transformations. congruent similar image vs pre-image pre-image: the figure before a transformation...

Transformations

Congruent

Similar

Image vs Pre-image• Pre-image: the figure before a transformation is

applied.

• Image: The figure resulting from a transformation.

• For example, if the reflection of point P in line l is P', then P' is called the image of point P under the reflection and P is the pre-image. Such a transformation is denoted rl (P) = P'.

Transformations

• A transformation is a change in position, shape or size of a figure.

• Transformations that create congruent figures are referred to as rigid motions.

• Types of transformations that create congruent figures are– Translations (slide)– Reflections (flip)– Rotations (turn)

• A transformation that creates a similar figure is – Dilation (enlarge/shrink)

Labels• It is common to label each corner

with letters, and to use a little dash (called a Prime) to mark each corner of the reflected image.

• Here the original is ABC and the reflected image is A'B'C' 

Math is easy.

Translation

• A translation is a transformation in which the location of the image is changed.

• Also referred to as a slide.

Translation

• Translation means moving…

• …without rotating, resizing or anything else, just moving.

• Every point of the shape must move:– The same distance– In the same direction

Translation• Example:

– If we want to say that the shape gets moved 3 Units in the "x" direction, and 4 Units in the "y" direction, we can write:

 (x, y) (x + 3, y + 4)

– This says "all the x and y coordinates will become x+3 and y+4"

Translation• A translation "slides" an object a fixed

distance in a given direction.

• The original object and its translation have the same shape and size, and they face in the same direction.

• A translation creates a figure that is congruent with the original figure and preserves distance (length) and orientation (lettering order).

TranslationExample:

• Describe how the following figure was translated.

TranslationExample:

• Describe how the following figure was translated.

Vector• A vector is a segment in the plane. One of its two

endpoints is designated as a starting point; while the other is simply called the endpoint.

• The length of a vector is, by definition, the length of its underlying segment.

• Visually, we distinguish a vector from its underlying segment by adding an arrow above the symbol. Thus, if the segment is AB (A and B being its endpoints), then the vector with starting point A and endpoint B is denoted by . Likewise, the vector with starting point B and endpoint A will be denoted by .

AB

BA

Vector (continued)

• Note that the arrowhead on the endpoint of a vector distinguishes it from the starting point. Here vector is on the left, and vector is on the right.

AB

BA

A A

B B

Vector (continued)

• A vector tells the direction the shape will move. The vector does not have to go through the shape or even touch the shape.

Concept

M

A

B

M

A

B

Concept

• Check out this links for additional information and practice with translations:

• http://www.harpercollege.edu/~skoswatt/RigidMotions/translation.html

• http://www.mathsisfun.com/geometry/translation.html

Concept• A translation can also move figures along a vector (a half line

with direction)

• We denote the red line by T(L)

• More formally, if G is a given figure in the plane, then we denote by T(G) the collection of all the points T(P), where P is a point in G. We call T(G) the image of G by T, and (as in the case of a point) we also say that T maps G to T(G).

Summary• Remember that translations have three basic

properties: – Translations map lines to lines, segments to

segments, rays to rays, angles to angles, – Lengths of segments are preserved, – Degrees of measures of angles are

preserved. • Remember you can use a simplified notation,

for example, M’ to represent the translated point M.

Summary (continued)

• Remember that a translation just moves a figure.

• It is the same exact shape but just in a different place.

• A vector shows where the figure is going to be moved.

• The image (new figure) is exactly the same as the original figure.

• The length of lines stays exactly the same.• The measure of angles stays exactly the same.

Reflection

• A reflection is a type of transformation in which the image is flipped over a line (or point).

• Also referred to as flip.

Reflection• Reflections are everywhere... in mirrors, glass,

and here in a lake. ...what do you notice ?

• Every point is the same distance from the central line.

• The reflection has the same size as the original image.

Reflection

• The central line is called the Mirror Line, and it doesn't matter what direction the mirror line goes, the reflected image is always the same size, it just faces the other way:

Reflection• How to do a reflection:

– For each corner of the shape:

1.Measure from the point to the mirror line (must hit the mirror line at a right angle)

2.Measure the same distance again on the other side and place a dot.

3.Then connect the new dots up!

ReflectionExample:

• How is the following figure reflected?

Rotation

• A rotation is a type of transformation in which the figure is turned around a fixed point.

Rotation

Example:

• In which direction has the following figure been rotated and in what direction?

Dilation

• A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.

Scale Factor

• If the scale factor is greater than 1, the image is an enlargement.If the scale factor is between 0 and 1, the image is a reduction.

DilationExample:

• What is the scale factor of the following dilation?