Transformations : Translations, Reflections & Rotations

Counterclockwise

(*not yet) In what direction do rotations go?

A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, Y

(*not yet) What are some letters of the alphabet that have reflectional symmetry?

H, I, N, O, S, X, Z

(*not yet) What are some letters of the alphabet that have rotational symmetry?

Find the equivalent rotation counterclockwise. For e.g., if you are asked to rotate 90 degrees clockwise, follow the rule for 270 counterclockwise

(*not yet) What do you do if the direction is to rotate clockwise?

Order and a number represents the number ways for the object to look the same. For example, a perfect square would have an order of 4, a circle with an H in the center would have an order of 2.

(*not yet) What does "order" of a number represent in relation to Rotational Symmetry?

(x,y) -> (-x,-y)

(*not yet) What is the rule for rotation of 180 degrees about the origin? Abbreviated R180 degrees.

(x,y) -> (y,-x)

(*not yet) What is the rule for rotation of 270 degrees about the origin? Abbreviated R270 degrees.

(x,y) -> (x,y)

(*not yet) What is the rule for rotation of 360 degrees about the origin? Abbreviated R360 degrees.

(x,y) -> (-y,x)

(*not yet) What is the rule for rotation of 90 degrees about the origin? Abbreviated R90.degrees.

If there there is a center point around which the object is rotated a certain number of degrees and the object looks the same.

(*not yet) When does an object have Rotational Symmetry?

All segments that are translated are parallel to each other

A key property to translations about all segments that are translated....

What is symmetry?

Definition 1: Symmetry is when two halves of a figure mirrors each other across a line. The line of symmetry is the line that divides the figure into two mirror images. A simple test is to fold the figure along the supposed line of symmetry and see if the two halves of the figure coincide. Definition 2: A set of points has line symmetry if and only if there is a line, l, such that the reflection through l of each point of the set is also a point of the set. Definition 3: If half the figure is a mirror image of the other half. Definition 4: A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line.

If there is a rule that applies, then use the rule to determine the points of reflection (that is the easiest and usually most accurate). If you are not using a specific rule, then graph the pre-image, measure the distance of each point to the line of reflection, then graph each point of the image the same distance but on the other side of the line of reflection.

How do you graph a reflection?

(x,y) -> (x + # moved to right or left on the x axis, y + # moved up or down on the y axis)...for example, if each point is moved 4 units right and 2 units down, then you can say that eac (x,y) pair in the original figure is mapped to (x',y'), where x' = x + 4 and y' = y - 2. The rule would be written: (x,y) -> (x + 4, y - 2)

How do you write the translation rule using an example of a square being moved 4 units right and 2 units down?

(x,y) -> (x,-y). The abbreviation is r x-axis

Reflection in the x-axis. Abbreviated Rx

(x,y) -> (-x,y).

Reflection in the y-axis. Abbreviated Ry

(x,y) -> (-y,-x)

Reflection in the y=-x. Abbreviate Ry=-x

(x,y) -> (y,x)

Reflection in the y=x. Abbreviated Ry=x

P(x,y) -> P'(x+a, y+b) or Ta,b or <a,b>; for example a translation that moves a point 4 units right and 3 units down can be written as follows: P(x,y) -> P'(x+4, y-3) (the algebraic rule) or T4,-3 (shorthand) or <4,-3> (vector notation) -- unless told otherwise, use the algebraic rule)

What are the three ways to write a translation?

When an image is changed in some way. The change is size, shape, or position.

What are transformations?

When the image is congruent to the pre-image, it is said to be isometric. If something is isometric then the following properties are preserved between the pre-image and its image: - distance (lengths of segments are the same) -angel measure (angles stay the same) -parallelism (things that were parallel are still parallel) -collinearity (points on a line, remain on the line)

What does it mean for a transformation (image of a pre-image) to be isometric?

"Flip" an image over a line (line of reflection) - it is isometric. They are usually over the x or y-axis, but may also be flipped over lines, such as y=x or x=2. A reflection over a line m is an isometric transformation in which each point of the original figure (pre-image) has an image that is the same distance from the line of reflection as the original point but is on the opposite side of the line. Because a reflection is an an isometric, the image does not change size or shape

What is a reflection?

Turning an object around a point. The point it is rotating around is called the point of rotation.

What is a rotation?

A translation is an isometric transformation that maps all points of a figure the same distance in the same direction. It is like a "slide" and moves an object without changing its size or rotating it.

What is a translation?

Ray BC is parallel to ray B'C'. This is the Special Translation Property - Translating an angle along one of its rays.

When translating an angle by a vector (e.g. <ABC by vector BA): 1.) <ABC equivalent to <A'B'C' (Isometry) and 2.) B, A, B' and A' are collinear (translation on right angle ray), then because the two angles are equal and formed on the same ray, then:

Translations, reflections and rotations are all called isometries because the image is congruent to the pre-image

Which of the following are are isometries: Translations, reflections, and/or rotations?