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It looks pretty close. But let's actually verify it. The distance between point C and point D, so the distance squared is going to be equal to the change in x's. So we could say 4 minus, so we're trying the distance between C and B. It's 4 minus 3 squared. Plus negative 2 minus 1 squared, which is equal to 1 squared plus negative 3 squared. And so our distance squared is equal to 10, or our distance is equal to the square root of 10. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
So we could say 4 minus, so we're trying the distance between C and B. It's 4 minus 3 squared. Plus negative 2 minus 1 squared, which is equal to 1 squared plus negative 3 squared. And so our distance squared is equal to 10, or our distance is equal to the square root of 10. So this is also, the distance right over here is the square root of 10. So this is on the circle. If we wanted to draw circle B, it would look something like this. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
And so our distance squared is equal to 10, or our distance is equal to the square root of 10. So this is also, the distance right over here is the square root of 10. So this is on the circle. If we wanted to draw circle B, it would look something like this. And once again, I'm hand-drawing it, so it's not perfect. But it would look something, I'm going to draw part of it, looks something like this. This is exactly a radius away. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
If we wanted to draw circle B, it would look something like this. And once again, I'm hand-drawing it, so it's not perfect. But it would look something, I'm going to draw part of it, looks something like this. This is exactly a radius away. So let me write, this is on circle B. Now let's look at this point, the point 5, 3. So I'll do that in pink. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
This is exactly a radius away. So let me write, this is on circle B. Now let's look at this point, the point 5, 3. So I'll do that in pink. So 1, 2, 3, 4, 5, 3. So this looks close, but let's verify just in case. So now our distance is equal to, let me just write it this way. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
So I'll do that in pink. So 1, 2, 3, 4, 5, 3. So this looks close, but let's verify just in case. So now our distance is equal to, let me just write it this way. Our distance squared is going to be our change in x squared. So 5 minus 3 squared plus 3 minus 1 squared. Change in y. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
So now our distance is equal to, let me just write it this way. Our distance squared is going to be our change in x squared. So 5 minus 3 squared plus 3 minus 1 squared. Change in y. 3 minus 1 squared. And so our distance is going to be equal to, actually I don't want to skip too many steps. You see this is 2 squared, which is 4, plus 2 squared, which is another 4. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
Change in y. 3 minus 1 squared. And so our distance is going to be equal to, actually I don't want to skip too many steps. You see this is 2 squared, which is 4, plus 2 squared, which is another 4. So our distance is going to be equal to the square root of 8, which is the same thing as the square root of 2 times 4, which is the same thing as 2 times the square root of 2. Square root of 4 is 2, and then of course you just have the 2 left in the radical. So this is a different distance away than square root of 10. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
You see this is 2 squared, which is 4, plus 2 squared, which is another 4. So our distance is going to be equal to the square root of 8, which is the same thing as the square root of 2 times 4, which is the same thing as 2 times the square root of 2. Square root of 4 is 2, and then of course you just have the 2 left in the radical. So this is a different distance away than square root of 10. So this one right over here is definitely not on circle B. And just eyeballing it, you can see that it's not going to be on circle A. This distance, just eyeballing it, is much further than 5 square roots of 2. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
So this is a different distance away than square root of 10. So this one right over here is definitely not on circle B. And just eyeballing it, you can see that it's not going to be on circle A. This distance, just eyeballing it, is much further than 5 square roots of 2. And that's also true for point C. Point C is much further than 5 square roots of 2. You can just look at that visually. They're much further than a radius away from A. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
This distance, just eyeballing it, is much further than 5 square roots of 2. And that's also true for point C. Point C is much further than 5 square roots of 2. You can just look at that visually. They're much further than a radius away from A. So this point right over here, this is neither. This is on neither circle. Now finally, we have the point negative 2 comma 8. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
They're much further than a radius away from A. So this point right over here, this is neither. This is on neither circle. Now finally, we have the point negative 2 comma 8. So let me find, I'm running out of colors. Let me see, I could use yellow again. Negative 2 comma 8. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
Now finally, we have the point negative 2 comma 8. So let me find, I'm running out of colors. Let me see, I could use yellow again. Negative 2 comma 8. So that's negative 2 comma 1, 2, 3, 4, 5, 6, 7, 8. So it's right over here. That is point E. Just eyeballing it, this distance, so it's clearly way too far. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
Negative 2 comma 8. So that's negative 2 comma 1, 2, 3, 4, 5, 6, 7, 8. So it's right over here. That is point E. Just eyeballing it, this distance, so it's clearly way too far. Just looking at it, just eyeballing it, it's clearly more than a radius away from B. So this isn't going to be on circle B. And also looking at it relative to point A, it looks much closer to point A. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
That is point E. Just eyeballing it, this distance, so it's clearly way too far. Just looking at it, just eyeballing it, it's clearly more than a radius away from B. So this isn't going to be on circle B. And also looking at it relative to point A, it looks much closer to point A. It doesn't even seem close than point P is. So it looks, just inspecting it, you could rule this one out that this is going to be neither. But we can verify this on our own if we like. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
And also looking at it relative to point A, it looks much closer to point A. It doesn't even seem close than point P is. So it looks, just inspecting it, you could rule this one out that this is going to be neither. But we can verify this on our own if we like. We can just find the distance between these two points. Our distance squared is going to be our change in x's. So negative 2 minus negative 5 squared plus our change in y. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
But we can verify this on our own if we like. We can just find the distance between these two points. Our distance squared is going to be our change in x's. So negative 2 minus negative 5 squared plus our change in y. So it's 8 minus 5 squared. And so this is our distance squared is going to be equal to negative 2 minus negative 5. That's negative 2 plus 5. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
So negative 2 minus negative 5 squared plus our change in y. So it's 8 minus 5 squared. And so this is our distance squared is going to be equal to negative 2 minus negative 5. That's negative 2 plus 5. So that's going to be 3 squared plus 3 squared. And you see that right over here, Pythagorean theorem. This distance right over here is 3. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
That's negative 2 plus 5. So that's going to be 3 squared plus 3 squared. And you see that right over here, Pythagorean theorem. This distance right over here is 3. This distance right over here, this is your change in x, is 3. Change in y is 3. 3 squared plus 3 squared is going to be the distance squared, the hypotenuse squared. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
This distance right over here is 3. This distance right over here, this is your change in x, is 3. Change in y is 3. 3 squared plus 3 squared is going to be the distance squared, the hypotenuse squared. So our distance squared is going to be, or I could say our distance, skip a few steps, is equal to the square root of, we could write this as 9 times 2, or the distance is equal to 3 times the square root of 2. The radius of circle A is 5 times the square root of 2, not 3 times the square root of 2. So this is actually going to be inside the circle. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
3 squared plus 3 squared is going to be the distance squared, the hypotenuse squared. So our distance squared is going to be, or I could say our distance, skip a few steps, is equal to the square root of, we could write this as 9 times 2, or the distance is equal to 3 times the square root of 2. The radius of circle A is 5 times the square root of 2, not 3 times the square root of 2. So this is actually going to be inside the circle. So if we want to draw circle A, it's going to look something like this. And point E is on the inside. Point D and point C are on the outside of circle A. | Recognizing points on a circle Analytic geometry Geometry Khan Academy.mp3 |
What they want us to prove is that their diagonals are perpendicular, that AC is perpendicular to BD. Now let's think about everything we know about a rhombus. First of all, a rhombus is a special case of a parallelogram. In a parallelogram, the opposite sides are parallel, so that side is parallel to that side, these two sides are parallel. And in a rhombus, not only are the opposite sides parallel, it's a parallelogram, but also all of the sides have equal length. So this side is equal to this side, which is equal to that side, which is equal to that side right over there. Now there's other interesting things we know about the diagonals of a parallelogram, which we know all rhombi are parallelograms. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
In a parallelogram, the opposite sides are parallel, so that side is parallel to that side, these two sides are parallel. And in a rhombus, not only are the opposite sides parallel, it's a parallelogram, but also all of the sides have equal length. So this side is equal to this side, which is equal to that side, which is equal to that side right over there. Now there's other interesting things we know about the diagonals of a parallelogram, which we know all rhombi are parallelograms. The other way around is not necessarily true. We know that for any parallelogram, and a rhombus is a parallelogram, that the diagonals bisect each other. So for example, let me label this point in the center, let me label it point E. We know that AE is going to be equal to EC, I'll put two slashes right over there, and we also know that EB is going to be equal to ED. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
Now there's other interesting things we know about the diagonals of a parallelogram, which we know all rhombi are parallelograms. The other way around is not necessarily true. We know that for any parallelogram, and a rhombus is a parallelogram, that the diagonals bisect each other. So for example, let me label this point in the center, let me label it point E. We know that AE is going to be equal to EC, I'll put two slashes right over there, and we also know that EB is going to be equal to ED. So this is all of what we know when someone just says that ABCD is a rhombus, based on other things that we've proven to ourselves. Now we need to prove that AC is perpendicular to BD. So an interesting way to prove it, and you can kind of look at it just by eyeballing, is if we can show that this triangle is congruent to this triangle, and that these two angles right over here correspond to each other, then they have to be the same, and they'll be supplementary, and then they'll be 90 degrees. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
So for example, let me label this point in the center, let me label it point E. We know that AE is going to be equal to EC, I'll put two slashes right over there, and we also know that EB is going to be equal to ED. So this is all of what we know when someone just says that ABCD is a rhombus, based on other things that we've proven to ourselves. Now we need to prove that AC is perpendicular to BD. So an interesting way to prove it, and you can kind of look at it just by eyeballing, is if we can show that this triangle is congruent to this triangle, and that these two angles right over here correspond to each other, then they have to be the same, and they'll be supplementary, and then they'll be 90 degrees. So let's just prove it to ourselves. So the first thing we see is we have a side, a side, and a side. So we can see that triangle, let me write it here, triangle, let me just add a new color, we see that triangle ABE is congruent to triangle CBE, and we know that by side, side, side congruency. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
So an interesting way to prove it, and you can kind of look at it just by eyeballing, is if we can show that this triangle is congruent to this triangle, and that these two angles right over here correspond to each other, then they have to be the same, and they'll be supplementary, and then they'll be 90 degrees. So let's just prove it to ourselves. So the first thing we see is we have a side, a side, and a side. So we can see that triangle, let me write it here, triangle, let me just add a new color, we see that triangle ABE is congruent to triangle CBE, and we know that by side, side, side congruency. We have a side, a side, and a side, a side, a side, and a side. And then once we know that, we know that all the corresponding angles are congruent, and in particular we know that AEB, we know that angle AEB is going to be congruent to angle, so AEB to CEB, to angle CEB, because they are corresponding angles of congruent, corresponding angles of congruent triangles. So this angle right over here is going to be equal to that angle over there. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
So we can see that triangle, let me write it here, triangle, let me just add a new color, we see that triangle ABE is congruent to triangle CBE, and we know that by side, side, side congruency. We have a side, a side, and a side, a side, a side, and a side. And then once we know that, we know that all the corresponding angles are congruent, and in particular we know that AEB, we know that angle AEB is going to be congruent to angle, so AEB to CEB, to angle CEB, because they are corresponding angles of congruent, corresponding angles of congruent triangles. So this angle right over here is going to be equal to that angle over there. And we also know that they are supplementary, and so they're both supplementary, so we also know, and let me write it this way, they're congruent and they are supplementary. So we have these two are going to have the same measure, and they need to add up to 180 degrees. So if I have two things that are the same thing, and they add up to 180 degrees, what does that tell me? | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
So this angle right over here is going to be equal to that angle over there. And we also know that they are supplementary, and so they're both supplementary, so we also know, and let me write it this way, they're congruent and they are supplementary. So we have these two are going to have the same measure, and they need to add up to 180 degrees. So if I have two things that are the same thing, and they add up to 180 degrees, what does that tell me? So that tells me that angle, the measure of angle AEB is equal to the measure of angle CEB, which is equal to, which must be equal to 90 degrees. They're the same measure and they are supplementary. So this is a right angle, and then this is a right angle. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
So if I have two things that are the same thing, and they add up to 180 degrees, what does that tell me? So that tells me that angle, the measure of angle AEB is equal to the measure of angle CEB, which is equal to, which must be equal to 90 degrees. They're the same measure and they are supplementary. So this is a right angle, and then this is a right angle. And obviously if this is a right angle, this angle down here is a vertical angle, that's going to be a right angle. If this is a right angle, this over here is going to be a vertical angle. And you see the diagonals intersect at a 90 degree angle. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
So this is a right angle, and then this is a right angle. And obviously if this is a right angle, this angle down here is a vertical angle, that's going to be a right angle. If this is a right angle, this over here is going to be a vertical angle. And you see the diagonals intersect at a 90 degree angle. So we've just proved, so this is interesting. In a parallelogram, the diagonals bisect each other. For a rhombus, where all the sides are equal, we've shown that not only do they bisect each other, but they're perpendicular bisectors of each other. | Proof Rhombus diagonals are perpendicular bisectors Quadrilaterals Geometry Khan Academy.mp3 |
So let's think about what they're asking. So if that's point C, I'm just going to redraw this line segment just to conceptualize what they're asking for. And that's point A. They're asking us to find some point B that the distance between C and B, so that's this distance right over here. So if this distance is x, then the distance between B and A is going to be 3 times that. It's going to be 3 times that. So this will be 3x. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
They're asking us to find some point B that the distance between C and B, so that's this distance right over here. So if this distance is x, then the distance between B and A is going to be 3 times that. It's going to be 3 times that. So this will be 3x. That the ratio of AB to BC is 3 to 1. So that would be the ratio. Let me write this down. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So this will be 3x. That the ratio of AB to BC is 3 to 1. So that would be the ratio. Let me write this down. It would be AB, that looks like an HB. It would be AB to BC is going to be equal to 3x to x, which is the same thing as 3 to 1. Which is the same thing as 3 to 1, if we wanted to write it a slightly different way. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
Let me write this down. It would be AB, that looks like an HB. It would be AB to BC is going to be equal to 3x to x, which is the same thing as 3 to 1. Which is the same thing as 3 to 1, if we wanted to write it a slightly different way. So how can we think about it? You might be tempted to say, oh, well, you could use the distance formula, find the distance, which by itself isn't completely uncomplicated. And then this will be one-fourth of the way. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
Which is the same thing as 3 to 1, if we wanted to write it a slightly different way. So how can we think about it? You might be tempted to say, oh, well, you could use the distance formula, find the distance, which by itself isn't completely uncomplicated. And then this will be one-fourth of the way. Because if you think about it, this entire distance is going to be 4x. This entire distance is going to be, let me draw that a little bit neater. This entire distance, if you have an x plus a 3x, is going to be 4x. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
And then this will be one-fourth of the way. Because if you think about it, this entire distance is going to be 4x. This entire distance is going to be, let me draw that a little bit neater. This entire distance, if you have an x plus a 3x, is going to be 4x. So you'd say, well, this is one out of the 4x's along the way. This is going to be one-fourth of the distance between the two points. So this is, let me write that down. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
This entire distance, if you have an x plus a 3x, is going to be 4x. So you'd say, well, this is one out of the 4x's along the way. This is going to be one-fourth of the distance between the two points. So this is, let me write that down. This is one-fourth of the way. One-fourth of the way between C and B. Going from C, sorry, going from C to A. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So this is, let me write that down. This is one-fourth of the way. One-fourth of the way between C and B. Going from C, sorry, going from C to A. B is going to be one-fourth of the way. So maybe you try to find the distance. And you say, oh, well, what are all the points that are one-fourth away? | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
Going from C, sorry, going from C to A. B is going to be one-fourth of the way. So maybe you try to find the distance. And you say, oh, well, what are all the points that are one-fourth away? But it has to be one-fourth of that distance away. But then it has to be on that line. But that makes it complicated. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
And you say, oh, well, what are all the points that are one-fourth away? But it has to be one-fourth of that distance away. But then it has to be on that line. But that makes it complicated. Because this line is at an incline. It's not just horizontal. It's not just vertical. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
But that makes it complicated. Because this line is at an incline. It's not just horizontal. It's not just vertical. What we can do, however, is break this problem down into the vertical change between A and C and the horizontal change between A and C. So for example, the horizontal change between A and C, A is at 9 right over here. And C is at negative 7. So this distance right over here is 9 minus negative 7, which is equal to 9 plus 7, which is equal to 16. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
It's not just vertical. What we can do, however, is break this problem down into the vertical change between A and C and the horizontal change between A and C. So for example, the horizontal change between A and C, A is at 9 right over here. And C is at negative 7. So this distance right over here is 9 minus negative 7, which is equal to 9 plus 7, which is equal to 16. And you see that here. 9 plus 7, this total distance is 16. That's the horizontal change going from A to C or going from C to A. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So this distance right over here is 9 minus negative 7, which is equal to 9 plus 7, which is equal to 16. And you see that here. 9 plus 7, this total distance is 16. That's the horizontal change going from A to C or going from C to A. And the vertical change, and you could even just count that, that's going to be 4. C is at 1. A is at 5. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
That's the horizontal change going from A to C or going from C to A. And the vertical change, and you could even just count that, that's going to be 4. C is at 1. A is at 5. Going from 1 to 5, you've changed vertically 4. So what we can say, going from C to B in each direction, in the vertical direction and the horizontal direction, we're going to go one-fourth of the way. So if we go one-fourth in the vertical direction, we're going to end up at y is equal to 2. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
A is at 5. Going from 1 to 5, you've changed vertically 4. So what we can say, going from C to B in each direction, in the vertical direction and the horizontal direction, we're going to go one-fourth of the way. So if we go one-fourth in the vertical direction, we're going to end up at y is equal to 2. So I'm just going, starting at C, one-fourth of the way. One-fourth of 4 is 1. So I've just moved up 1. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So if we go one-fourth in the vertical direction, we're going to end up at y is equal to 2. So I'm just going, starting at C, one-fourth of the way. One-fourth of 4 is 1. So I've just moved up 1. So our y is going to be equal to 2. And if we go one-fourth in the horizontal direction, one-fourth of 16 is 4. So we go 1, 2, 3, 4. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So I've just moved up 1. So our y is going to be equal to 2. And if we go one-fourth in the horizontal direction, one-fourth of 16 is 4. So we go 1, 2, 3, 4. So we end up right over here. Our x is negative 3. So we end up at that point right over there. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So we go 1, 2, 3, 4. So we end up right over here. Our x is negative 3. So we end up at that point right over there. We end up at this point. This is the point negative 3, 2. And if you were really careful with your drawing, you could have actually just drawn. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So we end up at that point right over there. We end up at this point. This is the point negative 3, 2. And if you were really careful with your drawing, you could have actually just drawn. Actually, you don't have to be that careful, since this is graph paper. You actually could have just said, hey, we're going to go one-fourth this way. And where does that intersect the line? | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
And if you were really careful with your drawing, you could have actually just drawn. Actually, you don't have to be that careful, since this is graph paper. You actually could have just said, hey, we're going to go one-fourth this way. And where does that intersect the line? Hey, it intersects the line right over there. Or you could have said, we're going to go one-fourth this way. Where does that intersect the line? | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
And where does that intersect the line? Hey, it intersects the line right over there. Or you could have said, we're going to go one-fourth this way. Where does that intersect the line? And that would have let you figure it out either way. So this point right over here is B. It is one-fourth of the way between C and A. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
Where does that intersect the line? And that would have let you figure it out either way. So this point right over here is B. It is one-fourth of the way between C and A. Or another way of thinking about it, the distance between C and B, which we haven't even figured out, we could do that using the distance formula or the Pythagorean theorem, which it really is. This distance, the distance CB, is one-third the distance BA. Or BA, the ratio of AB to BC is 3 to 1. | Finding a point part way between two points Analytic geometry Geometry Khan Academy.mp3 |
So I'm gonna use the translation tool. So I'm gonna start with a translation. So it says translate by, and this is gonna say how much do we translate the X coordinates and how much do we translate the Y coordinates. Let's see, if I wanna map, if I wanna get point W to correspond to this point right over here, which it seems like it should, I would have to go from X equals two to X equals negative five. So my X would have to decrease by seven. So let me type that in, negative seven. And then, let's see, on the Y side, and we saw that. | Formal translation tool example Transformations Geometry Khan Academy.mp3 |
Let's see, if I wanna map, if I wanna get point W to correspond to this point right over here, which it seems like it should, I would have to go from X equals two to X equals negative five. So my X would have to decrease by seven. So let me type that in, negative seven. And then, let's see, on the Y side, and we saw that. So by just typing in negative seven here, we've moved it to the left by seven. And now in the Y, in the Y axis, I need to move it down by, let's see, one, two, three, four. So my Y coordinates, I need to move it down by four. | Formal translation tool example Transformations Geometry Khan Academy.mp3 |
And then, let's see, on the Y side, and we saw that. So by just typing in negative seven here, we've moved it to the left by seven. And now in the Y, in the Y axis, I need to move it down by, let's see, one, two, three, four. So my Y coordinates, I need to move it down by four. And let's see what happens. And it looks like I was able to successfully translate it. By translating X by negative seven, every point here, every point on this has been shifted to the left by seven and has been shifted down by four, and I was able to get onto this triangle. | Formal translation tool example Transformations Geometry Khan Academy.mp3 |
The laws of nature are but the mathematical thoughts of God. And this is a quote by Euclid of Alexandria, who was a Greek mathematician and philosopher who lived about 300 years before Christ. And the reason why I include this quote is because Euclid is considered to be the father, the father of geometry. And it is a neat quote. Regardless of your views of God, whether or not God exists or the nature of God, it says something very fundamental about nature. The laws of nature are but the mathematical thoughts of God, that math underpins all of the laws of nature. And the word geometry itself has Greek roots. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And it is a neat quote. Regardless of your views of God, whether or not God exists or the nature of God, it says something very fundamental about nature. The laws of nature are but the mathematical thoughts of God, that math underpins all of the laws of nature. And the word geometry itself has Greek roots. Geo comes from Greek for earth. Metry comes from Greek for measurement. You're probably used to something like the metric system. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And the word geometry itself has Greek roots. Geo comes from Greek for earth. Metry comes from Greek for measurement. You're probably used to something like the metric system. And Euclid is considered to be the father of geometry, not because he was the first person who studied geometry. You could imagine the very first humans might have studied geometry. They might have looked at two twigs on the ground that looked something like that. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
You're probably used to something like the metric system. And Euclid is considered to be the father of geometry, not because he was the first person who studied geometry. You could imagine the very first humans might have studied geometry. They might have looked at two twigs on the ground that looked something like that. And they might have looked at another pair of twigs that looked like that and said, this is a bigger opening. What is the relationship here? Or they might have looked at a tree that had a branch that came off like that. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
They might have looked at two twigs on the ground that looked something like that. And they might have looked at another pair of twigs that looked like that and said, this is a bigger opening. What is the relationship here? Or they might have looked at a tree that had a branch that came off like that. And they said, oh, there's something similar about this opening here and this opening here. Or they might have asked themselves, what is the ratio? Or what is the relationship between the distance around a circle and the distance across it? | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
Or they might have looked at a tree that had a branch that came off like that. And they said, oh, there's something similar about this opening here and this opening here. Or they might have asked themselves, what is the ratio? Or what is the relationship between the distance around a circle and the distance across it? And is that the same for all circles? And is there a way for us to feel really good that that is definitely true? And then once you got to the early Greeks, they started to get even more thoughtful, essentially, about geometric things when you talk about Greek mathematicians like Pythagoras, who came before Euclid. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
Or what is the relationship between the distance around a circle and the distance across it? And is that the same for all circles? And is there a way for us to feel really good that that is definitely true? And then once you got to the early Greeks, they started to get even more thoughtful, essentially, about geometric things when you talk about Greek mathematicians like Pythagoras, who came before Euclid. But the reason why Euclid is considered to be the father of geometry and why we often talk about Euclidean geometry is around 300 BC. And this right over here is a picture of Euclid painted by Raphael. And no one really knows what Euclid looked like even when he was born or when he died. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And then once you got to the early Greeks, they started to get even more thoughtful, essentially, about geometric things when you talk about Greek mathematicians like Pythagoras, who came before Euclid. But the reason why Euclid is considered to be the father of geometry and why we often talk about Euclidean geometry is around 300 BC. And this right over here is a picture of Euclid painted by Raphael. And no one really knows what Euclid looked like even when he was born or when he died. So this is just Raphael's impression of what Euclid might have looked like while he was teaching in Alexandria. But what made Euclid the father of geometry is really his writing of Euclid's elements. And what the elements were were essentially a 13-volume textbook, and arguably the most famous textbook of all time. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And no one really knows what Euclid looked like even when he was born or when he died. So this is just Raphael's impression of what Euclid might have looked like while he was teaching in Alexandria. But what made Euclid the father of geometry is really his writing of Euclid's elements. And what the elements were were essentially a 13-volume textbook, and arguably the most famous textbook of all time. And what he did in those 13 volumes is he essentially did a rigorous, thoughtful, logical march through geometry and number theory, and then also solid geometry, so geometry in three dimensions. And this right over here is the frontispiece for the English version of, or the first translation of the English version of Euclid's elements. And this was done in 1570. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And what the elements were were essentially a 13-volume textbook, and arguably the most famous textbook of all time. And what he did in those 13 volumes is he essentially did a rigorous, thoughtful, logical march through geometry and number theory, and then also solid geometry, so geometry in three dimensions. And this right over here is the frontispiece for the English version of, or the first translation of the English version of Euclid's elements. And this was done in 1570. But it was obviously first written in Greek. And then during much of the Middle Ages, that knowledge was shepherded by the Arabs. And it was translated into Arabic. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And this was done in 1570. But it was obviously first written in Greek. And then during much of the Middle Ages, that knowledge was shepherded by the Arabs. And it was translated into Arabic. And then eventually in the late Middle Ages, translated into Latin, and then obviously eventually English. And when I say that he did a rigorous march, what Euclid did is he didn't just say, well, I think if you take the length of one side of a right triangle and the length of the other side of the right triangle, it's going to be the same as the square of the hypotenuse, all of these other things. And we'll go into depth about what all of these things are meant. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And it was translated into Arabic. And then eventually in the late Middle Ages, translated into Latin, and then obviously eventually English. And when I say that he did a rigorous march, what Euclid did is he didn't just say, well, I think if you take the length of one side of a right triangle and the length of the other side of the right triangle, it's going to be the same as the square of the hypotenuse, all of these other things. And we'll go into depth about what all of these things are meant. He says, I don't want to just feel good that it's probably true. I want to prove to myself that it is true. And so what he did in Elements, especially the six books that are concerned with planar geometry, in fact, he did all of them, but from a geometrical point of view, he started with basic assumptions. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And we'll go into depth about what all of these things are meant. He says, I don't want to just feel good that it's probably true. I want to prove to myself that it is true. And so what he did in Elements, especially the six books that are concerned with planar geometry, in fact, he did all of them, but from a geometrical point of view, he started with basic assumptions. So he started with basic assumptions. And those basic assumptions in geometric speak are called axioms or postulates. And from them, he proved, he deduced other statements or propositions, or these are sometimes called theorems. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And so what he did in Elements, especially the six books that are concerned with planar geometry, in fact, he did all of them, but from a geometrical point of view, he started with basic assumptions. So he started with basic assumptions. And those basic assumptions in geometric speak are called axioms or postulates. And from them, he proved, he deduced other statements or propositions, or these are sometimes called theorems. And then he says, now I know if this is true and this is true, this must be true. And he could also prove that other things cannot be true. So then he could prove that this is not going to be the true. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And from them, he proved, he deduced other statements or propositions, or these are sometimes called theorems. And then he says, now I know if this is true and this is true, this must be true. And he could also prove that other things cannot be true. So then he could prove that this is not going to be the true. He didn't just say, well, every circle I've said has this property. He says, I've now proven that this is true. And then from there, we can go and deduce other propositions or theorems. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
So then he could prove that this is not going to be the true. He didn't just say, well, every circle I've said has this property. He says, I've now proven that this is true. And then from there, we can go and deduce other propositions or theorems. And we could use some of our original axioms to do that. And what's special about that is no one had really done that before. Rigorously proven beyond a shadow of a doubt, across a whole broad sweep of knowledge. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And then from there, we can go and deduce other propositions or theorems. And we could use some of our original axioms to do that. And what's special about that is no one had really done that before. Rigorously proven beyond a shadow of a doubt, across a whole broad sweep of knowledge. So not just one proof here or there. He did it for an entire set of knowledge that we're talking about, a rigorous march through a subject so that he could build this scaffold of axioms and postulates and theorems and propositions. And theorems and propositions are essentially the same thing. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
Rigorously proven beyond a shadow of a doubt, across a whole broad sweep of knowledge. So not just one proof here or there. He did it for an entire set of knowledge that we're talking about, a rigorous march through a subject so that he could build this scaffold of axioms and postulates and theorems and propositions. And theorems and propositions are essentially the same thing. And essentially, for about 2,000 years after Euclid, so this is unbelievable shelf life for a textbook, people didn't view you as educated if you did not read and understand Euclid's elements. And Euclid's elements, the book itself, was the second most printed book in the Western world after the Bible. This is a math textbook. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
And theorems and propositions are essentially the same thing. And essentially, for about 2,000 years after Euclid, so this is unbelievable shelf life for a textbook, people didn't view you as educated if you did not read and understand Euclid's elements. And Euclid's elements, the book itself, was the second most printed book in the Western world after the Bible. This is a math textbook. It was second only to the Bible. When the first printing presses came out, they said, OK, let's print the Bible. What do we print next? | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
This is a math textbook. It was second only to the Bible. When the first printing presses came out, they said, OK, let's print the Bible. What do we print next? Let's print Euclid's elements. And to show that this is relevant into the fairly recent past, although some would, whether or not you argue that about 150, 160 years ago is a recent past, this right here is a direct quote from Abraham Lincoln, obviously one of the great American presidents. I like this picture of Abraham Lincoln. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
What do we print next? Let's print Euclid's elements. And to show that this is relevant into the fairly recent past, although some would, whether or not you argue that about 150, 160 years ago is a recent past, this right here is a direct quote from Abraham Lincoln, obviously one of the great American presidents. I like this picture of Abraham Lincoln. This is actually a photograph of Lincoln in his late 30s. But he was a huge fan of Euclid's elements. He would actually use it to fine tune his mind. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
I like this picture of Abraham Lincoln. This is actually a photograph of Lincoln in his late 30s. But he was a huge fan of Euclid's elements. He would actually use it to fine tune his mind. While he was riding his horse, he would read Euclid's element. While he was in the White House, he would read Euclid's element. But this is a direct quote from Lincoln. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
He would actually use it to fine tune his mind. While he was riding his horse, he would read Euclid's element. While he was in the White House, he would read Euclid's element. But this is a direct quote from Lincoln. In the course of my law reading, I constantly came upon the word demonstrate. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, what do I do when I demonstrate more than when I reason or prove? | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
But this is a direct quote from Lincoln. In the course of my law reading, I constantly came upon the word demonstrate. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, what do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof? So in Lincoln's thing, there's this word demonstration that kind of means something more, proving beyond doubt, something more rigorous, more than just simple kind of feeling good about something or reasoning through it. I consulted Webster's Dictionary. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
I said to myself, what do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof? So in Lincoln's thing, there's this word demonstration that kind of means something more, proving beyond doubt, something more rigorous, more than just simple kind of feeling good about something or reasoning through it. I consulted Webster's Dictionary. So Webster's Dictionary was around even around when Lincoln was around. They told of certain proof, proof beyond the possibility of doubt. But I could form no idea of what sort of proof that was. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
I consulted Webster's Dictionary. So Webster's Dictionary was around even around when Lincoln was around. They told of certain proof, proof beyond the possibility of doubt. But I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
But I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said, Lincoln, he's talking to himself, at last I said, Lincoln, you never can make a lawyer if you do not understand what demonstrate means. And I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at site, so the six books concerned with planar geometry. I then found out what demonstrate means and went back to my law studies. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
You might as well have defined blue to a blind man. At last I said, Lincoln, he's talking to himself, at last I said, Lincoln, you never can make a lawyer if you do not understand what demonstrate means. And I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at site, so the six books concerned with planar geometry. I then found out what demonstrate means and went back to my law studies. So one of the greatest American presidents of all time felt that in order to be a great lawyer, he had to understand, be able to prove any proposition in the six books of Euclid's elements at site. And also, once he was in the White House, he continued to do this to make him, in his mind, to fine tune his mind to become a great president. And so what we're going to be doing in the geometry playlist is essentially that. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
I then found out what demonstrate means and went back to my law studies. So one of the greatest American presidents of all time felt that in order to be a great lawyer, he had to understand, be able to prove any proposition in the six books of Euclid's elements at site. And also, once he was in the White House, he continued to do this to make him, in his mind, to fine tune his mind to become a great president. And so what we're going to be doing in the geometry playlist is essentially that. What we're going to study is we're going to think about how do we really tightly, rigorously prove things. We're essentially going to be, in a slightly more modern form, be studying what Euclid studied 2,300 years ago to really tighten our reasoning of different statements and being able to make sure that when we say something, we can really prove what we're saying. This is really some of the most fundamental real mathematics that you will do. | Euclid as the father of geometry Introduction to Euclidean geometry Geometry Khan Academy.mp3 |
So we have a circle here, and they specified some points for us. This little orangish, or I guess maroonish red point right over here is the center of the circle, and then this blue point is a point that happens to sit on the circle. And so with that information, I want you to pause the video and see if you can figure out the equation for this circle. All right, let's work through this together. So let's first think about the center of the circle, and the center of the circle is just going to be the coordinates of that point. So the x-coordinate is negative one, and then the y-coordinate is one. So center is negative one, comma, one. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
All right, let's work through this together. So let's first think about the center of the circle, and the center of the circle is just going to be the coordinates of that point. So the x-coordinate is negative one, and then the y-coordinate is one. So center is negative one, comma, one. And now let's think about what the radius of the circle is. Well, the radius is going to be the distance between the center and any point on the circle. So for example, this distance, the distance of that line, let's see, I can do a thicker version of that. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
So center is negative one, comma, one. And now let's think about what the radius of the circle is. Well, the radius is going to be the distance between the center and any point on the circle. So for example, this distance, the distance of that line, let's see, I can do a thicker version of that. This line right over there, something strange about my, something strange about my pen tool is making that very thin. Let me do it one more time. Okay, that's better. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
So for example, this distance, the distance of that line, let's see, I can do a thicker version of that. This line right over there, something strange about my, something strange about my pen tool is making that very thin. Let me do it one more time. Okay, that's better. The distance of that line right over there, that is going to be the radius. So how can we figure that out? Well, we can set up a right triangle and essentially use the distance formula which comes from the Pythagorean theorem. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
Okay, that's better. The distance of that line right over there, that is going to be the radius. So how can we figure that out? Well, we can set up a right triangle and essentially use the distance formula which comes from the Pythagorean theorem. To figure out the length of that line, so this is the radius, we could figure out a change in x. So if we look at our change in x right over here, our change in x as we go from the center to this point, so this is our change in x, and then we could say that this is our change in y. That right over there is our change in y. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
Well, we can set up a right triangle and essentially use the distance formula which comes from the Pythagorean theorem. To figure out the length of that line, so this is the radius, we could figure out a change in x. So if we look at our change in x right over here, our change in x as we go from the center to this point, so this is our change in x, and then we could say that this is our change in y. That right over there is our change in y. And so our change in x squared plus our change in y squared is going to be our radius squared. That comes straight out of the Pythagorean theorem. This is a right triangle. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
That right over there is our change in y. And so our change in x squared plus our change in y squared is going to be our radius squared. That comes straight out of the Pythagorean theorem. This is a right triangle. And so we can say that r squared is going to be equal to our change in x squared plus our change in y squared. Plus our change in y squared. Now what is our change in x going to be? | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
This is a right triangle. And so we can say that r squared is going to be equal to our change in x squared plus our change in y squared. Plus our change in y squared. Now what is our change in x going to be? Our change in x is going to be equal to, well when we go from the radius to this point over here, our x goes from negative one to six. So you could view it as our ending x minus our starting x. So negative one minus, sorry, six minus negative one is equal to seven. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
Now what is our change in x going to be? Our change in x is going to be equal to, well when we go from the radius to this point over here, our x goes from negative one to six. So you could view it as our ending x minus our starting x. So negative one minus, sorry, six minus negative one is equal to seven. So let me, so we have our change in x, this right over here, is equal to seven. If we viewed this as the start point and this as the end point, it would be negative seven, but we really care about the absolute value of the change in x, and once you square it, it all becomes a positive anyway. So our change in x right over here is going to be positive seven. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
So negative one minus, sorry, six minus negative one is equal to seven. So let me, so we have our change in x, this right over here, is equal to seven. If we viewed this as the start point and this as the end point, it would be negative seven, but we really care about the absolute value of the change in x, and once you square it, it all becomes a positive anyway. So our change in x right over here is going to be positive seven. And our change in y, well, we are starting at, we are starting at y is equal to one, and we are going to y is equal to negative four, so it would be negative four minus one, which is equal to negative five. And so our change in y is negative five. You could view this distance right over here as the absolute value of our change in y, which of course would be the absolute value of five. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
So our change in x right over here is going to be positive seven. And our change in y, well, we are starting at, we are starting at y is equal to one, and we are going to y is equal to negative four, so it would be negative four minus one, which is equal to negative five. And so our change in y is negative five. You could view this distance right over here as the absolute value of our change in y, which of course would be the absolute value of five. But once you square it, it doesn't matter, the negative sign goes away. And so this is going to simplify to seven squared, change in x squared is 49. Change in y squared, negative five squared is 25. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
You could view this distance right over here as the absolute value of our change in y, which of course would be the absolute value of five. But once you square it, it doesn't matter, the negative sign goes away. And so this is going to simplify to seven squared, change in x squared is 49. Change in y squared, negative five squared is 25. So we get r squared, we get r squared is equal to 49 plus 25. So what's 49 plus 25? Let's see, that's going to be 54, was it 74? | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
Change in y squared, negative five squared is 25. So we get r squared, we get r squared is equal to 49 plus 25. So what's 49 plus 25? Let's see, that's going to be 54, was it 74? R squared is equal to 74. Did I do that right? Yep, 74. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
Let's see, that's going to be 54, was it 74? R squared is equal to 74. Did I do that right? Yep, 74. And so now we can write the equation for the circle. The circle is going to be all of the points that are, well, let me write, so if r squared is equal to 74, r is equal to the square root of 74. And so the equation of the circle is going to be all points x comma y that are this far away from the center. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
Yep, 74. And so now we can write the equation for the circle. The circle is going to be all of the points that are, well, let me write, so if r squared is equal to 74, r is equal to the square root of 74. And so the equation of the circle is going to be all points x comma y that are this far away from the center. And so what are those points going to be? Well, the distance is going to be x minus the x coordinate of the center, x minus negative one squared. Let me do that in blue color. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
And so the equation of the circle is going to be all points x comma y that are this far away from the center. And so what are those points going to be? Well, the distance is going to be x minus the x coordinate of the center, x minus negative one squared. Let me do that in blue color. Minus negative one squared plus y minus, y minus the y coordinate of the center, y minus one squared, is equal, is going to be equal to r squared, is going to be equal to the length of the radius squared. Well, r squared we already know is going to be 74. 74. | Writing standard equation of a circle Mathematics II High School Math Khan Academy.mp3 |
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