9.1 – Translate Figures and Use Vectors

Transformation: Moves or changes a figure

Preimage: Original figure

Image: Transformed figure

Isometry: A congruent transformation

“P prime”Point becomes 'P P

Translation:

An isometry that moves every point a certain distance in a certain direction

P

Q

'P

'Q

Translation:

Note: ' 'PP QQ and ' 'PP QQ≅

Motion Rule:

Moves each point left, right, down, or up

( )( , ) #, #x y x y→ + +

Left or

Right

Downor Up

Use the translation ( )( , ) 2, 5x y x y→ + −

What is the image of D(4, 7)?

D (4 + 2, 7 – 5)

(6, 2)

'D =

'D =

'D

+2

–5

Use the translation ( )( , ) 7, 4x y x y→ − +

What is the image of R(2, –4)?

R

(2 – 7, –4 + 4)

(–5, 0)

'R =

'R =

'R

–7

+4

Use the translation ( )( , ) 4, 6x y x y→ + −

What is the preimage of (–5, 3)?

M(–5 – 4, 3 + 6)

(–9, 9)

M =

M =

'M–4

+6

'M

Use the translation ( )( , ) 5, x y x y→ −

What is the preimage of (4, –1)?

A

(4 + 5, –1)

(9, –1)

A=

A=

'A +5

'A

The vertices of ABC are A(–1, 1), B(4, –1), and C(2, 4). Graph the image of the triangle using prime notation.

( )( , ) 3, 5x y x y→ − +

A

B

C

'C

'A'B

(–1 – 3, 1 + 5)

(–4, 6)

'A =

'A =

(4 – 3, –1 + 5)

(1, 4)

'B =

'B =

(2 – 3, 4 + 5)

(–1, 9)

'C =

'C =

The vertices of ABC are A(–1, 1), B(4, –1), and C(2, 4). Graph the image of the triangle using prime notation.

( )( , ) , 3x y x y→ −

A

B

C

'C

'A'B

(–1 , 1 – 3)

(–1, –2)

'A =

'A =

(4, –1 – 3)

(4, –4)

'B =

'B =

(2, 4 – 3)

(2, 1)

'C =

'C =

is the image of ABC after a translation. Write a rule for the translation.

' ' 'A B C

( )( , ) 5, 3x y x y→ − +

–5

+3

is the image of ABC after a translation. Write a rule for the translation.

' ' 'A B C

( )( , ) 2, 5x y x y→ + −

+2

–5

Vector:

Translates a shape in direction and magnitude, or size.

FGWritten:

Where F is the initial point and G is the terminal point.

Component form:

Vector:

< x, y >

5, 3

Name the vector and write its component form.

5, 1−

+5–1

JD

Name the vector and write its component form.

7, 3− −

–7

–3

DR

Name the vector and write its component form.

0, 4−

–4

RS

Use the point P(5, –2). Find the component form of the vector that describes the translation to '.P

P

'(2, 0)P

'P

–3+2

3, 2−

Use the point P(5, –2). Find the component form of the vector that describes the translation to '.P

P'P

–10–2

10, 2− −

'( 5, 4)P − −

Find the value of each variable in the translation.

a = 80°

2b = 8b = 4

5d = 100d = 20°

c = 13

Find the value of each variable in the translation.

a = 180 – 90 – 31

a = 59°

b – 5 = 12b = 17

3c + 2 = 203c = 18

c = 6

9.3 – Perform Reflections

Reflection:

Transformation that uses a line like a mirror to reflect an image

Line of Reflection:

Mirror line in a reflection

A reflection in a line m maps every point P in the plane to a point , such that:

• If P is not on m, then m is the perpendicular bisector of

• If P is on m, then

'.PP

'P

'P P=

P

Reflect point P(5, 7) in the given line.

'P

P(5, 7) becomes'(5, 7)P −

A reflection in the x-axis changes (x, y) into _______(x, –y)

x – axis

P

Reflect point P(5, 7) in the given line.

'P

P(5, 7) becomes'( 5, 7)P −

A reflection in the y-axis changes (x, y) into _______(–x, y)

y – axis

P

Reflect point P(5, 7) in the given line.

'P

P(5, 7) becomes'(7, 5)P

A reflection in the y = x changes (x, y) into _______(y, x)

y = x

Graph the reflection of the polygon in the given line.

x – axis

'C

'A

'B

Graph the reflection of the polygon in the given line.

y – axis

'C

'A

'B

'D

'C

'A

'B

Graph the reflection of the polygon in the given line.

y = x

(–1 , –3)

(–3, –1)

A=

'A =

(2, –4 )

(–4, 2)

B =

'B =

(3, 0)

(0, 3)

C =

'C =

Graph the reflection of the polygon in the given line.

x – axis

'C

'A

'B

'D

Graph the reflection of the polygon in the given line.

x = –1

'C'A

'B

Graph the reflection of the polygon in the given line.

y = 2'C

'A =

'B

'D=