Transformations and the Coordinate Plane Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin.

Transformations and the Coordinate Plane

Eleanor Roosevelt High School

Geometry

Mr. Chin-Sung Lin

The Coordinates of a Point in a Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Two intersecting lines determine a plane. The coordinate plane is determined by a horizontal line, the x-axis, and a vertical line, the y-axis, which are perpendicular and intersect at a point called the origin

X

Y

O

Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Every point on a plane can be described by two numbers, called the coordinates of the point, usually written as an ordered pair (x, y)

X

(x, y)Y

O

Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

The x-coordinate or the abscissa, is the distance from the point to the y-axis. The y-coordinate or the ordinate is the distance from the point to the x-axis. Point O, the origin, has the coordinates (0, 0)

X

(x, y)Y

O (0, 0)x

y

Postulates of Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Two points are on the same horizontal line if and only if they have the same y-coordinates

X

(x2, y)Y

O

(x1, y)


ERHS Math Geometry

Mr. Chin-Sung Lin

The length of a horizontal line segment is the absolute value of the difference of the x-coordinates

d = |x2 – x1|

X

(x2, y)Y

O

(x1, y)


ERHS Math Geometry

Mr. Chin-Sung Lin

Two points are on the same vertical line if and only if they have the same x-coordinates

X

(x, y2)

Y

O

(x, y1)


ERHS Math Geometry

Mr. Chin-Sung Lin

The length of a vertical line segment is the absolute value of the difference of the y-coordinates

d = |y2 – y1|

X

(x, y2)

Y

O

(x, y1)


ERHS Math Geometry

Mr. Chin-Sung Lin

Each vertical line is perpendicular to each horizontal line

X

Y

O

Locating a Point in the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

1. From the origin, move to the right if the x-coordinate is positive or to the left if the x-coordinate is negative. If it is 0, there is no movement

2. From the point on the x-axis, move up if the y-coordinate is positive or down if the y-coordinate is negative. If it is 0, there is no movement

X

(x, y)Y

Ox

y

Finding the Coordinates of a Point

ERHS Math Geometry

Mr. Chin-Sung Lin

1. From the point, move along a vertical line to the x-axis.The number on the x-axis is the x-coordinate of the point

2. From the point, move along a horizontal line to the y-axis.The number on the y-axis is the y-coordinate of the point

X

(x, y)Y

O x

y

Graphing on the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Graph the following points: A(4, 1), B(1, 5), C(-2,1). Then draw ∆ ABC and find its area

X

Y

O


ERHS Math Geometry

Mr. Chin-Sung Lin


A (4, 1)C (-2, 1)

B (1, 5)

D (1, 1)X

Y

O


ERHS Math Geometry

Mr. Chin-Sung Lin


AC = | 4 – (-2) | = 6

BD = | 5 – 1 | = 4

Area = ½ (AC)(BD)

= ½ (6)(4) = 12

A (4, 1)C (-2, 1)

B (1, 5)

D (1, 1)X

Y

O

Line Reflections

ERHS Math Geometry

Mr. Chin-Sung Lin

Line Reflections

ERHS Math Geometry

Mr. Chin-Sung Lin

Y

Line of Reflection

Line Reflection (Object & Image)

Transformation

ERHS Math Geometry

Mr. Chin-Sung Lin

A one-to-one correspondence between two sets of points, S and S’, such that every point in set S corresponds to one and only one point in set S’, called its image, and every point in S’ is the image of one and only one point in S, called its preimage

S S’

A Reflection in Line k

ERHS Math Geometry

Mr. Chin-Sung Lin

1. If point P is not on k, then the image of P is P’ where k is the perpendicular bisector of PP’

2. If point P is on k, the image of P is P

P’

kP

P

Theorem of Line Reflection - Distance

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a line reflection, distance is preserved

Given: Under a reflection in line k, the image of A is A’ and the image of B is B’

Prove: AB = A’B’

B’

k

B

A’A


ERHS Math Geometry

Mr. Chin-Sung Lin




B’

k

B

A’AC

D


ERHS Math Geometry

Mr. Chin-Sung Lin




B’

k

B

A’A

SASC

D


ERHS Math Geometry

Mr. Chin-Sung Lin




B’

k

B

A’A

CPCTCC

D


ERHS Math Geometry

Mr. Chin-Sung Lin




B’

k

B

A’A

SASC

D


ERHS Math Geometry

Mr. Chin-Sung Lin




B’

k

B

A’A

CPCTCC

D


ERHS Math Geometry

Mr. Chin-Sung Lin

Since distance is preserved under a line reflection, the image of a triangle is a congruent triangle

B’

k

B

A’A

C C’

M’M

D D’

SSS

Corollaries of Line Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a line reflection, angle measure is preserved

Under a line reflection, collinearity is preserved

Under a line reflection, midpoint is preserved

B’

k

B

A’A

C C’

M’M

D D’

Notation of Line Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

We use rk as a symbol for the image under a reflection in line k

rk (A) = A’

rk (∆ ABC ) = ∆ A’B’C’

B’

k

B

A’A

C C’

Construction of Line Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

If rk (AC) = A’C’, construct A’C’

k

A

C


ERHS Math Geometry

Mr. Chin-Sung Lin

Construct the perpendicular line from A to k. Let the point of intersection be M

k

A

C

M


ERHS Math Geometry

Mr. Chin-Sung Lin

Construct the perpendicular line from C to k. Let the point of intersection be N

k

A

CN

M


ERHS Math Geometry

Mr. Chin-Sung Lin

Construct A’ on AM such that AM = A’M

Construct C’ on CN such that CN = C’N

k

A’A

C C’

N

M


ERHS Math Geometry

Mr. Chin-Sung Lin

Draw A’C’

k

A’A

C C’

N

M

Line Symmetry in Nature

ERHS Math Geometry

Mr. Chin-Sung Lin

Line Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

A figure has line symmetry when the figure is its own image under a line reflection

This line of reflection is a line of symmetry, or an axis of symmetry

Line Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

It is possible for a figure to have more than one axis of symmetry

Line Reflections in the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Reflection in the y-axis

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a reflection in the y-axis, the image of P(a, b) is P’(-a, b)

y

Ox

P(a, b)

Q(0, b)

P’(-a, b)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the y-axis, where A(-3, 3), B(-4, 1), and C(-1, 1), draw ry-axis (∆ ABC ) = ∆ A’B’C’

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the y-axis, where A(-3, 3), B(-4, 1), and C(-1, 1), draw ry-axis (∆ ABC ) = ∆ A’B’C’

y

Ox

B’(4, 1)

A’(3, 3)

C’(1, 1)B(-4, 1)

A(-3, 3)

C(-1, 1)

Reflection in the x-axis

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a reflection in the x-axis, the image of P(a, b) is P’(a, -b)

y

Ox

P(a, b)

Q(a, 0)

P’(a, -b)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the x-axis, where A(3, 3), B(4, 1), and C(1, 1), draw rx-axis (∆ ABC ) = ∆ A’B’C’

y

Ox

B(4, 1)

A(3, 3)

C(1, 1)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the x-axis, where A(3, 3), B(4, 1), and C(1, 1), draw rx-axis (∆ ABC ) = ∆ A’B’C’

y

Ox

B(4, 1)

A(3, 3)

C(1, 1)

A’(3, -3)

B’(4, -1)C’(1, -1)

Reflection in the Line y = x

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a reflection in the y = x, the image of P(a, b) is P’(b, a)

y

O x

P(a, b)

Q(a, a)

P’(b, a)

R(b, b)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the x-axis, where A(2, 2), B(1, 4), and C(-1, 1), draw ry=x (∆ ABC ) = ∆ A’B’C’

y

O x

B(1, 4)

A(2, 2)

C(-1, 1)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the x-axis, where A(2, 2), B(1, 4), and C(-1, 1), draw ry=x (∆ ABC ) = ∆ A’B’C’

* Point A is a fixed point since it is on the line of reflection

y

O x

B(1, 4)

A(2, 2)=A’(2, 2)

C(-1, 1) B’(4, 1)

C’(1, -1)

Point Reflections in the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

A Point Reflection in P

ERHS Math Geometry

Mr. Chin-Sung Lin

1. If point A is not point P, then the image of A is A’ and P the midpoint of AA’

2. The point P is its own imagey

Ox

A

A’

P

Theorem of Point Reflections

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a point reflection, distance is preserved

y

Ox

A

P

B

B’

A’


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: Under a reflection in point P, the image of A is A’ and the image of B is B’

Prove: AB = A’B’y

Ox

A

P

B

B’

A’


ERHS Math Geometry

Mr. Chin-Sung Lin



Ox

A

P

B

B’

A’

SAS


ERHS Math Geometry

Mr. Chin-Sung Lin



Ox

A

P

B

B’

A’

CPCTC

Properties of Point Reflections

ERHS Math Geometry

Mr. Chin-Sung Lin

1. Under a point reflection, angle measure is preserved

2. Under a point reflection, collinearity is preserved

3. Under a point reflection, midpoint is preservedy

Ox

A

A’

P

Notation of Point Reflections

ERHS Math Geometry

Mr. Chin-Sung Lin

We use Rp as a symbol for the image under a reflection in point P

Rp (A) = B means “The image of A under a reflection in point P is B.”

R(1,2) (A) = A’ means “The image of A under a reflection in point (1, 2) is A’.”

Point Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

A figure has point symmetry if the figure is its own image under a reflection in a point

Point Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

Other examples of figures that have point symmetry are letters such as S and N and numbers such as 8

S N 8

Reflection in the Origin

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a reflection in the origin, the image of P(a, b) is P’(-a, -b)

RO (a, b) = (-a, -b)

y

O

x

P(a, b)

P’(-a, -b)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the origin, where A(-3, 3), B(-4, 1), and C(-1, 1), draw RO (∆ ABC ) = ∆ A’B’C’

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)


ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC is reflected in the origin, where A(-3, 3), B(-4, 1), and C(-1, 1), draw RO (∆ ABC ) = ∆ A’B’C’

y

Ox

B’(4, -1)

A’(3, -3)

C’(1, -1)

B(-4, 1)

A(-3, 3)

C(-1, 1)

Reflection in the point

ERHS Math Geometry

Mr. Chin-Sung Lin

(A) What are the coordinates of B, the image of A(-3, 2) under a reflection in the origin?

(B) What are the coordinates of C, the image of A(-3, 2) under a reflection in the x-axis?

(C) What are the coordinates of D, the image of C under a reflection in the y-axis?

(D) Does a reflection in the origin give the same result as a reflection in the x-axis followed by a reflection in the y-axis? Justify your answer.

Translations in the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

A translation is a transformation of the plane that moves every point in the plane the same distance in the same direction

y

Ox

B’

B

C’

C

A’

A

Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

In the coordinate plane, the distance is given in terms of horizontal distance (change in the x-coordinates) and vertical distance (change in the y-coordinates)

y

Ox

B’

B

C’

C

A’

A

x-coor. y-coor.

Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

A translation of a units in the horizontal direction and b units in the vertical direction is a transformation of the plane such that the image of P(x, y) is P’(x + a, y + b)

y

x

P’(x + a, y + b)

P(x, y)a

b

Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

The image of P(x, y) is P’(x + a, y + b),

if the translation moves a point to the right, a > 0

if the translation moves a point to the left, a < 0

if the translation moves a point up, b > 0

if the translation moves a point down, b < 0

y

x

P’(x + a, y + b)

P(x, y)a

b

Theorem of Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a translation, distance is preserved

y

Ox

B

A

B’

A’


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: A translation in which the image of A(x1,y1) is A’(x1+a, y1+b) and the image of B(x2, y2) is B’(x2+a, y2+b)


y

Ox

A’ (x1+a, y1+b)

B (x2, y2)

A (x1, y1)B’ (x2+a, y2+b)


ERHS Math Geometry

Mr. Chin-Sung Lin



y

Ox

B (x2, y2)

A (x1, y1)B’ (x2+a, y2+b)

A’ (x1+a, y1+b)


ERHS Math Geometry

Mr. Chin-Sung Lin



y

Ox

B’ (x2+a, y2+b)

A’ (x1+a, y1+b)

|x1-x2|

|x1-x2||y1-y2|

|y1-y2|

B (x2, y2)

A (x1, y1)


ERHS Math Geometry

Mr. Chin-Sung Lin



y

Ox

B (x2, y2)

A (x1, y1)B’ (x2+a, y2+b)

A’ (x1+a, y1+b)

|x1-x2|

|x1-x2||y1-y2|

|y1-y2|

SAS & CPCTC

Properties of Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

1. Under a translation, angle measure is preserved

2. Under a translation, collinearity is preserved

3. Under a translation, midpoint is preserved

y

Ox

A’ (x1+a, y1+b)

B (x2, y2)

A (x1, y1)B’ (x2+a, y2+b)

Notation of Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

We use Ta, b as a symbol for the image under a translation of a units in the horizontal direction and b units in the vertical direction

Ta, b (x, y) = (x + a, y + b)

Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under T7,1

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

Translation

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under T7,1

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

B’(3, 2)

A’(4, 4)

C’(6, 2)

Translational Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

A figure has translational symmetry if the image of every point of the figure is a point of the figure

Translational Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

Rotations in the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

A rotation is a transformation of a plane about a fixed point P through an angle of d degrees such that:

1. For A, a point that is not the fixed point P, if the image of A is A’, then PA = PA’ and m APA’ = d

2. The image of the center of rotation P is P

y

O x

P

A’

Ad

Theorem of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

Distance is preserved under a rotation about a fixed point

P

A’

A

B’B

Theorem of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

Given: P is the center of rotation. If A is rotated about P to A’, and B is rotated the same number of degrees to B’


P

A’

A

B’B

d

d

Theorem of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin



P

A’

A

B’B mAPA’ = mBPB’

mAPB = mA’PB’

Theorem of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin



P

A’

A

B’B

SAS

Theorem of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin



P

A’

A

B’B

CPCTC

Properties of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

1. Under a rotation, angle measure is preserved

2. Under a rotation, collinearity is preserved

3. Under a rotation, midpoint is preserved

P

A’

A

B’B

Notation of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

We use RP, d as a symbol for the image under a rotation of d degrees about point P

A rotation in the counterclockwise direction is called a positive rotation

A rotation in the clockwise direction is called a negative rotation

RO, 30o (A) = B the image of A under a rotation of 30° degrees about the origin is B

Notation of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

The symbol R is used to designate both a point reflection and a rotation

1. When the symbol R is followed by a letter that designates a point, it represents a reflection in that point (e.g., RP)

2. When the symbol R is followed by both a letter that designates a point and the number of degrees, it represents a rotation of the given number of degrees about the given point (e.g., RO, 30o)

3. When the symbol R is followed by the number of degrees, it represents a rotation of the given number of degrees about the origin (e.g., R90o)

Theorem of Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a counterclockwise rotation of 90° about the origin, the image of P(a, b) is P’(–b, a)

RO,90°(x, y) = (-y, x) or R 90°(x, y) = (-y, x)

Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under RO,90o

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under RO,90o

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

A’(-3, -3)

B’(-1, -4)

C’(-1, -1)

Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A”B”C”, the image of ∆ ABC under RO,180o

y

x

B(-4, 1)

A(-3, 3)

C(-1, 1)

Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin


y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

A’(-3, -3)

B’(-1, -4)

C’(-1, -1) C”(1, -1)

A”(3, -3)

B”(4, -1)

Rotation

ERHS Math Geometry

Mr. Chin-Sung Lin


y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

C”(1, -1)

A”(3, -3)

B”(4, -1)

Rotation 180o = Point Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

∆ A”B”C”, the image of ∆ ABC under RO,180o is the same as the image of ∆ ABC under point reflection RO

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

C”(1, -1)

A”(3, -3)

B”(4, -1)

Rotational Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

A figure is said to have rotational symmetry if the figure is its own image under a rotation and the center of rotation is the only fixed point

Rotational Symmetry

ERHS Math Geometry

Mr. Chin-Sung Lin

Many letters, as well as designs in the shapes of wheels, stars, and polygons, have rotational symmetry

S H 8Z N

Glide Reflections in the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Composition of Transformations

ERHS Math Geometry

Mr. Chin-Sung Lin

When two transformations are performed, one following the other, we have a composition of transformations

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

A glide reflection is a composition of transformations of the plane that consists of a line reflection and a translation in the direction of the line of reflection performed in either order

y

x

BB’

AA’

C’ C

B”

A”

C”

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

A glide reflection is a composition of transformations of the plane that consists of a line reflection and a translation in the direction of the line of reflection performed in either order y

x

B

B’

A

A’

C’

C

B”

A”

C”

Theorem of Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a glide reflection, distance is preserved

y

x

BB’

AA’

C’ C

B”

A”

C”

Properties of Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

1. Under a glide reflection, angle measure is preserved

2. Under a glide reflection, collinearity is preserved

3. Under a glide reflection, midpoint is preserved

y

x

BB’

AA’

C’ C

B”

A”

C”

Isometry

ERHS Math Geometry

Mr. Chin-Sung Lin

An isometry is a transformation that preserves distance

All five transformations:

1. line reflection,

2. point reflection,

3. translation,

4. rotation, and

5. glide reflection.

Each of these transformations is called an isometry

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under ry-axis, and ∆ A”B”C”, the image of ∆ A’B’C’ under T0, –4

y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin


y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

A’(3, 3)

B’(4, 1)C’(1, 1)

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin


y

Ox

B(-4, 1)

A(-3, 3)

C(-1, 1)

A’(3, 3)

B’(4, 1)C’(1, 1)

A”(3, –1)

B”(4, –3)C”(1, – 3)

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

The vertices of ∆PQR are P(2, 1), Q(4, 1), and R(4, 3)

1. Find ∆P’Q’R’, the image of ∆PQR under ry=x followed by T–3, –3

2. Find ∆P”Q”R”, the image of ∆PQR under T–3, –3

followed by ry=x

3. Are ∆P’Q’R’ and ∆P”Q”R” the same triangle?

4. Are ry=x followed by T–3, –3 and T–3, –3 followed by ry=x the same glide reflection? Explain

5. Write a rule for this glide reflection

Dilations in the Coordinate Plane

ERHS Math Geometry

Mr. Chin-Sung Lin

Dilation

ERHS Math Geometry

Mr. Chin-Sung Lin

A dilation of k is a transformation of the plane such that:

1. The image of point O, the center of dilation, is O

2. When k is positive and the image of P is P’, then OP and OP’ are the same ray and OP’ = kOP

3. When k is negative and the image of P is P’, then OP and OP’ are opposite rays and OP’ = -kOP.

y

Ox

P

P’

k > 0

P’k < 0

Notation of Dilations

ERHS Math Geometry

Mr. Chin-Sung Lin

We use Dk as a symbol for the image under a dilation of k with center at the origin

P (x, y) P’ (kx, ky) or Dk (x, y) = (kx, ky)

D2 (3, 4) = (6, 8)

Dilation

ERHS Math Geometry

Mr. Chin-Sung Lin

Under a dilation about a fix point, distance is not preserved, and angle measurement is preserved

Dilation is not an isometry

y

Ox

A

A’

B

B’

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(2, 1), B(1, 3), and C(3, 2), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under D2

y

Ox

B(1, 3)

A(2, 1)

C(3, 2)

Glide Reflection

ERHS Math Geometry

Mr. Chin-Sung Lin

If ∆ ABC has vertices A(2, 1), B(1, 3), and C(3, 2), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under D2

y

Ox

B(1, 3)

A(2, 1)

C(3, 2)

B’(2, 6)

A’(4, 2)

C’(6, 4)

Transformations as Functions

ERHS Math Geometry

Mr. Chin-Sung Lin

Functions

ERHS Math Geometry

Mr. Chin-Sung Lin

A function is a set of ordered pairs in which no two pairs have the same first element

The set of first elements is the domain of the function and the set of second elements is the range

Domain Range

Transformations as Functions

ERHS Math Geometry

Mr. Chin-Sung Lin

Transformation can be viewed as a one-to-one function

S S’

Notations of Functions

ERHS Math Geometry

Mr. Chin-Sung Lin

For example, y = x + 1 is a function f, it can represented as:

y = x + 1

f(x) = x + 1

f: x -> x + 1

f = { (x, y) | y = x + 1}

y and f(x) both represent the second element of the ordered pair


ERHS Math Geometry

Mr. Chin-Sung Lin

When two transformations are performed, one (f) following the other (g), we have a composition of transformations

y = g( f(x) ) or y = g o f


ERHS Math Geometry

Mr. Chin-Sung Lin

A’ is the image of A(2, 5) under a reflection in the line y = x followed by the translation T2,0, we can write

T2, 0 (ry = x (A)) = A’ or T2, 0 o ry = x (A) = A’

A’ = T2, 0 (ry = x (2, 5)) = T2, 0 o ry = x (2, 5)

= T2, 0 (5, 2)

= (7, 2)

Orientation

ERHS Math Geometry

Mr. Chin-Sung Lin

In a figure, the vertices, when traced from A to B to C to …. are in the clockwise or the counter-clockwise direction, called the orientation of the points

A

C

B Clockwise Orientation

Direct Isometry

ERHS Math Geometry

Mr. Chin-Sung Lin

A direct isometry is a transformation that preserves distance and orientation

The following three transformations:

1. point reflection,

2. translation, and

3. rotation

each of these transformations is direct isometry

Opposite Isometry

ERHS Math Geometry

Mr. Chin-Sung Lin

An opposite isometry is a transformation that preserves distance , but changes the orientation

The following two transformations:

1. line reflection, and

2. glide reflection

each of these transformations is opposite isometry

Q & A

ERHS Math Geometry

Mr. Chin-Sung Lin

The End

ERHS Math Geometry

Mr. Chin-Sung Lin

Transformations and the Coordinate Plane Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin.

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