Geometry Honors: 9.1 Translate Figures and Use Vectors to 9.4 Perform Reflections

9.1 - Translate Figures and Use Vectors 9.2 - Use Properties of Matrices 9.3 - Identify Symmetry 9.4 - Perform Reflections

Transformation

an operation that maps an original figure (pre-image) onto a new figure (image)

Isometry

transformation that perseveres length and angle measure (congruent transformation)

Four Transformations

1. Translations (slide)

2. Reflections (flip)

3. Rotations (turn)

4. Dilation (enlarge/reduce)

A pre-image is plotted…

solid

An image is plotted…

dotted

Translation

to vertically and/or horizontally slide a figure

Vector

a quantity with both direction and magnitude

Component Form

a form a vector that names the vector first and then adds an "= ⟨horizontal component, vertical component⟩"

Vector Form

⟨horizontal component, vertical component⟩

Coordinate Form

(x,y) → (x+2, y-5)

Matrix

a rectangular arrangement of numbers in rows and columns

Element

each number in the matrix is called this

Dimensions

number of rows and number of columns in a matrix, rows x columns

Addition/Subtraction with Matrices

matrices need the same dimensions to do this to the corresponding elements

Matrix Multiplication

The product of two matrices A and B is defined only when the number of columns in A is equal to the number of rows in B

How to multiply matrices

1. Determine the dimensions of your answer matrix

2. Multiply the elements in the 1st row of the 1st matrix by every element of the columns of the 2nd matrix

3. Continue multiplying the element of the rows of the first matrix by every element of the columns of the 2nd matrix

4. Move to the 2nd row of the 1st matrix and start the process again, multiplying by every column in the 2nd matrix

Line Symmetry

a figure can be mapped onto itself by a reflection in a line (mirror image)

Rotational Symmetry

figure is mapped onto itself by rotating the figure less than 360° about a center point

Point Symmetry

figure is mapped onto itself by rotating the figure 180° about a center point (looks the same when flipped upside down)

Angle of Rotation

360° divided by the number of lines of symmetry

Reflection

it is a flip over a line of reflection where each point is the same distance from the line of reflection

x-axis reflection rule

(x,y) → (x,-y)

y-axis reflection rule

(x,y) → (-x,y)

y=x reflection rule

(x,y) → (y,x)

y=-x reflection rule

(x,y) → (-y,-x)

How do you graph a reflection if none of the rules apply?

Take the difference between the point and the line of reflection and then add or subtract that difference from the line of reflection