Module 2: Transformations

Rotation of 90° clockwise

(x, y) → (y, −x)

Rotation of 180°

(x, y) → (−x, −y)

Rotation of 90° counterclockwise

(x, y) → (−y, x)

Translation Rule

(x,y) -> (x+a, y+b)

To rotate a point about the origin (point 0,0) 180* CW

(x,y) becomes (-x, -y)

To reflect a point over the y-axis

(x,y) becomes (-x, y)

To reflect a point over the line y= -x

(x,y) becomes (-y, -x)

To rotate a point about the origin (point 0,0) 270* CW

(x,y) becomes (-y, x)

To reflect a point over the x-axis

(x,y) becomes (x, -y)

To rotate a point about the origin (point 0,0) 360* CW

(x,y) becomes (x,y)

To rotate a point about the origin (point 0,0) 90* CW

(x,y) becomes (y, -x)

To reflect a point over the line y=x

(x,y) becomes (y,x)

slope formula

(y₂- y₁) / (x₂- x₁)

180* clockwise is the same as...

180* counter clockwise

90* clockwise is the same as...

270* counter clockwise

Rotational symmetry formula

360 divided by the number of sides on the shape

360* clockwise is the same as...

360* counter clockwise AND will look like the original position

hexagon

6 sided polygon

270* clockwise is the same as...

90* counter clockwise

Pre-image

A figure before a transformation has taken place.

How is a line of symmetry drawn through a figure?

A line of symmetry is a line drawn through a figure that creates two congruent halves.

line of symmetry

A line that divides a figure into two congruent halves that are mirror images of each other.

Reflection Line

A line that is the perpendicular bisector of the segment with endpoints at a pre-image point and the image of that point after a reflection.

focal point

A location, floating or fixed, where all rigging is directed for anchor points.

Function (geometry)

A special relationship where each input has a single output. It is often written as "f(x)" where x is the input value. Example: f(x) = x/2 ("f of x equals x divided by 2") It is a function because each input "x" has a single output "x/2": • f(2) = 1.

Reflection

A transformation that "flips" a figure over a line of reflection

rigid motion

A transformation that preserves distance and angle measures

Translation

A transformation that slides each point of a figure the same distance in the same direction.

Rotation

A transformation that turns a figure about a fixed point through a given angle and a given direction.

What rule will a translation tell you?

A translation rule will tell you how the image has moved.

Vector

A vector represents not only the magnitude of an object but also the direction. In other words, vectors are used to describe objects in motion. A vector has a fixed length.

standard form of a linear equation

Ax + By = C

How does the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions?

Congruence triangles have the same size and shape Rigid motion keeps the same distance between the points that are transformed A rigid motion can be sliding,rotating, or flipping SAS-2 corresponding sides+2 corresponding angles that are congruent.

Reflect across a horizontal or vertical line

Count the distance between each point on the figure and the line of reflection. Then, for each point, count this same distance from the line of reflection to find the corresponding point.

How are translations represented as a function?

Every translation is a function of a direction and a distance, it is represented by a direction and a distance

Hypotenuse-Leg-Theorem (HL)

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Side-Side-Side (SSS)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Angle-Side-Angle (ASA)

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Side-Angle-Side (SAS)

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Can a translation be seen as a function?

It can also be seen as a function, which is a type of input/output machine.

What rule does (x.y) -> (x+1, y-2) mean?

It means all x- coordinates of the pre-image are moved 1 unit right while all y- coordinates are moved 2 units down to create the image

Reflection

Mirror image over given line

Rotation

Moving a shape around a given point (the origin is point (0,0)

What is the relationship between a reflection and a rigid motion?

Reflection is one of the four basic rigid motions. Four basic rigid motions: 1.Reflection 2.Glide Reflection 3.Rotation 4.Translation

How are reflections represented as a function?

Reflections are a type of transformation that move an entire curve such that its mirror image lies on the other side of the x or y axis. A vertical reflection is given by the equation y=−f(x) and results in the curve being "reflected" across the x-axis. A horizontal reflection is given by the equation y=f(−x) and results in the curve being "reflected" across the y-axis.

Can a reflection be seen as a function?

Remember that a reflection can also be seen as a function, which is a type of input/output machine. f you had a point at (2, 3) and wanted to reflect it using the rule (x,y) → (−x,y), you could place 2 in for x, and you would get −2 for your x-coordinate since the rule says to take the opposite of the x-coordinate. If we place 3 in for y and follow the rule, our output is 3 since the y-coordinate remains unchanged. Our new coordinate would be (−2, 3).

slope

Rise over run

How are rotations represented as a function?

Rotation Rule for 90 Degrees Clockwise Around Origin: (x, y) —>(y, -x) Rotation Rule for 90 Degrees Counterclockwise Around Origin: (x, y) —>(-y, x) Rotation Rule for 180 Degrees Around Origin: (x, y)—>(-x, -y)

When an image is rotated around the origin of the coordinate plane, there are three rules that may be applied to the pre-image in order to find the position of the image. What are the three rules?

Rotation of 90° clockwise: (x, y) → (y, −x) Rotation of 90° counterclockwise: (x, y) → (−y, x) Rotation of 180°: (x, y) → (−x, −y)

How does a rotational symmetry happen?

Rotational symmetry happens when a figure turns around its center point, and the image looks like the pre-image before turning a full 360 degrees.

What is the relationship between a rotation and a rigid motion?

Rotations are rigid motions.

Translation

Sliding all the points of a shape according to given directions (up/down, right / left)

rotational symmetry

The ability of a figure to be rotated less than a full circle and exactly match its original image

Angle of Rotation

The amount of rotation (in degrees) of a figure about a fixed point such as the origin.

How can the angle of rotation be found?

The angle of rotation can be found by dividing 360 degrees by the order of rotation.

Transformation

The mapping, or movement, of all points of a figure in a plane according to a common operation, such as translation, reflection or rotation.

The number of times the image looks like the pre-image is known as what?

The order of rotation

fixed point

The performing of arithmetical calculations without regard to the position of the radix point. The relative position of the point has to be controlled during calculations.

Image

The result of a transformation.

How do you connect the ideas of congruency and rigid motion?

The rigid motion concept can be used to define congruency. Congruency is a relation between shapes or figures. One shape is congruent to another if, and only if, you can turn the one shape into the other with a rigid motion.

Dilation

Transformation that changes the size of a figure, but not the shape.

What is the relationship between a translation and a rigid motion?

Translation is a form of rigid motion. it involves the action of moving an object to a different point without altering its shape and size. Reflection, rotation, translation and glide reflection are the four types of rigid motion that exist.

Line x=2

X = lines are always vertical

Line y =2

Y= lines are always horizontal

Isometry

a distance preserving length and angles; map of a geometric figure to another location using a reflection, rotation or translation. M -> M' indicates an isometry of the figure M to a new location M'. M and M' remain congruent.

corresponding parts

matching parts of congruent polygons

clockwise rotation

rotating in the direction that clock hands move, to the right

counterclockwise rotation

rotating opposite the direction that clock hands move, to the left

If you place 2 in for x and then, by following the rule of x + 1, add 1 to the x-coordinate, what is the output?

the output is 2 + 1 which equals 3.

If you place 3 in for y and then, by following the rule of y − 2, subtract 2 from the y-coordinate, what is the output?

the output is 3 − 2 which equals 1.

Primes

the small hash mark that you put next to the point labels of the image (ie: A')

translational symmetry

the type of symmetry for which there is a translation that maps a figure onto itself

point slope form

y - y1 = m(x - x1)

slope-intercept form

y = mx + b

If you had a point at (2, 3) and wanted to translate it using the rule (x + 1, y − 2) how would you do it?

you first use the x-coordinate rule of x+1 as the input to find the output.