GSE Geometry Unit 1

Transformations

Isometry/Rigid Transformations

Preserve size and shape

The three isometries

Translation, Rotation, Reflection

Reflection

Mirroring a point or figure across a line

Reflection over y=x

(x,y) →(y,x)

Reflection over y = -x

(x,y) →(-y,-x)

Reflection over y = A

Count right(if +A)/left(if -A) from the pre-points to the line of reflection (A) and count that same number of times over that line again. Y stays the same, x changes

Reflection over x = A

Count up(if +A)/down(if -A) from the pre-points to the line of reflection (A) and count that same number of times over that line again. X stays the same, y changes

Reflect over x-axis

Count up(if +A)/down(if -A) from the pre-points to the x-axis and count that same number over the x-axis. X stays the same, y changes

Reflect over y-axis

Count right(if +A)/left(if -A) from the pre-points to the y-axis and count that same number over the y-axis. Y stays the same, x changes

Reflection Coordinate Notation

(x,y) →(x +/- A, y +/- A)

Reflection Formula Notation

rx-axis, ry-axis, ry=x, ry=-x, rA

Rotation

Turning of a figure over a point on a plane

180 degrees rule

(x,y) →(-x,-y)

90 degrees clockwise/270 degrees counterclockwise rule

(x,y) → (y,-x)

90 degrees counterclockwise/270 degrees clockwise

(x,y) →(-y,x)

Rotation Function notation

Rx, +/- degrees ; x = point of rotation; + degrees = Clockwise, - degrees = Counterclockwise

Translation

Moving a point or figure across the plane

Translate (x,y) → (A, 0)

if +A, move right/add, if -A, move left/subtract; Y doesn’t change because y = 0 in this formula.

Translate (x,y) →(0,B)

if +B, move up/add, if -B, move down/subtract; X doesn’t change because x = 0 in this formula.

Translation Coordinate Notation

(x,y) →(x+/- a, y+/- b)

Translation Function Notation

Ta,b

Dilation

change of size, not shape; NOT ISOMETRY

Scale factor (K)

determines the size change; if K > 1, it’s an enlargement. If K < 1, it’s a reduction. If K = 1, No change

Center of dilation

If you connect the corresponding vertices of the image and pre-image, they will meet at a point.

Dilation Coordinate Notation

(x,y) →\[x(a), y(b)\]

Dilation Function Notation

Dx/y

Composition

A sum of all translations in a problem or a combination; represented by small oval in the middle of every transformation function present; WORK LEFT TO RIGHT.