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Reflections - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A reflection is an isometry, which means the original and image are congruent, that can be described as a "flip". To perform a geometry reflection, a line of reflection is needed; the resulting orientation of the two figures are opposite. Corresponding parts of the figures are the same distance from the line of reflection. Ordered pair rules reflect over the x-axis: (x, -y), y-axis: (-x, y), line y=x: (y, x).

When we're talking about transformations, there are 4 different types one of which is a reflection. Now, what is a reflection? Well, reflection is an isometry which means it's rigid transformation which means its shapes are congruent after the reflection. Secondly its image has opposite orientations. Well that's a little difficult to understand so let's take a look at two different pictures.
Here I have an original and I reflected this quadrilateral over this dotted little line onto the other side, so if we were to go in clockwise direction, after a I would have d after d I would have vertex c after vertex c I would have vertex b and after vertex b I would have vertex a. Now, if we go over to our image, if I start with a the next vertex is b not d, so that's what we mean when we say the image has opposite orientations because when you reflect it you're going to have a different order of vertices that's going to be important when you get to Chemistry.
Next, a point and its image are equidistant from the line of reflection so again let's go back to our little example here and if I picked this vertex c and if I drew a perpendicular from vertex c to the line of reflection, that's going to be the same distance as its image which is c prime, so the distance from c prime to that line of reflection will be congruent, so that's how you know how far to reflect and in what direction over the line that is given to you.
And the last thing is you can kind of consider this a flip that will be the non mathematical way of describing a reflection.
There are 3 types of reflections that you need to know about in regards to the order pair rules. The first one is xy is mapped onto -x, y. Well to figure out what type of reflection this is let's write a little, draw a little sketch here, so I'm going to make an x axis and a y axis and I'm just going to pick some random point in the first quadrant here and we're going to call this point a and let's say point a has x coordinate 3 and y coordinate 5. Now according to this order pair rule I'm going to take my x and takes its opposite and I'm going to keep my y exactly the same so the opposite of 3 is going to be -3 so I'm going to have to write my new image a prime over on this side of the y axis so what is our line of reflection? Well it's pretty clear that a kept the same y coordinate but it's x coordinate was taken the opposite of so we're going to categorize this as reflection over the x axis excuse me y axis. I was thinking that the x coordinates changed but since the x coordinates are changing that means that the y axis is our line of reflection.
Secondly we have xy is mapped onto x and -y which means if we go back to our original point, x is going to stay the same and our y is going to be taken the opposite of so the opposite of 5 would be -5 so now our point a double prime is going to be at 3,-5 so it looks like our line of reflection is the x axis so what we're saying reflection over the x-axis.
The final one that you should know by heart is xy is mapped onto yx. What this one does, it takes every x coordinate and then makes that into a y coordinate and does the same for your y coordinates, every y coordinate becomes an x coordinate so if we have a, our a since we have a prime and a double prime our a triple prime is going to be at 5, 3 so what I'm going to do is I'm going to write 3, 5 right here. Now 5, 3 is going to be lower and more to the right I'm going to write a 1, 2, 3 tripple prime at 5, 3 now this one is a little tricky to figure out what exactly is a line of reflection, now give you a hint is a diagonal line with the slope of 1 and the y intercept of 0 so if we're to drawn this in that line right there is the line y=x so if you have a point on this side of the line y=x when you switch your x and y coordinates it's going to flip up to this side. So we'll say this reflection over the line y=x. So that way in your quiz when your teacher says what does this mapping do x, y -xy well you can say it's going to be reflected over y axis. This one when you keep x the same and take the opposite of y you know that's a reflection over the x axis and last when you switch the x's and y's that will reflect it over line y equals x.

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