Geometry Chapter 9 study Guide

Transformation

1. a mapping that results in a change in position, shape or size of an image.

Pre-image

1. the original figure

   

Image

1. the resulting figure 

   

Rigid motion

a transformation that preserves distance and angle measures

What are the three types of transformations:

1. translations, reflections, and rotations

Translation

 a transformation that maps all points of a figure the same distance in the same direction

Notation:

1. The part that explains how the transformation is. Like: “pre-image  ∆ABC

        image∆A’B’C’

Prime:

the apostrophe ‘ after a letter showing you its an image.

Translation notation:

1.  T(3,-2)(ABC) = A’B’C’

How to calculate translations from coordinates:

1. B’ - B

Reflection:

1. a transformation across a line m, called the line of reflection. It contains the following properties:

• If a point A is on line m,  then the image of A is itself (that is A = A’)

• If a point B in not on line m then m is the perpendicular bisector of BB'

• Reflections preserve distance

• Reflections preserve angle measures 

• Reflections map each point of the preimage to one and only one corresponding point of its image

• Reflections are rigid motions 

• Every corresponding point on the image and preimage are equidistant from the line of reflection 

  

Reflection notation example:

  Ry=x(ABC) = A’B’C’

Ry=-1 is what type of line:

1.  a horizontal line 

Remember:

1. when flipping over the axis, the prime is the opposite -2 → 2 and 2 → -2. Y axis = x coordinate changes, X axis = Y coordinate changes

X = 0 is:

Y axis

Y = 0 →

1. x axis 

A rotation:

1. A rotation of x° about a point Q, called the center of rotation is a transformation with these two properties:

• The image of Q is itself (Q’ = Q) 

• For any other point V, QV’ = QV and m∠VQV’ = x°

  

Angle of rotation:

1. the number of degrees a figure rotates

Remember:

1. A rotation about a point is a rigid number. You write the x° rotation of ∆UVW about point Q as r( x°,Q) (∆UVW) = ∆U’V’W’ where x° is rotation and Q is the center of rotation.

Remember:

1. unless stated otherwise, rotations are counterclockwise.

Rules for rotation 90°:

1.  r(90°,O)   (x-y) = (-y,x)

Rules for rotation of 270°:

1.  r(270°,O)(x,y) = (y, -x)

Degrees that have rules that require the switching of x and y:

90° and 270°

Rules for 180° and 360°:

Remember:

1. A rigid motion can be expressed as a composition of reflections 

Isometry:

 a transformation that preserves distance or length.

Examples of Isometries:

1. Translation, rotations and reflections 

   

Theorem 9.1

1. The composition  of two or more isometries is an isometry. There are only 4 kinds of Isometries: 

   1. Translations

   2. Reflections 

   3. Rotations

   4. Glide Reflection

      

Theorem 9.2 - Reflections Across Parallel Lines:

A composition of reflections across two parallel lines is a translation.

Remember:

1. (Rm ° Rl) (↑) is the same as saying Rm(RL(↑) )

Which comes first in “(Rm ° Rl)

1.  Rl

Theorem 9.3 - Reflections Across intersecting Lines:

1.  A composition of reflections across two intersecting lines is a rotation about the point of intersecting lines.  

   

Glide reflection:

1. a composition of a translation (a glide) and a reflection across a line parallel to the direction of the transition. 

   

Remember:

1. A dilation with center of dilation C and scale factor n,n > 0, can be written as D(n,c)

Dilation:

1.  a transformation with the following properties:

   1. The image of C is itself )that is C’ = C)

   2. For any other point R,R’ is on CR and CR’ + n✖CR or n = CR'CR

   3. Dilations preserve angle measures 

      

What are dilations similar to:

 similar figures

Scale factor =

1. image/preimage

Remember:

1. we will consider all centers of dilations in the coordinate plane to be the origin.

 the scale factor is > 1

1. dilation is an enlargement 

If dilation is between 0 and 1

1. dilatation is a reduction 

   

Remember:

1. you can find the dilation image of point P(x,y) by multiplying the coordinates of P by the scale factor n 

Image =

1.  scale factor ✖ preimage length