Transformations
You say you want a transformation, well, you know we all want to change the world
Do you recall the definition of symmetry?
In the previous section you looked at pictures that already had some type of symmetry and you had to determine what rule was used to map one set of points onto its corresponding points. The rules that we looked at were reflections, rotations, and translations. These rules are all types of transformations.
Transformations are functions that take a set of points then transform each point either by moving, flipping, turning, shrinking, or other types of motion, to create a set of image points. If you remember the idea of a function, it is a relation between a set of inputs so that each input has one output. In geometry we are using a set of points as the inputs and the output is the set of image points. We usually use capital letters as the set of input points (also called the pre-image) and the set of output points are named with capital letters with the addition of an apostrophe and are spoken with the word “prime”.
Pt. A $\rightarrow$ Pt. A’ or f (A) = A’
Verbally, we could say:
“Point A is mapped by a function f onto point A prime.”
Instead of looking at figures that already have a rule set, suppose we wanted to set the rule ourselves. I mean, I don’t want to be kept down by someone else’s rules! I am my own person! Don’t let the people in control keep you down! (Who ever those people happen to be.)
On each interactive of this section, the labeled figure with regular capital letters (i.e. Pt. A) is the original figure, also called the pre-image, and the figure labeled with an apostrophe (i.e. Pt. A’) is the resulting figure called the image after the transformation.
Translations
Tap to open interactive!
a. If you had to translate a point on a piece of paper by the same vector, explain how you would do this transformation.
b. What does it mean to translate a figure? Does it change size? shape? orientation?
Note: Orientation is the way in which a figure is named. We would name the original figure, quadrilateral BECK named in a clockwise direction. Is the image figure named B’E’C’K’ in a clockwise or counter-clockwise direction ?
c. If the image figure retains the same shape and same size (side lengths and angle measures) as the original image, then the transformation is called an isometry. Is a translation an isometry?
d. Make the length of the vector as close to zero as you can. Where is the image figure? Why does this make sense?
Reflections
Tap to open interactive!
2. Reflections – Line n is referred to as the line of reflection. Figure SLY is reflected over line n.
Grab points (open dots) to move line n. As you move the line, observe the effects on the image figure. Also, grab points of the original figure SLY to see how the image is affected.
a. If you had a point and a line on your paper, explain how you would reflect the point over the line using a ruler.
b. What does it mean to reflect a figure? Does it change size? shape? orientation?
c. Is a reflection an isometry?
d. What is true about the image point if the original point is on the line of reflection?
e. What happens to the image figure if the line of reflection goes through the original figure?
Rotations
Tap to open interactive!
3. Rotations – Figure WILCO is rotated counter-clockwise about D by ∠XAY.
Grab the point X (open dot) on the circle at the top left to change the angle of rotation. As you move the point, observe the effects on the image figure. Also, grab points of the original figure WILCO to see the effects on the image figure.
Also, grab and drag point D. This also has a great effect on the image figure.
a. If you had to rotate a point around a central point on paper by 135°, explain how you would do a rotation using a ruler. This one is a little trickier; if you need hints tap the buttons in the widget.
b. What does it mean to rotate a figure? Does it change size? shape? orientation?
c. Name two different ways that you could make the image figure partially overlap the original figure.
Dilations
Tap to open interactive!
4. Dilations – ALTJ is dilated by a factor of k from a center point P.
Grab the slider bar in the top left to change the factor of dilation. As you move the slider, observe the effects on the image figure.
Also, grab the center of dilation, point P. Note the effect on the image figure by changing the center of dilation.
a. If you had to dilate a point away from a center point by a factor of 2, explain how you would draw the dilation on a piece of paper.
b. What does it mean to dilate a figure? Does it change size? shape? orientation? How does the value of k affect the dilation?
c. Move point P. Describe how the position of the center of dilation affects the image figure
5. Which characteristics are preserved under each type of transformation? Check those that apply for each.
| Translations | Reflections | Rotations | Dilations | |
| Orientation Naming a figure clockwise or counter−clockwise. | ||||
| Angle Measure Comparing ∠ABC with ∠A’B’C’ | ||||
| Segment Length Segment AB vs. Segment A’B’ | ||||
| Betweenness/Collinearity Points between and on the same line as A and B remain between and on the same line as A’ and B’. |
Transformations that create images which maintain angle measure, segment length, and betweenness of points are called isometries.
An isometry is a transformation which creates a image figure that is congruent to the pre-image.
Two figures are congruent if all their corresponding sides and angles are equal. Or we could say, two figures are congruent if one figure could be mapped onto the other figure by any number of isometries. So congruent is like “equal to” but it is stronger than just equal. In fact, congruent has a stronger symbol than equal ( = ); the symbol for congruent is $\cong$.
Explain it to me like I’m a five-year-old: If one figure can be picked up and flipped (reflected) , turned (rotated), or slid (translated) onto the other so that all corresponding parts match up perfectly, then the figures are congruent.
