delta. It's a shorthand for change in. It means change in y. Delta y means change in y. Over change in x.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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This triangle is the Greek letter delta. It's a shorthand for change in. It means change in y. Delta y means change in y. Over change in x. And if you're dealing with a line, this right over here is constant. Constant for a line. What I want to do in this video is to actually prove that using similar triangles from geometry.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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Over change in x. And if you're dealing with a line, this right over here is constant. Constant for a line. What I want to do in this video is to actually prove that using similar triangles from geometry. So let's think about two sets of two points. So let's say that's a point there. Let me do it in a different color.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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What I want to do in this video is to actually prove that using similar triangles from geometry. So let's think about two sets of two points. So let's say that's a point there. Let me do it in a different color. Let me start at this point. And let me end up at that point. So what is our change in x between these two points?
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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Let me do it in a different color. Let me start at this point. And let me end up at that point. So what is our change in x between these two points? So this point's x value is right over here. This point's x value is right over here. So our change in x is going to be that right over there.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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So what is our change in x between these two points? So this point's x value is right over here. This point's x value is right over here. So our change in x is going to be that right over there. And what's our change in y? Well, this point's y value is right over here. This point's y value is right over here.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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So our change in x is going to be that right over there. And what's our change in y? Well, this point's y value is right over here. This point's y value is right over here. So this height or this height is our change in y. So that is our change in y. Now, let's look at two other points.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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This point's y value is right over here. So this height or this height is our change in y. So that is our change in y. Now, let's look at two other points. Let's say I have this point and this point right over here. And let's do the same exercise. What's the change in x?
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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Now, let's look at two other points. Let's say I have this point and this point right over here. And let's do the same exercise. What's the change in x? Well, let's see. If we're going this point's x value is here. This point's x value is here.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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What's the change in x? Well, let's see. If we're going this point's x value is here. This point's x value is here. So if we start here and we go this far, this would be the change in x between this point and this point. And this is going to be the change. Let me do that in the same green color.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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This point's x value is here. So if we start here and we go this far, this would be the change in x between this point and this point. And this is going to be the change. Let me do that in the same green color. So this is going to be the change in x between those two points. And our change in y, well, this y value is here. This y value is up here.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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Let me do that in the same green color. So this is going to be the change in x between those two points. And our change in y, well, this y value is here. This y value is up here. So our change in y is going to be that right over here. So what I need to show, I'm just picking two arbitrary points. I need to show that the ratio of this change in y to this change of x is going to be the same as the ratio of this change in y to this change of x.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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This y value is up here. So our change in y is going to be that right over here. So what I need to show, I'm just picking two arbitrary points. I need to show that the ratio of this change in y to this change of x is going to be the same as the ratio of this change in y to this change of x. Or the ratio of this purple side to this green side is going to be the same as the ratio of this purple side to this green side. Remember, I'm just picking two sets of arbitrary points here. And the way that I will show it is through similarity.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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I need to show that the ratio of this change in y to this change of x is going to be the same as the ratio of this change in y to this change of x. Or the ratio of this purple side to this green side is going to be the same as the ratio of this purple side to this green side. Remember, I'm just picking two sets of arbitrary points here. And the way that I will show it is through similarity. If I can show that this triangle is similar to this triangle, then we are all set up. And just as a reminder of what similarity is, two triangles are similar. And there's multiple ways of thinking about it.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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And the way that I will show it is through similarity. If I can show that this triangle is similar to this triangle, then we are all set up. And just as a reminder of what similarity is, two triangles are similar. And there's multiple ways of thinking about it. So you're similar if and only if all corresponding, or I should say all three angles, are the same or are congruent. So all three, and let me be careful here. They don't have to be the same exact angle.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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And there's multiple ways of thinking about it. So you're similar if and only if all corresponding, or I should say all three angles, are the same or are congruent. So all three, and let me be careful here. They don't have to be the same exact angle. The corresponding angles have to be the same. So corresponding, I always misspell it. Corresponding angles are going to be equal.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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They don't have to be the same exact angle. The corresponding angles have to be the same. So corresponding, I always misspell it. Corresponding angles are going to be equal. Or we could say they are congruent. So for example, if I have this triangle right over here, and this is 30, this is 60, and this is 90. And then I have this triangle right over here.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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Corresponding angles are going to be equal. Or we could say they are congruent. So for example, if I have this triangle right over here, and this is 30, this is 60, and this is 90. And then I have this triangle right over here. I'll try to draw it. So I have this triangle where this is 30 degrees, this is 60 degrees, and this is 90 degrees. Even though their side lengths are different, these are going to be similar triangles.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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And then I have this triangle right over here. I'll try to draw it. So I have this triangle where this is 30 degrees, this is 60 degrees, and this is 90 degrees. Even though their side lengths are different, these are going to be similar triangles. They're essentially scaled up versions of each other. All the corresponding angles. 60 corresponds to this 60.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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Even though their side lengths are different, these are going to be similar triangles. They're essentially scaled up versions of each other. All the corresponding angles. 60 corresponds to this 60. 30 corresponds to this 30. And 90 corresponds to this one. So these two triangles are similar.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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60 corresponds to this 60. 30 corresponds to this 30. And 90 corresponds to this one. So these two triangles are similar. And what's neat about similar triangles, if you can establish that two triangles are similar, then the ratio between corresponding sides is going to be the same. So if these two are similar, then the ratio of this side to this side is going to be the same as the ratio of this side to this side. And so you could see why that will be useful in proving that the slope is constant here.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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So these two triangles are similar. And what's neat about similar triangles, if you can establish that two triangles are similar, then the ratio between corresponding sides is going to be the same. So if these two are similar, then the ratio of this side to this side is going to be the same as the ratio of this side to this side. And so you could see why that will be useful in proving that the slope is constant here. Because all we have to do is, look, if these two triangles are similar, then the ratio between corresponding sides is always going to be the same. We've picked two arbitrary sets of points. Then this would be true really for any two arbitrary sets of points across the line.
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Similar triangles to prove that the slope is constant for a line Algebra I Khan Academy.mp3
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