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Let's construct an angle theta. Let's call this angle right over here theta. Let's just say, for the sake of argument, that this angle is just the exact right measure. If you look at the arc that subtends this angle, that seems like a very fancy word. Let me draw the angle. If you look at the arc that subtends the angle, that's a fancy word, but that's really just talking about the arc along the circle that intersects the two sides of the angles. This arc right over here subtends the angle theta.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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If you look at the arc that subtends this angle, that seems like a very fancy word. Let me draw the angle. If you look at the arc that subtends the angle, that's a fancy word, but that's really just talking about the arc along the circle that intersects the two sides of the angles. This arc right over here subtends the angle theta. Let me write that down. Subtends this arc. Subtends angle theta.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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This arc right over here subtends the angle theta. Let me write that down. Subtends this arc. Subtends angle theta. Let's say theta is the exact right size. This arc is also the same length as the radius of the circle. This arc is also of length r. Given that, if you were defining a new type of angle measurement, and you wanted to call it a radian, which is very close to a radius, how many radians would you define this angle to be?
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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Subtends angle theta. Let's say theta is the exact right size. This arc is also the same length as the radius of the circle. This arc is also of length r. Given that, if you were defining a new type of angle measurement, and you wanted to call it a radian, which is very close to a radius, how many radians would you define this angle to be? The most obvious one, if you view a radian as another way of saying radiuses or radii, you say, look, this is subtended by an arc of one radius, so why don't we call this right over here one radian? Which is exactly how a radian is defined. When you have a circle and you have an angle of one radian, the arc that subtends it is exactly one radius long, which you can imagine might be a little bit useful as we start to interpret more and more types of circles.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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This arc is also of length r. Given that, if you were defining a new type of angle measurement, and you wanted to call it a radian, which is very close to a radius, how many radians would you define this angle to be? The most obvious one, if you view a radian as another way of saying radiuses or radii, you say, look, this is subtended by an arc of one radius, so why don't we call this right over here one radian? Which is exactly how a radian is defined. When you have a circle and you have an angle of one radian, the arc that subtends it is exactly one radius long, which you can imagine might be a little bit useful as we start to interpret more and more types of circles. When you give a degree, you really have to do a little bit of math and think about the circumference and all of that to think about how many radiuses are subtending that angle. Here, the angle in radians tells you exactly how many arc lengths that is subtending the angle. Let's do a couple of thought experiments here.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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When you have a circle and you have an angle of one radian, the arc that subtends it is exactly one radius long, which you can imagine might be a little bit useful as we start to interpret more and more types of circles. When you give a degree, you really have to do a little bit of math and think about the circumference and all of that to think about how many radiuses are subtending that angle. Here, the angle in radians tells you exactly how many arc lengths that is subtending the angle. Let's do a couple of thought experiments here. Given that, what would be the angle in radians if we were to go... Let me draw another circle here. Let me draw another circle here. That's the center.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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Let's do a couple of thought experiments here. Given that, what would be the angle in radians if we were to go... Let me draw another circle here. Let me draw another circle here. That's the center. We'll start right over there. What would happen if I had an angle? What angle, if I wanted to measure in radians, what angle would this be in radians?
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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That's the center. We'll start right over there. What would happen if I had an angle? What angle, if I wanted to measure in radians, what angle would this be in radians? You can almost think of it as radiuses. What would that angle be? Going one full revolution in degrees, that would be 360 degrees.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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What angle, if I wanted to measure in radians, what angle would this be in radians? You can almost think of it as radiuses. What would that angle be? Going one full revolution in degrees, that would be 360 degrees. Based on this definition, what would this be in radians? Let's think about the arc that subtends this angle. The arc that subtends this angle is the entire circumference of this circle.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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Going one full revolution in degrees, that would be 360 degrees. Based on this definition, what would this be in radians? Let's think about the arc that subtends this angle. The arc that subtends this angle is the entire circumference of this circle. It's the entire circumference of this circle. What's the circumference of a circle in terms of radiuses? If this has length r, if the radius is length r, what's the circumference of the circle in terms of r?
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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The arc that subtends this angle is the entire circumference of this circle. It's the entire circumference of this circle. What's the circumference of a circle in terms of radiuses? If this has length r, if the radius is length r, what's the circumference of the circle in terms of r? We know that. That's going to be 2 pi r. Going back to this angle, the length of the arc that subtends this angle is how many radiuses this is? What's 2 pi radiuses this is?
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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If this has length r, if the radius is length r, what's the circumference of the circle in terms of r? We know that. That's going to be 2 pi r. Going back to this angle, the length of the arc that subtends this angle is how many radiuses this is? What's 2 pi radiuses this is? It's 2 pi times r. This angle right over here, I'll call this a different angle, x. x in this case is going to be 2 pi radians. It is subtended by an arc length of 2 pi radiuses. If the radius was one unit, then this would be 2 pi times 1, 2 pi radiuses.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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What's 2 pi radiuses this is? It's 2 pi times r. This angle right over here, I'll call this a different angle, x. x in this case is going to be 2 pi radians. It is subtended by an arc length of 2 pi radiuses. If the radius was one unit, then this would be 2 pi times 1, 2 pi radiuses. Given that, let's start to think about how we can convert between radians and degrees and vice versa. If I were to have, and we can just follow up over here, if we do one full revolution, that is 2 pi radians, how many degrees is this going to be equal to? We already know this.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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If the radius was one unit, then this would be 2 pi times 1, 2 pi radiuses. Given that, let's start to think about how we can convert between radians and degrees and vice versa. If I were to have, and we can just follow up over here, if we do one full revolution, that is 2 pi radians, how many degrees is this going to be equal to? We already know this. A full revolution in degrees is 360 degrees. I could either write out the word degrees or I can use this little degree notation there. Actually, let me write out the word degrees.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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We already know this. A full revolution in degrees is 360 degrees. I could either write out the word degrees or I can use this little degree notation there. Actually, let me write out the word degrees. It might make things a little bit clearer that we're using units in both cases. If we wanted to simplify this a little bit, we could divide both sides by 2, in which case we would get on the left-hand side, we would get pi radians would be equal to how many degrees? It would be equal to 180 degrees.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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Actually, let me write out the word degrees. It might make things a little bit clearer that we're using units in both cases. If we wanted to simplify this a little bit, we could divide both sides by 2, in which case we would get on the left-hand side, we would get pi radians would be equal to how many degrees? It would be equal to 180 degrees. 180 degrees. I could write it that way or I could write it that way. You see over here, this is 180 degrees.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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It would be equal to 180 degrees. 180 degrees. I could write it that way or I could write it that way. You see over here, this is 180 degrees. You also see if you were to draw a circle around here, we've gone halfway around the circle. The arc length or the arc that subtends the angle is half the circumference. Half the circumference are pi radiuses.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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You see over here, this is 180 degrees. You also see if you were to draw a circle around here, we've gone halfway around the circle. The arc length or the arc that subtends the angle is half the circumference. Half the circumference are pi radiuses. We call this pi radians. Pi radians is 180 degrees. From this, we can come up with conversions.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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Half the circumference are pi radiuses. We call this pi radians. Pi radians is 180 degrees. From this, we can come up with conversions. One radian would be how many degrees? To do that, we would just have to divide both sides by pi. On the left-hand side, you'd be left with 1.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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From this, we can come up with conversions. One radian would be how many degrees? To do that, we would just have to divide both sides by pi. On the left-hand side, you'd be left with 1. I'll just write it singular now. One radian is equal to, I'm just dividing both sides. Let me make it clear what I'm doing here just to show you.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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On the left-hand side, you'd be left with 1. I'll just write it singular now. One radian is equal to, I'm just dividing both sides. Let me make it clear what I'm doing here just to show you. This isn't some voodoo. I'm just dividing both sides by pi here. On the left-hand side, you're left with 1.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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Let me make it clear what I'm doing here just to show you. This isn't some voodoo. I'm just dividing both sides by pi here. On the left-hand side, you're left with 1. On the right-hand side, you're left with 180 over pi degrees. One radian is equal to 180 over pi degrees, which is starting to make it an interesting way to convert them. Let's think about it the other way.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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On the left-hand side, you're left with 1. On the right-hand side, you're left with 180 over pi degrees. One radian is equal to 180 over pi degrees, which is starting to make it an interesting way to convert them. Let's think about it the other way. If I were to have one degree, how many radians is that? Let's start off with, let me rewrite this thing over here. We said pi radians is equal to 180 degrees.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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Let's think about it the other way. If I were to have one degree, how many radians is that? Let's start off with, let me rewrite this thing over here. We said pi radians is equal to 180 degrees. Now we want to think about one degree. Let's solve for one degree. One degree, we can divide both sides by 180.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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We said pi radians is equal to 180 degrees. Now we want to think about one degree. Let's solve for one degree. One degree, we can divide both sides by 180. We are left with pi over 180 radians is equal to one degree. Pi over 180 radians is equal to one degree. This might seem confusing and daunting, and it was for me the first time I was exposed to it, especially because we're not exposed to this in our everyday life.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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One degree, we can divide both sides by 180. We are left with pi over 180 radians is equal to one degree. Pi over 180 radians is equal to one degree. This might seem confusing and daunting, and it was for me the first time I was exposed to it, especially because we're not exposed to this in our everyday life. What we're going to see over the next few examples is that as long as we keep in mind this whole idea that 2 pi radians is equal to 360 degrees or that pi radians is equal to 180 degrees, which is the two things that I do keep in my mind, we can always re-derive these two things. You might say, hey, how do I remember if it's pi over 180 or 180 over pi to convert the two things? Well, just remember, which is hopefully intuitive, that 2 pi radians is equal to 360 degrees.
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Introduction to radians Unit circle definition of trig functions Trigonometry Khan Academy.mp3
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So over here I have triangle BDC, it's inside of triangle AEC. They both share this angle right over there, so that gives us one angle. We need two to get to angle-angle, which gives us similarity. And we know that these two lines are parallel. We know if two lines are parallel and we have a transversal, that corresponding angles are going to be congruent. So that angle is going to correspond to that angle right over there. And we're done.
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And we know that these two lines are parallel. We know if two lines are parallel and we have a transversal, that corresponding angles are going to be congruent. So that angle is going to correspond to that angle right over there. And we're done. We have one angle in triangle AEC that is congruent to another angle in BDC. And then we have this angle that's obviously congruent to itself that's in both triangles. So both triangles have a pair of corresponding angles that are congruent, so they must be similar.
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And we're done. We have one angle in triangle AEC that is congruent to another angle in BDC. And then we have this angle that's obviously congruent to itself that's in both triangles. So both triangles have a pair of corresponding angles that are congruent, so they must be similar. So we can write triangle ACE is going to be similar to triangle, and we want to get the letters in the right order. So where the blue angle is here, the blue angle there is vertex B. Then we go to the white angle, C. And then we go to the unlabeled angle right over there, BCD.
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So both triangles have a pair of corresponding angles that are congruent, so they must be similar. So we can write triangle ACE is going to be similar to triangle, and we want to get the letters in the right order. So where the blue angle is here, the blue angle there is vertex B. Then we go to the white angle, C. And then we go to the unlabeled angle right over there, BCD. So we did that first one. Now let's do this one right over here. This is kind of similar, but at least it looks just superficially looking at it, that YZ is definitely not parallel to ST.
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Then we go to the white angle, C. And then we go to the unlabeled angle right over there, BCD. So we did that first one. Now let's do this one right over here. This is kind of similar, but at least it looks just superficially looking at it, that YZ is definitely not parallel to ST. So we won't be able to do this corresponding angle argument, especially because they didn't even label it as parallel. And so you don't want to look at things just by the way they look. You definitely want to say, what am I given and what am I not given?
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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This is kind of similar, but at least it looks just superficially looking at it, that YZ is definitely not parallel to ST. So we won't be able to do this corresponding angle argument, especially because they didn't even label it as parallel. And so you don't want to look at things just by the way they look. You definitely want to say, what am I given and what am I not given? If these weren't labeled parallel, we wouldn't be able to make the statement, even if they looked parallel. One thing we do have is that we have this angle right here that's common to the inner triangle and to the outer triangle. And they've given us a bunch of sides.
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You definitely want to say, what am I given and what am I not given? If these weren't labeled parallel, we wouldn't be able to make the statement, even if they looked parallel. One thing we do have is that we have this angle right here that's common to the inner triangle and to the outer triangle. And they've given us a bunch of sides. So maybe we can use SAS for similarity, meaning if we can show the ratio of the sides on either side of this angle, if they have the same ratio from the smaller triangle to the larger triangle, then we can show similarity. So let's go, and we have to go on either side of this angle right over here. Let's look at the shorter side on either side of this angle.
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And they've given us a bunch of sides. So maybe we can use SAS for similarity, meaning if we can show the ratio of the sides on either side of this angle, if they have the same ratio from the smaller triangle to the larger triangle, then we can show similarity. So let's go, and we have to go on either side of this angle right over here. Let's look at the shorter side on either side of this angle. So the shorter side is 2. And let's look at the shorter side on either side of the angle for the larger triangle. Well, then the shorter side is on the right-hand side, and that's going to be xt.
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Let's look at the shorter side on either side of this angle. So the shorter side is 2. And let's look at the shorter side on either side of the angle for the larger triangle. Well, then the shorter side is on the right-hand side, and that's going to be xt. So what we want to compare is the ratio between, let me write it this way, we want to see is xy over xt, is that equal to the ratio of the longer side, or if we're looking relative to this angle, the longer of the two, not necessarily the longest of the triangle, although it looks like that as well, is that equal to the ratio of xz over the longer of the two sides when you're looking at this angle right here on either side of that angle for the larger triangle over xs. And it's a little confusing because we've kind of flipped which side, but I'm just thinking about the shorter side on either side of this angle in between and then the longer side on either side of this angle. So these are the shorter sides for the smaller triangle and the larger triangle.
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Well, then the shorter side is on the right-hand side, and that's going to be xt. So what we want to compare is the ratio between, let me write it this way, we want to see is xy over xt, is that equal to the ratio of the longer side, or if we're looking relative to this angle, the longer of the two, not necessarily the longest of the triangle, although it looks like that as well, is that equal to the ratio of xz over the longer of the two sides when you're looking at this angle right here on either side of that angle for the larger triangle over xs. And it's a little confusing because we've kind of flipped which side, but I'm just thinking about the shorter side on either side of this angle in between and then the longer side on either side of this angle. So these are the shorter sides for the smaller triangle and the larger triangle. These are the longer sides for the smaller triangle and the larger triangle. And we see xy, this is 2, xt is 3 plus 1 is 4, xz is 3, and xs is 6. So you have 2 over 4, which is 1 half, which is the same thing as 3, 6.
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So these are the shorter sides for the smaller triangle and the larger triangle. These are the longer sides for the smaller triangle and the larger triangle. And we see xy, this is 2, xt is 3 plus 1 is 4, xz is 3, and xs is 6. So you have 2 over 4, which is 1 half, which is the same thing as 3, 6. So the ratio between the shorter sides on either side of the angle and the longer sides on either side of the angle for both triangles, the ratio is the same. So by SAS we know that the two triangles are congruent. But we have to be careful on how we state the triangles.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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So you have 2 over 4, which is 1 half, which is the same thing as 3, 6. So the ratio between the shorter sides on either side of the angle and the longer sides on either side of the angle for both triangles, the ratio is the same. So by SAS we know that the two triangles are congruent. But we have to be careful on how we state the triangles. We want to make sure we get the corresponding sides. So we could say that triangle, and I'm running out of space here, let me write it right above here. We can write that triangle xyz is similar to triangle.
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But we have to be careful on how we state the triangles. We want to make sure we get the corresponding sides. So we could say that triangle, and I'm running out of space here, let me write it right above here. We can write that triangle xyz is similar to triangle. So we started up at x, which is the vertex at the angle, and we went to the shorter side first. So now we want to start at x and go to the shorter side of the large triangle. So you go to xts.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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We can write that triangle xyz is similar to triangle. So we started up at x, which is the vertex at the angle, and we went to the shorter side first. So now we want to start at x and go to the shorter side of the large triangle. So you go to xts. xyz is similar to xts. Now let's look at this right over here. So in our larger triangle we have a right angle here, but we really know nothing about what's going on with any of these smaller triangles in terms of their actual angles.
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So you go to xts. xyz is similar to xts. Now let's look at this right over here. So in our larger triangle we have a right angle here, but we really know nothing about what's going on with any of these smaller triangles in terms of their actual angles. Even though this looks like a right angle, we cannot assume it. And it shares, if we look at this smaller triangle right over here, it shares one side with the larger triangle, but that's not enough to do anything. And then this triangle over here also shares another side, but that also doesn't do anything.
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So in our larger triangle we have a right angle here, but we really know nothing about what's going on with any of these smaller triangles in terms of their actual angles. Even though this looks like a right angle, we cannot assume it. And it shares, if we look at this smaller triangle right over here, it shares one side with the larger triangle, but that's not enough to do anything. And then this triangle over here also shares another side, but that also doesn't do anything. So we really can't make any statement here about any kind of similarity. So there's no similarity going on here. If they gave us, well, there are some shared angles.
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And then this triangle over here also shares another side, but that also doesn't do anything. So we really can't make any statement here about any kind of similarity. So there's no similarity going on here. If they gave us, well, there are some shared angles. This guy, they both share that angle. The larger triangle and the smaller triangle. So there could be a statement of similarity we could make if we knew that this definitely was a right angle.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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If they gave us, well, there are some shared angles. This guy, they both share that angle. The larger triangle and the smaller triangle. So there could be a statement of similarity we could make if we knew that this definitely was a right angle. Then we could make some interesting statements about similarity. But right now we can't really do anything as is. Let's try this one out, or this pair right over here.
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So there could be a statement of similarity we could make if we knew that this definitely was a right angle. Then we could make some interesting statements about similarity. But right now we can't really do anything as is. Let's try this one out, or this pair right over here. So these are the first ones that we've actually separated out the triangles. So they've given us the three sides of both triangles. So let's just figure out if the ratios between corresponding sides are a constant.
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Let's try this one out, or this pair right over here. So these are the first ones that we've actually separated out the triangles. So they've given us the three sides of both triangles. So let's just figure out if the ratios between corresponding sides are a constant. So let's start with the short side. So the short side here is 3. The shortest side here is 9 square roots of 3.
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So let's just figure out if the ratios between corresponding sides are a constant. So let's start with the short side. So the short side here is 3. The shortest side here is 9 square roots of 3. So we want to see whether the ratio of 3 to 9 square roots of 3 is equal to the next longest side over here is 3 square roots of 3, is equal to 3 square roots of 3, over the next longest side over here, which is 27. And then see if that's going to be equal to the ratio of the longest side. So the longest side here is 6.
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The shortest side here is 9 square roots of 3. So we want to see whether the ratio of 3 to 9 square roots of 3 is equal to the next longest side over here is 3 square roots of 3, is equal to 3 square roots of 3, over the next longest side over here, which is 27. And then see if that's going to be equal to the ratio of the longest side. So the longest side here is 6. And then the longest side over here is 18 square roots of 3. So this is going to give us, let's see, this is 3. Let me do this in a neutral color.
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So the longest side here is 6. And then the longest side over here is 18 square roots of 3. So this is going to give us, let's see, this is 3. Let me do this in a neutral color. So this becomes 1 over 3 square roots of 3. This becomes 1 over square roots of 3, root 3 over 9, which seems like a different number, but we want to be careful here. And then this right over here, this becomes, this is a, if you divide the numerator and denominator by 6, this becomes a 1 and this becomes 3 square roots of 3.
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Let me do this in a neutral color. So this becomes 1 over 3 square roots of 3. This becomes 1 over square roots of 3, root 3 over 9, which seems like a different number, but we want to be careful here. And then this right over here, this becomes, this is a, if you divide the numerator and denominator by 6, this becomes a 1 and this becomes 3 square roots of 3. So you get 1 square roots of 3, 1 over 3 root 3 needs to be equal to 1, needs to be equal to square root of 3 over 9, which needs to be equal to 1 over 3 square roots of 3. At first they don't look equal, but we can actually rationalize this denominator right over here. We can show that 1 over 3 square roots of 3, if you multiply it by square root of 3 over square root of 3, this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 times 3 is 9.
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And then this right over here, this becomes, this is a, if you divide the numerator and denominator by 6, this becomes a 1 and this becomes 3 square roots of 3. So you get 1 square roots of 3, 1 over 3 root 3 needs to be equal to 1, needs to be equal to square root of 3 over 9, which needs to be equal to 1 over 3 square roots of 3. At first they don't look equal, but we can actually rationalize this denominator right over here. We can show that 1 over 3 square roots of 3, if you multiply it by square root of 3 over square root of 3, this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 times 3 is 9. So these actually are all the same. This is actually saying, this is 1 over 3 root 3, which is the same thing as square root of 3 over 9, which is this right over here, which is the same thing as 1 over 3 root 3. So actually these are similar triangles.
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We can show that 1 over 3 square roots of 3, if you multiply it by square root of 3 over square root of 3, this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 times 3 is 9. So these actually are all the same. This is actually saying, this is 1 over 3 root 3, which is the same thing as square root of 3 over 9, which is this right over here, which is the same thing as 1 over 3 root 3. So actually these are similar triangles. So we can actually say it, and I'll make sure I get the order right. So I'll start with E, which is between the blue and the magenta side. So that's between the blue and the magenta side, that is H right over here.
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So actually these are similar triangles. So we can actually say it, and I'll make sure I get the order right. So I'll start with E, which is between the blue and the magenta side. So that's between the blue and the magenta side, that is H right over here. So triangle E, I'll do it like this. Triangle E, and then I'll go along the blue side, F. Then I'll go along the blue side over here. Oh, sorry, let me do it this way.
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So that's between the blue and the magenta side, that is H right over here. So triangle E, I'll do it like this. Triangle E, and then I'll go along the blue side, F. Then I'll go along the blue side over here. Oh, sorry, let me do it this way. Actually, let me just write it this way. E, triangle E, F, G, we know is similar to triangle. So E is between the blue and the magenta side.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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Oh, sorry, let me do it this way. Actually, let me just write it this way. E, triangle E, F, G, we know is similar to triangle. So E is between the blue and the magenta side. Blue and magenta side, that is H. And then we go along the blue side to F, go along the blue side to I. And then you went along the orange side to G. And then you go along the orange side to J. So triangle E, F, G is similar to triangle H, I, J by side, side, side similarity.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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So E is between the blue and the magenta side. Blue and magenta side, that is H. And then we go along the blue side to F, go along the blue side to I. And then you went along the orange side to G. And then you go along the orange side to J. So triangle E, F, G is similar to triangle H, I, J by side, side, side similarity. They're not congruent sides. They all have just the same ratio or the same scaling factor. Now let's do this last one right over here.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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So triangle E, F, G is similar to triangle H, I, J by side, side, side similarity. They're not congruent sides. They all have just the same ratio or the same scaling factor. Now let's do this last one right over here. So we have an angle that's congruent to another angle right over there. And we have two sides. And so it might be tempting to use side, angle, side, because we have side, angle, side here.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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Now let's do this last one right over here. So we have an angle that's congruent to another angle right over there. And we have two sides. And so it might be tempting to use side, angle, side, because we have side, angle, side here. And even the ratios look kind of tempting, because 4 times 2 is 8, 5 times 2 is 10. But it's tricky here, because they aren't the same corresponding sides. In order to use side, angle, side, the two sides that have the same corresponding ratios, they have to be on either side of the angle.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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And so it might be tempting to use side, angle, side, because we have side, angle, side here. And even the ratios look kind of tempting, because 4 times 2 is 8, 5 times 2 is 10. But it's tricky here, because they aren't the same corresponding sides. In order to use side, angle, side, the two sides that have the same corresponding ratios, they have to be on either side of the angle. So in this case, they are on either side of the angle. In this case, the 4 is on one side of the angle, but the 5 is not. So because if this 5 was over here, then we could make an argument for similarity.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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In order to use side, angle, side, the two sides that have the same corresponding ratios, they have to be on either side of the angle. So in this case, they are on either side of the angle. In this case, the 4 is on one side of the angle, but the 5 is not. So because if this 5 was over here, then we could make an argument for similarity. But with this 5 not being on the other side of the angle, it's not sandwiching the angle with the 4, we can't use side, angle, side. And frankly, there's nothing that we can do over here. So we can't make some strong statement about similarity for this last one.
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Similar triangle example problems Similarity Geometry Khan Academy.mp3
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And in this example, P doesn't sit on this circle. P is outside of the circle. So we want to draw something like, and actually let me get my straight edge out. So my straight edge, and I have my controls up here. It is on the Khan Academy exercise, constructing a line tangent to a circle. So I have that up there. So let me add a straight edge.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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So my straight edge, and I have my controls up here. It is on the Khan Academy exercise, constructing a line tangent to a circle. So I have that up there. So let me add a straight edge. Once again, you could try to eyeball it. I'm going to go through P. You want to be tangent to the circle. So hey, yeah, maybe something like that.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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So let me add a straight edge. Once again, you could try to eyeball it. I'm going to go through P. You want to be tangent to the circle. So hey, yeah, maybe something like that. That looks pretty good. But like we've said in other geometric constructions videos, that's just eyeballing it. We don't know how precise that actually is.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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So hey, yeah, maybe something like that. That looks pretty good. But like we've said in other geometric constructions videos, that's just eyeballing it. We don't know how precise that actually is. So what if we had a compass and a straight edge? How do we make something more precise? And actually, in the process of doing it, we'll make really fun patterns.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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We don't know how precise that actually is. So what if we had a compass and a straight edge? How do we make something more precise? And actually, in the process of doing it, we'll make really fun patterns. So how do we do that? Well, what I'm going to do is I'm going to try to construct a circle that has the segment PC as a diameter. So let me draw that.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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And actually, in the process of doing it, we'll make really fun patterns. So how do we do that? Well, what I'm going to do is I'm going to try to construct a circle that has the segment PC as a diameter. So let me draw that. So I want to construct a circle that has segment PC as a diameter. And in order to do that, I need to figure out where the center of that circle is. And you'll see in a second why it's useful to have this circle that has PC as a diameter.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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So let me draw that. So I want to construct a circle that has segment PC as a diameter. And in order to do that, I need to figure out where the center of that circle is. And you'll see in a second why it's useful to have this circle that has PC as a diameter. So where is the center? It looks like it's roughly here. But how do we actually construct it?
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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And you'll see in a second why it's useful to have this circle that has PC as a diameter. So where is the center? It looks like it's roughly here. But how do we actually construct it? To do that, we're going to have to figure out the midpoint of segment PC or CP. And to do that, what I'm going to do is construct two circles. I'll just make them a little bit larger.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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But how do we actually construct it? To do that, we're going to have to figure out the midpoint of segment PC or CP. And to do that, what I'm going to do is construct two circles. I'll just make them a little bit larger. So one centered at C. I'll make it reasonably large, maybe that big. And then I'm going to construct another circle of the same radius. It's that large one that I just constructed.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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I'll just make them a little bit larger. So one centered at C. I'll make it reasonably large, maybe that big. And then I'm going to construct another circle of the same radius. It's that large one that I just constructed. So it's that same radius. But I'm now going to center it at P. Now why is this interesting? Why is what I just did interesting?
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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It's that large one that I just constructed. So it's that same radius. But I'm now going to center it at P. Now why is this interesting? Why is what I just did interesting? Well, where these two larger circles intersect are going to be equidistant to P and C. How do we know that? Well, all the points on this circle are equidistant to C. All the points on this circle are equidistant to P. And these circles have the same radius. And at this point right over here, they are equidistant to both of them because it sits on both circles.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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Why is what I just did interesting? Well, where these two larger circles intersect are going to be equidistant to P and C. How do we know that? Well, all the points on this circle are equidistant to C. All the points on this circle are equidistant to P. And these circles have the same radius. And at this point right over here, they are equidistant to both of them because it sits on both circles. So this point is equidistant to both of them. And so is this point right over here. And so they both sit on the perpendicular bisector of this segment, of the segment CP.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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And at this point right over here, they are equidistant to both of them because it sits on both circles. So this point is equidistant to both of them. And so is this point right over here. And so they both sit on the perpendicular bisector of this segment, of the segment CP. So actually, let me draw that. So if I draw a line that looks something like this, this right over here is a perpendicular bisector of line of segment CP. Now what we really just care about is that it is bisecting it because we wanted to find the midpoint.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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And so they both sit on the perpendicular bisector of this segment, of the segment CP. So actually, let me draw that. So if I draw a line that looks something like this, this right over here is a perpendicular bisector of line of segment CP. Now what we really just care about is that it is bisecting it because we wanted to find the midpoint. And now that we have found the midpoint of our segment, we're ready to construct that circle I talked about. A circle centered at the midpoint. And it has a diameter.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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Now what we really just care about is that it is bisecting it because we wanted to find the midpoint. And now that we have found the midpoint of our segment, we're ready to construct that circle I talked about. A circle centered at the midpoint. And it has a diameter. It has CP as a diameter. So I got that far. But why did I go through all of this trouble to do that?
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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And it has a diameter. It has CP as a diameter. So I got that far. But why did I go through all of this trouble to do that? Well, now we're going to use the idea that if you have a triangle embedded in a circle where one side of the triangle is a diameter, then you're going to have a right triangle. Well, what am I talking about? Well, let me actually just draw the triangle.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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But why did I go through all of this trouble to do that? Well, now we're going to use the idea that if you have a triangle embedded in a circle where one side of the triangle is a diameter, then you're going to have a right triangle. Well, what am I talking about? Well, let me actually just draw the triangle. So let me add a straight edge here. So I'm going to draw a triangle. So it has one side as a diameter.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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Well, let me actually just draw the triangle. So let me add a straight edge here. So I'm going to draw a triangle. So it has one side as a diameter. So we're going to embed it in this circle that I just constructed. The circle centered at this midpoint, this circle right over here. CP is clearly a diameter.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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So it has one side as a diameter. So we're going to embed it in this circle that I just constructed. The circle centered at this midpoint, this circle right over here. CP is clearly a diameter. And I'm going to embed it in CP. But I'm going to put it at this point right over here. Because this point sits on this circle centered at C. And I'm going to draw another line.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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CP is clearly a diameter. And I'm going to embed it in CP. But I'm going to put it at this point right over here. Because this point sits on this circle centered at C. And I'm going to draw another line. So this other line, let me put that there and this there. So my claim is that this is a right triangle. How do I make this claim?
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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Because this point sits on this circle centered at C. And I'm going to draw another line. So this other line, let me put that there and this there. So my claim is that this is a right triangle. How do I make this claim? And I've proved it in other videos. Is it's embedded in a circle. It's embedded in this circle right over here that's highlighted in yellow.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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How do I make this claim? And I've proved it in other videos. Is it's embedded in a circle. It's embedded in this circle right over here that's highlighted in yellow. It has a diameter. And it has a diameter for one of its sides. And that side is actually its hypotenuse.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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It's embedded in this circle right over here that's highlighted in yellow. It has a diameter. And it has a diameter for one of its sides. And that side is actually its hypotenuse. And we prove it in other videos. So this is a right triangle. Well, why is this useful for proving, or why is this useful for constructing a tangent line to this circle over here, to circle centered at C?
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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And that side is actually its hypotenuse. And we prove it in other videos. So this is a right triangle. Well, why is this useful for proving, or why is this useful for constructing a tangent line to this circle over here, to circle centered at C? Well, this side right over here that I have in orange, that I'm highlighting right now, that's a radius of C. And if that forms a right angle with this line right over here, then this line right over here must be tangent. To really make it look tangent, I'll elongate it a little bit just like that. And there you go.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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Well, why is this useful for proving, or why is this useful for constructing a tangent line to this circle over here, to circle centered at C? Well, this side right over here that I have in orange, that I'm highlighting right now, that's a radius of C. And if that forms a right angle with this line right over here, then this line right over here must be tangent. To really make it look tangent, I'll elongate it a little bit just like that. And there you go. You should feel good that this looks like it's truly intersecting at a right angle, and that this segment that I'm highlighting in orange right now is truly tangent. So once again, a lot more trouble than just eyeballing it and trying to draw it. And actually, a lot of you all probably could have eyeballed this segment right over here.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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And there you go. You should feel good that this looks like it's truly intersecting at a right angle, and that this segment that I'm highlighting in orange right now is truly tangent. So once again, a lot more trouble than just eyeballing it and trying to draw it. And actually, a lot of you all probably could have eyeballed this segment right over here. But if you're doing something on a larger scale, you want to be more precise, it's useful to be able to do these constructions. And frankly, in the process here, while you're dealing with the compass and the ruler or the straight edge, you get a new appreciation for what these tools can do. And you're also making some pretty fun patterns and designs.
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Another example using compass and straightedge for tangent line Geometry Khan Academy.mp3
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Fair enough. So based on that, they're going to ask us some questions, and I encourage you to pause this video and see if you can figure out the answers to these questions on your own before I work through them. So the first question they say is, well, what's A prime, C prime? This is really, what's the length of segment A prime, C prime? So they want the length of this right over here. How do we figure that out? Well, the key realization here is a reflection is a rigid transformation, rigid transformation, which is a very fancy word, but it's really just saying that it's a transformation where the length between corresponding points don't change.
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Finding measures using rigid transformations.mp3
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This is really, what's the length of segment A prime, C prime? So they want the length of this right over here. How do we figure that out? Well, the key realization here is a reflection is a rigid transformation, rigid transformation, which is a very fancy word, but it's really just saying that it's a transformation where the length between corresponding points don't change. So if we're talking about a shape like a triangle, the angle measures won't change, the perimeter won't change, and the area won't change. So we're gonna use the fact that the length between corresponding points won't change, so the length between A prime and C prime is gonna be the same as the length between A and C. So A prime, C prime is going to be equal to AC, which is equal to, they tell us, right over there. That's this corresponding side of the triangle.
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Finding measures using rigid transformations.mp3
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Well, the key realization here is a reflection is a rigid transformation, rigid transformation, which is a very fancy word, but it's really just saying that it's a transformation where the length between corresponding points don't change. So if we're talking about a shape like a triangle, the angle measures won't change, the perimeter won't change, and the area won't change. So we're gonna use the fact that the length between corresponding points won't change, so the length between A prime and C prime is gonna be the same as the length between A and C. So A prime, C prime is going to be equal to AC, which is equal to, they tell us, right over there. That's this corresponding side of the triangle. That has a length of three. So we answered the first question, and maybe that gave you a good clue, and so I encourage you to keep pausing the video when you feel like you can have a go at it. All right, the next question is, what is the measure of angle B prime?
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Finding measures using rigid transformations.mp3
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That's this corresponding side of the triangle. That has a length of three. So we answered the first question, and maybe that gave you a good clue, and so I encourage you to keep pausing the video when you feel like you can have a go at it. All right, the next question is, what is the measure of angle B prime? So that's this angle right over here, and we're gonna use the exact same property. Angle B prime corresponds to angle B. It underwent a rigid transformation of a reflection.
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Finding measures using rigid transformations.mp3
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All right, the next question is, what is the measure of angle B prime? So that's this angle right over here, and we're gonna use the exact same property. Angle B prime corresponds to angle B. It underwent a rigid transformation of a reflection. This would also be true if we had a translation or if we had a rotation, and so right over here, the measure of angle B prime would be the same as the measure of angle B, but what is that going to be equal to? Well, we can use the fact that if we call that measure, let's just call that X, X plus 53 degrees, we'll do it all in degrees, plus 90 degrees, this right angle here, well, the sum of the interior angles of a triangle add up to 180 degrees, and so what do we have? We could subtract, let's see, 53 plus 90 is X, plus 143 degrees is equal to 180 degrees, and so subtract 143 degrees from both sides, you get X is equal to, let's see, 180, 80 minus 40 would be 40, 80 minus 43 would be 37 degrees.
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Finding measures using rigid transformations.mp3
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It underwent a rigid transformation of a reflection. This would also be true if we had a translation or if we had a rotation, and so right over here, the measure of angle B prime would be the same as the measure of angle B, but what is that going to be equal to? Well, we can use the fact that if we call that measure, let's just call that X, X plus 53 degrees, we'll do it all in degrees, plus 90 degrees, this right angle here, well, the sum of the interior angles of a triangle add up to 180 degrees, and so what do we have? We could subtract, let's see, 53 plus 90 is X, plus 143 degrees is equal to 180 degrees, and so subtract 143 degrees from both sides, you get X is equal to, let's see, 180, 80 minus 40 would be 40, 80 minus 43 would be 37 degrees. X is equal to 37 degrees, so that is 37 degrees. If that's 37 degrees, then this is also going to be 37 degrees. Next, they ask us, what is the area of triangle ABC, ABC?
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Finding measures using rigid transformations.mp3
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We could subtract, let's see, 53 plus 90 is X, plus 143 degrees is equal to 180 degrees, and so subtract 143 degrees from both sides, you get X is equal to, let's see, 180, 80 minus 40 would be 40, 80 minus 43 would be 37 degrees. X is equal to 37 degrees, so that is 37 degrees. If that's 37 degrees, then this is also going to be 37 degrees. Next, they ask us, what is the area of triangle ABC, ABC? Well, it's gonna have the same area as A prime, B prime, C prime, and so a couple of ways we could think about it. We could try to find the area of A prime, B prime, C prime based on the fact that we already know that this length is three and this is a right triangle, or we can use the fact that this length right over here, four, from A prime to B prime, is gonna be the same thing as this length right over here, from A prime to B prime, which is four. And so the area of this triangle, especially this is a right triangle, it's quite straightforward, it's the base times the height times 1 1⁄2.
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Finding measures using rigid transformations.mp3
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Next, they ask us, what is the area of triangle ABC, ABC? Well, it's gonna have the same area as A prime, B prime, C prime, and so a couple of ways we could think about it. We could try to find the area of A prime, B prime, C prime based on the fact that we already know that this length is three and this is a right triangle, or we can use the fact that this length right over here, four, from A prime to B prime, is gonna be the same thing as this length right over here, from A prime to B prime, which is four. And so the area of this triangle, especially this is a right triangle, it's quite straightforward, it's the base times the height times 1 1⁄2. So this area is gonna be 1 1⁄2 times the base, four, times the height, three, which is equal to 1⁄2, which is equal to six square units. And then last but not least, what's the perimeter of triangle A prime, B prime, C prime? Well, here we just use the Pythagorean theorem to figure out the length of this hypotenuse, and we know that this is a length of three based on the whole rigid transformation and lengths are preserved.
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Finding measures using rigid transformations.mp3
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And so the area of this triangle, especially this is a right triangle, it's quite straightforward, it's the base times the height times 1 1⁄2. So this area is gonna be 1 1⁄2 times the base, four, times the height, three, which is equal to 1⁄2, which is equal to six square units. And then last but not least, what's the perimeter of triangle A prime, B prime, C prime? Well, here we just use the Pythagorean theorem to figure out the length of this hypotenuse, and we know that this is a length of three based on the whole rigid transformation and lengths are preserved. And so you might immediately recognize that if you have a right triangle where one side is three and another side is four, that the hypotenuse is five, three, four, five triangles, or you could just use the Pythagorean theorem. You say three squared plus four squared, four squared is equal to, let's just say, the hypotenuse, the hypotenuse squared. Well, three squared plus four squared, that's nine plus 16.
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Finding measures using rigid transformations.mp3
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Well, here we just use the Pythagorean theorem to figure out the length of this hypotenuse, and we know that this is a length of three based on the whole rigid transformation and lengths are preserved. And so you might immediately recognize that if you have a right triangle where one side is three and another side is four, that the hypotenuse is five, three, four, five triangles, or you could just use the Pythagorean theorem. You say three squared plus four squared, four squared is equal to, let's just say, the hypotenuse, the hypotenuse squared. Well, three squared plus four squared, that's nine plus 16. 25 is equal to the hypotenuse squared, and so the hypotenuse right over here will be equal to five. And so they're not asking us the length of the hypotenuse, they wanna know the perimeter. So it's gonna be four plus three plus five, which is equal to 12, the perimeter of either of those triangles, because it's just one's the image of the other under a rigid transformation, they're gonna have the same perimeter, the same area.
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Finding measures using rigid transformations.mp3
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Pause this video and see if you can figure that out on your own. All right, now let's work through this together. So let's see what we can figure out. We see that segment DC is parallel to segment AB, that's what these little arrows tell us. And so you can view this segment AC as something of a transversal across those parallel lines. And we know that alternate interior angles would be congruent. So we know, for example, that the measure of this angle is the same as the measure of this angle, or that those angles are congruent.
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Proving triangle congruence Congruence High school geometry Khan Academy.mp3
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We see that segment DC is parallel to segment AB, that's what these little arrows tell us. And so you can view this segment AC as something of a transversal across those parallel lines. And we know that alternate interior angles would be congruent. So we know, for example, that the measure of this angle is the same as the measure of this angle, or that those angles are congruent. We also know that both of these triangles, both triangle DCA and triangle BAC, they share this side, which by reflexivity is going to be congruent to itself. So in both triangles, we have an angle and a side that are congruent, but can we figure out anything else? Well, you might be tempted to make a similar argument thinking that this is parallel to that because it looks parallel, but you can't make that assumption just based on how it looks.
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Proving triangle congruence Congruence High school geometry Khan Academy.mp3
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So we know, for example, that the measure of this angle is the same as the measure of this angle, or that those angles are congruent. We also know that both of these triangles, both triangle DCA and triangle BAC, they share this side, which by reflexivity is going to be congruent to itself. So in both triangles, we have an angle and a side that are congruent, but can we figure out anything else? Well, you might be tempted to make a similar argument thinking that this is parallel to that because it looks parallel, but you can't make that assumption just based on how it looks. If you did know that, then you would be able to make some other assumptions about some other angles here and maybe prove congruency. But it turns out given the information that we have, we can't just assume that because something looks parallel, or because something looks congruent that they are. And so based on just the information given, we actually can't prove congruency.
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Proving triangle congruence Congruence High school geometry Khan Academy.mp3
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Well, you might be tempted to make a similar argument thinking that this is parallel to that because it looks parallel, but you can't make that assumption just based on how it looks. If you did know that, then you would be able to make some other assumptions about some other angles here and maybe prove congruency. But it turns out given the information that we have, we can't just assume that because something looks parallel, or because something looks congruent that they are. And so based on just the information given, we actually can't prove congruency. Now, let me ask you a slightly different question. Let's say that we did give you a little bit more information. Let's say we told you that the measure of this angle right over here is 31 degrees, and the measure of this angle right over here is 31 degrees.
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Proving triangle congruence Congruence High school geometry Khan Academy.mp3
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