Main content

### Course: High school geometry > Unit 1

Lesson 5: Reflections# Reflecting points

We can plot points after reflecting them across a line, like the x-axis or y-axis. Reflections create mirror images of points, keeping the same distance from the line. When we reflect across the y-axis, the image point is the same height, but has the opposite position from left to right.

## Want to join the conversation?

- What does prime mean when for example they say C prime?(33 votes)
- In the context of geometric transformations, the prime symbol (') denotes the image of a point as a result of a transformation. For example, point A' is the image of point A, point B' is the image of point B, and point C' is the image of point C.

Eventually, you might see prime symbols in other mathematical contexts as well. For example, in calculus, if f is a function, then f' denotes the derivative (instantaneous rate of change) of the function f.(44 votes)

- Is there a video that shows how to reflect a point over a diagonal line? I am really confused because I thought that there was a formula to do so.(19 votes)
- Very late here, but here it is: https://youtu.be/kj3ZfOQGKdE(1 vote)

- not as bad as rotating...rotation confuses me(14 votes)
- 90° clockwise rotation: (x,y) becomes (y,-x)

90° counterclockwise rotation: (x,y) becomes (-y,x)

180° clockwise and counterclockwise rotation: (x, y) becomes (-x,-y)

270° clockwise rotation: (x,y) becomes (-y,x)

270° counterclockwise rotation: (x,y) becomes (y,-x)(23 votes)

- how do you identify the reflection of -2,6 ?(7 votes)
- Depends what axis you are reflecting over, if you are reflecting over x axis, change the y positive or negative sign to the opposite sign, if it reflects over y axis, change the sign of the x.

Hope this helps(17 votes)

- if it helps reference stranger things. the Upside Down mirrors the actual world. Then imagine Will Byers from stranger things as a shape/ figure undergoing reflection. Simplified, i would describe a trip to the Upside Down a reflection in mathematical terms(12 votes)
- can i get a little bit of explanation?(3 votes)
- Some basic rules:

- Reflection over x-axis:

(x, y) ---> (x, -y)

- Reflection over y-axis

(x, y) ---> (-x, y)

- Reflection over line where y = x

(x, y) ---> (y, x)

- Reflection over line where y = -x

(x, y) ---> (-y, -x)(19 votes)

- Thanks! there's actually a way to do it algebraically too, if you don't know it already you'll need it: Across the x-axis -(x,y)- (x,-y)

Across the y-axis- (x,y) - (-x, y)

Just follow this rule and you won't even need a graph to know a reflection!(6 votes)

- This was fairly easy thank you for making this so i can understand it better.(5 votes)
- are you exposed to be in high school ? if so I'm in elementary school(2 votes)
- You don't have to be in a specific grade to learn the concepts. This is done on purpose so students can learn at their own pace. With that said, don't feel bad if you do not understand some of the stuff at this level.(6 votes)

- Is reflecting the same thing as rotating 180 degrees about point "x" for example?(2 votes)
- That depends on the shape. In the case of a square, for example, you could rotate it 90 degrees about the center and get the same thing. However, if the shape does not have a line of symmetry, and is rotated 180 degrees around "point x", you might not get the same thing.(6 votes)

## Video transcript

- [Instructor] We're asked
to plot the image of point A under a reflection across the line l. So we have our line l here, and so we wanna plot the image of here, we wanna plot the image of point A under a reflection across line l. Well, one way to think about it is point A is exactly one, two, three,
four units to the right of l. And so its reflection is going to be four units to the left of l. So if we go one, two, three, four, that would be the image of point A. We could maybe denote that as A prime. So if you're doing this on
the Khan Academy exercise, you would actually just click
on a point right over there, and it would show up. But this would be the reflection of point A across the line l. Let's do another example. So here we're asked plot
the image of point B under a reflection across the x-axis. Alright, so this is point B, and we're going to reflect it across the x-axis right over here. So to go from B to the x-axis, it's exactly five units below the x-axis. One, two, three, four, five. So if we were to reflect
across the x-axis, essentially create its mirror image, it's going to be five
units above the x-axis. One, two, three, four, five. So that's where the image would be. Maybe we could denote that with a B prime. We are reflecting across the x-axis. Let's do another example. So here they tell us point
C prime is the image of C, which is at the coordinates negative four comma negative two, under a
reflection across the y-axis. What are the coordinates of C prime? So pause this video and see if you can figure
it out on your own. So there's several ways to approach it. It doesn't hurt to do
a quick visual diagram. So that could be my x-axis. This would be my y-axis. And it's the point negative
four comma negative two, so that might look like this. Negative four, negative two. So this is the point C right over here. And we wanna reflect across the y-axis. So we wanna reflect across the y-axis, which I am coloring it
in red right over here. So let's see. The point C is four to
the left of the y-axis. So its reflection is going to be four to the right of the y-axis. So let me do it like this. So instead of being four to the left, we wanna go four to the
right, so plus four. So where would that put our C prime? So our C prime would be right over there. And what would its coordinates be? Well, it would have the same y-coordinate, so C prime would have a
y-coordinate of negative two. But what would its x-coordinate be? Well, instead of it being negative four, it gets flipped over the y-axis, so now it's gonna have a
x-coordinate of positive four. So the coordinates here would
be four comma negative two. Four comma negative two. You might've been able
to do this in your head. Although, for me, even if
I try to do it in my head, I would still have this
visualization going on in my head. Negative four comma negative two. I'm sitting there in the third quadrant. If I'm flipping over the y-axis, my y-coordinate wouldn't change, but my x-coordinate
would become the opposite and I would end up in the fourth quadrant, and that's exactly what happened. Y-coordinate did not change,
but then my x-coordinate, since I'm flipping over the y-axis, it became the negative of this, so the opposite of negative
four which is positive four.