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2.6 Investigation - Properties of Transformations

Author:marentsen
Topic:Geometry

#1

ABC was reflected over line ED to create A'B'C'. Click ABC and move it around. You can also drag points A, B, C, D, or E and watch how the image changes. Once you are done, reset it by refreshing the page.

1a

Drag Triangle ABC closer to line DE. What do you notice about Triangle A'B'C'?

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1b

Let's take a look at something called "Orientation". Starting at point A, lets name the triangle by going CLOCKWISE. What would the triangle be called?

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1c

So the Pre-Image is called Triangle ABC going clockwise. How would you name the IMAGE when starting at A and going CLOCKWISE.

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1d

We say orientation is the same if the order in which we say the points in the pre-image is the same as with the Image. For example, XYZ would have the same orientation as X'Y'Z'. Look back at the triangles above from the reflection. Is the orientation the same, or reversed?

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Reflections with Distances

1e

This is the same situation as above where Triangle ABC is reflected over line DE, but this time you can see the measure of the segments from each Pre-Image point to its Image. As you drag the triangles or points A,B,C,D, or E you should see that the measures of AA', BB', and CC' change. Did each point move the same distance after the reflection? (another way to ask this is "are the lengths of AA', BB', and CC' all the same)?

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1f

Is it possible to make the distance between A and A' 0? (Is AA' ever 0)? If so, where does point A need to be to make that happen?

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Next lets look at Rotations. Use the applet below to observe some of the properties of Rotations.

Rotations - Quadrilateral FGHI was rotated 135 degrees about point J.

2a

Get familiar with rotations by clicking on Quad FGHI and dragging it to different places. Look at how Quad F'G'H'I' changes in relation to point J. Next you can try dragging any of the following points: F, G, H, I or J. Check the orientation of the pre-image and image above. Is the orientation of Quad FGHI and Quad F'G'H'I' the same or reversed?

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I created line segments connecting each Pre-image point with it's image. The lengths of the segments are shown below.

2b

Let's take a look at the distances between pre-images and their image points. Did each point move the same distance after the rotation? (another way to ask this is "are the lengths of FF', GG', HH' and II' all the same)?

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2c

Is it possible to make the distance between F and F' 0? (Is FF' ever 0)? If so, where does point A need to be to make that happen?

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3

Next let's look at Translations.

Parallelogram KLMN was translated by Vector PQ (moved the direction the arrow is pointing, and the length of the vector (arrow)).

3a

Get familiar with translations by clicking on Parallelogram KLMN and dragging it to different places. Then drag point Q to a new location. What happens to the image when you move point Q?

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3b

Now look at the orientation of the parallelogram after the Translation.

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Translations with distances

3b

Let's take a look at the distances between pre-images and their image points. Did each point move the same distance after the rotation? (another way to ask this is "are the lengths of KK', LL', MM' and NN' all the same)?

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3c

Is it possible to make the distance between K and K' 0? (Is KK' ever 0)? If so, where does point A need to be to make that happen?

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  • A
  • B
  • C
Check my answer (3)

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