Distances in the coordinate plane (between reflected points)
About this video
The video finds distances between points that are reflections of each other on the coordinate plane. R(2, -6) and P(2, 6) differ only in the sign of y, so they are 12 units apart; N(3, 5) and L(-3, 5) differ only in the sign of x, so they are 6 units apart. It checks by counting grid units and by taking the absolute value of the difference.
You'll be able to
- find the coordinates of a point reflected across an axis by changing one sign.
- find the distance between two points that share an x- or y-coordinate.
- use absolute value of a difference to find a distance.
Key words
- coordinates
- The pair (x, y) that gives a point's position on the coordinate plane.
- reflection
- A flip across an axis; changing the sign of one coordinate reflects a point.
- absolute value
- The distance of a number from zero, always zero or positive.
Transcript
- [Instructor] We're told Point R has the same coordinates as Point P, except its Y coordinate has the opposite sign. Point R is at two, negative six. What is the diff... 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. One is we could 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. So 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. So let me set up some axes here, make sure I can fit it. I have to fit six above and six below.
So 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. So... And then, let me 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 one, two, three, four, five, positive six there. And then one, two, three, four, five, negative six right over there. So Point R at two, negative six. Two, negative six is right over there. That is Point R. And then Point P, two, six.
Two, 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 one, two, three, four, five, six, seven, eight, nine, 10, 11, 12. So 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, so 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'd have gotten negative 12. Absolute value still 12. Now, you can do distance in two dimensions on a coordinate plane like this when one of the coordinates stays the same.
In 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 had to find the distance in one dimension. Let's now do another example. It says Point N has the same coordinates as Point L, except now it's X coordinate has the opposite sign.
So 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 going to have the same Y coordinate, five. Okay, so we could 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. So 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 confident and to visualize it all. So if this is my Y-axis, Y-axis, and this is my 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. One, two, three, four, five. And then, I'll go one, two, three and then negative one, negative two, negative three. Point N, I will do in pink. Point N is at three, five.
So three, five is right over there. That is Point N. And then, we figured out point L is at negative three. So we took the opposite of the X coordinate, which reflects it over the Y-axis, so this is L right over here, and you could see that distance. It's one, two, three, four, five and six.
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