[G.CO.4] Congruence #4

Objective

Common Core Text:

  • [G.CO.4] Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments

Said Differently:

    • That's actually a pretty good way of saying it

Example

Define Reflection

    1. Every point is the same distance from the line of reflection as its image

    2. Every point and its image create a line segment that is perpendicular to the line of reflection

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Explanation

By now, you know what translations, rotations, and reflections are. However, you would probably have to talk about things like "spin around" or "mirror image", which aren't mathematically solid terms. Here, you will learn precise definitions of translation, rotation, and reflection.

Reflection

Here we have a parallelogram being reflected across a line. The line has a slope of 1. Lets notice some things

Length from E to A = 2

Length from E to A' (the image of A) = 2

Length from F to B = 2

Length from F to B' (the image of B) = 2

You can see that every point is the same distance from the line of reflection as its image

The second thing is trickier. Let's connect the points to their images. In the picture below, we've connected A to A' (the image of A) and B to B' (the image of B).

The thing to notice is that these new lines are perpendicular to the line of reflection. This means every point and its image create a line segment that is perpendicular to the line of reflection

Putting our findings together, we have

  1. Every point is the same distance from the line of reflection as its image

  2. Every point and its image create a line segment that is perpendicular to the line of reflection

To be fancy we can go one step further. Notice that the line of reflection cuts line segments AA' and BB' in half.

Because points and their images are the same distance from the line of reflection, this will happen for any line segment between a point and its image. Therefore, instead of the 2 findings above, we could combine our findings and simply say that

  • The line of reflection is a perpendicular bisector to all line segments bound by a point and its image

(Notice that this doesn't apply to points that lie on the line of reflection) In this picture, E doesn't have an image because it lies on the line of reflection (or you could say that E and E' are the same point)

Translation

You already know how translations work. They move the shape

  • some distance

  • in some direction.

In math, when things have a length and a direction, we like to use little arrows called vectors. In this picture, our vector is EF.

Let's translate the shape by this vector.

Connecting the points with their images

We can see that for translation

  1. The line segments between the points and their images are the same length as a given vector

  2. The line segments between the points and their images are parallel to a given vector

Rotation

Rotations need:

  • a point to rotate around, called the center of rotation. In this picture, our center of rotation is E, which is on the origin.

  • an angle that tells how much it rotates in a circle, called the angle of rotation. In this picture, our angle of rotation will be 90° counter clockwise.

Rotating, we get the following

Lets connect points and their images to the center of rotation

Notice that

  • ∠BEB' = 90°

  • ∠DED' = 90°

In fact, any point and its image connected to the origin will create an angle equal to the given angle of rotation

Also notice that line segments

So our findings for rotation are that

  • Each point is the same distance from the center of rotation as its image

  • Each angle created by a point, the center of rotation, and its image is equal to the given angle of rotation

In Summary