Distances in the coordinate plane (between reflected points) | Khan Academy
All Khan Academy transcripts →
Summary
This video explains how to calculate the distance between two points on a coordinate plane when one of the coordinates is the same and the other has the opposite sign. It demonstrates this concept with two examples, one where the Y-coordinates have opposite signs and another where the X-coordinates have opposite signs, using both graphical and absolute value methods to find the distance.
Key points- When two points have the same X-coordinate and their Y-coordinates have opposite signs, the distance between them can be found by summing the absolute values of their Y-coordinates.
- Graphing the points can help visualize the distance, especially when the points are directly above or below each other on the coordinate plane.
- The distance between two points can be calculated using the absolute value of the difference between their differing coordinates.
- When two points have the same Y-coordinate and their X-coordinates have opposite signs, the distance between them can be found by summing the absolute values of their X-coordinates.
- Reflecting a point over the Y-axis changes the sign of its X-coordinate while keeping the Y-coordinate the same.
- Reflecting a point over the X-axis changes the sign of its Y-coordinate while keeping the X-coordinate the same.
Chapters
Notable quotes (4)
“as point P, except it's Y coordinate has the opposite sign.”
“what is the distance between points P and R?”
“but it's Y coordinate, instead of being negative six is going to be positive six.”
“six minus negative six.”
0:00- [Tutor] We're told point R has the same coordinates as point P, except it's Y coordinate has the opposite sign. Point R is at two, common negative six. What is the different, what is the distance between points P and R? So pause this video and have a go at this before we do this together. Okay, so there's two ways we can do it and they're good to check on each other.
0:22One is we can just think about, "Okay, what is going to be the coordinates of point P?" So P is going to have the coordinates, let's see, it has, it's the same coordinates except it's Y coordinate has the opposite sign. So it's X coordinate is going to be the same, but it's Y coordinate, instead of being negative six is going to be positive six.
0:41So one way to think about the distance between these two points, they're going to be right above each other, right above and below each other. This one is six units below the X-axis. This is six units above the X-axis. So the distance should be 12. But let's graph that to really make sure that we feel comfortable with this.
1:02So let me set up some axes here. Make sure I can fit it. I have to fit six above and six below. So something, something like, let me make sure I can, so something like this should do the trick. And I only have to plot one point here, so actually I don't even have to have it that long.
1:25So, and then let, make, make that my Y-axis. So that's Y. This is X. Right over here we have one, two, right over there. And then we have 1, 2, 3, 4, 5, positive six there. And then 1, 2, 3, 4, 5, negative six right over there. So point R at two comma negative six, two comma negative six is right over there.
1:55That is point R. And then point P, two comma six. Two comma six, right over there, point P. And we already said these are going to be right above and below each other and we can see that. And if we just wanted to count the grid lines, we go 1, 2, 3, 4, 5, 6, 7 8 9 10 11 12.
2:17So this distance is definitely, this distance right over here is definitely 12. Another way is we reason six below and six above. Another way is you could take the absolute value of the difference of these two numbers. So if I just took the Y values here, the absolute value of the distance, I could say six minus negative six.
2:44So that's going to be six plus six, which is equal to 12. You take the absolute value of that, that's still 12. If you went the other way around, if you did negative six, minus six, you would've gotten negative 12. Absolute value, still 12. Now you can do distance on in two dimensions on a coordinate plane like this when one of the coordinates stays the same.
3:05In future videos, you'll learn how to take distance when two, when both of the coordinates are different. But here the X coordinate was the same. And so we just have to find the distance in one dimension. Let's now do another example. It says point N has the same coordinate as point L, except now it's X coordinate has the opposite sign.
3:25So this is the, instead of the Y coordinate having the opposite sign, the X coordinate does. So in this scenario, let's see, point N. Point N has the same, so this is N right over here. So point L, point L is going to have the opposite coordinates on the X. So point L is going to be negative three, but it's gonna have the same Y coordinate, five, okay?
3:51So we can see that these are both right to left of each other. I'll graph these in a second to validate that. But really what we're doing is we're going from three to the left of the Y-axis to three to the right. So in total we need to travel six to the right to go from point N to point L.
4:08So the distance should be six. We could also do what we did in the last one. We could say, "Well, what's the absolute value of negative three minus three?" So the absolute value of negative three minus three? Well, that's the absolute value of negative six. That is also six. But let's graph this to feel, to feel confident and to visualize it all.
4:27So if this is my Y-axis. Y-axis, and this is my X-axis, you have to go up to Y equals five, and my Xs are going to go positive and negative three. So maybe this is my X-axis right over here. And I'm going to go, let me just do one unit per, 1 2 3 4 5. And then I'll go 1, 2, 3, and then negative one, negative two, negative three.
5:02Point N I will do in pink. Point N is at three five. So three comma five is right over there. That is point N. And then we figured out point L is at negative three. So, oh, we took the opposite of the X coordinate, which reflects it over the Y-axis. So this is L right over here and you can see that distance.
5:23It's 1, 2, 3, 4, 5, and six.
Works on any YouTube video.
Swap youtube.com for 2outube.com in the address bar — or paste a link.
Read another video