CONCEPT 1 -Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular line, parallel lines, and line segment.

THE ISOMETRIC TRANSFORMATIONS

(1) THE REFLECTION

EXPLANATION

A reflection over a line m (notation R_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 isometry, the image does not change size or shape.  The line of reflection is the perpendicular bisector of the segment joining every point and its image.

DEFINITION

A reflection in a line m is a isometric transformation that maps every point P in the plane to a point P’, so that the following properties are true;
1. If point P is NOT on line m, then line m is the perpendicular bisector of .	2. If point P is ON line m, then P = P’

PROPERTIES

ISOMETRIC PROPERTIES - Because a reflection is an isometric transformation the following properties are preserved between the pre-image and its image:

• Distance (lengths of segments are the same)
• Angle measure  (angles stay the same)
• Parallelism (things that were parallel are still parallel)
• Collinearity (points on a line, remain on the line)

After a reflection, the pre-image and image are identical.

TRANSFORMATION PROPERTIES – Because a reflection is a transformation that maps all points the perpendicular distance on the opposite side of the line of reflection the following properties are present.

• DISTANCES ARE DIFFERENT -- Points in the plane move different distances, depending on their distance from the line of reflection.  Points farther away from the line of reflection move a greater distance than those closer to the line of reflection.  Notice how AA’ is greater than BB’.  Notice that because line m is the perpendicular to ,  and  they are all parallel to each other.

• ORIENTATION IS REVERSED – The pre-image has a reversed orientation than its image.  Orientation is the order of the points about the shape.  In DABC the points in a clockwise direction come in the order of A – B – C but in the image the points in a clockwise direction come in the order of A’ – C’ – B’.  This occurs because a reflection creates the mirror image.

• SPECIAL POINTS – Points on the line of reflection do not move at all under the reflection.  The pre-image (D) = image (D’) when the point is on the line of reflection. 

(2) THE ROTATION

EXPLANATION

A rotation is an isometric transformation that turns a figure about a fixed point called the center of rotation.  Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation (notation R_{center, degree}).

An object and its rotation are the same shape and size, but the figures may be turned in different directions.  

DEFINITION

A rotation about a Point O through Θ degrees is an isometric transformation that maps every point P in the plane to a point P’, so that the following properties are true;
1. If point P is NOT point O, then OP = OP’ and m∠QPQ’ = Θ°.	2. If point P IS point O, then P = P’.  The center of rotation is the ONLY point in the plane that is unaffected by a rotation.

ROTATION DIRECTION

One full rotation is 360°, this would return all points in the plane to their original location.  Because a rotation can go in two directions along the same arc we need to define positive and negative rotation values.  COUNTERCLOCKWISE IS A POSITIVE DIRECTION, and clockwise is a negative direction.

Students often want to know why this ‘backwards’ relationship happens….. my belief is that the rotation direction comes from the coordinate plane,  In the coordinate plane, the initial arm of the angle is the positive x axis, and the terminal arm opens in a counterclockwise direction from there.

EQUIVALENT ROTATIONS

Because angles are formed along an arc of a circle there are two ways to get to the same location, a positive direction and a negative direction.  A 60° rotation is the same as a -300° rotation.  Co-terminal angles can be calculated using the formula, a coterminal angle = initial angle + 360n, where n is an integer.

60° = 60° + (1)360° = 420°                             60° = 60° + (2)360° = 780°

60° = 60° + (-1)360° = -300°                           60° = 60° + (-2)360° = -660°

Conterminal angle = initial angle + 360n

PROPERTIES

ISOMETRIC PROPERTIES - Because a rotation is an isometric transformation the following properties are preserved between the pre-image and its image:

• Distance (lengths of segments are the same)
• Angle measure  (angles stay the same)
• Parallelism (things that were parallel are still parallel)
• Collinearity (points on a line, remain on the line)

After a rotation, the pre-image and image are identical.

TRANSFORMATION PROPERTIES – Because a rotation is a transformation that maps all points along an arc the following properties are present.

• DISTANCES ARE DIFFERENT -- Points in the plane move different distances, depending on their distance from the center of rotation.  A point farther away from the center of rotation maps a greater distance than those points closer to the center of rotation.  Notice that different from a reflection, ,  and  are NOT parallel to each other.

• ORIENTATION IS THE SAME – The pre-image has the same orientation as its image.  Orientation is the order of the points about the shape.  In DABC the points in a clockwise direction come in the order of A – B – C and in the image the points in a clockwise direction come in the order of A’ – B’ – C’.  They are the same!! (Sometimes students want to say that a rotation changes the orientation of the shape because the shape has been turned to create a different look of the shape…. this is NOT an orientation change!)

• SPECIAL POINTS – The center of rotation is the only point in the plane that is unchanged,  O = O’.

SPECIAL ROTATION – ROTATION OF 180°

A rotation of 180° maps A to A’ such that:             a) m∠AOA’ = 180° (from definition of rotation)               b) OA = OA’ (from definition of rotation)               c) Ray  and Ray  are opposite rays.  (They form a line.)                 is the same line as

(3) THE TRANSLATION

EXPLANATION

A translation slides an object a fixed distance in a given direction.  When working in the plane this is usually represented by an arrow, the arrow provides both distance and direction of the translation.  When working on the coordinate plane, a vector is used to describe the fixed distance and the given direction often denoted by <x,y>.  The x value describes the effect on the x coordinates (right or left) and the y value describes the effect on the y coordinates (up or down).

The pre-image and image have the same shape and size.

DEFINITION

A translation is an isometric transformation that maps every two points A and B in the plane to points A’ and B’, so that the following properties are true;
1. AA’ = BB’ (a fixed distance).   2.  (a fixed direction).

PROPERTIES

ISOMETRIC PROPERTIES - Because a translation is an isometric transformation the following properties are preserved between the pre-image and its image:

• Distance (lengths of segments are the same)
• Angle measure  (angles stay the same)
• Parallelism (things that were parallel are still parallel)
• Collinearity (points on a line, remain on the line)

After a translation, the pre-image and image are identical.

TRANSFORMATION PROPERTIES – Because a translation is a transformation that maps all a fixed distance and in a fixed direction the following properties are present.

• DISTANCES ARE THE SAME -- Points in the plane all map the exact same distance.  Notice how AA’ is EQUAL TO BB’.

• ORIENTATION IS THE SAME – The pre-image has the same orientation as its image.  In DABC the points in a clockwise direction come in the order of A – B – C and in the image the points in a clockwise direction come in the order of A’ – B’ – C’. 

• SPECIAL POINTS – There are NO special points, ALL POINTS IN THE PLANE MOVE!!! 

SPECIAL TRANSLATION PROPERTY – TRANSLATING AN ANGLE ALONG ONE OF ITS RAYS