12-2 Translations

Holt Geometry

I CANI CAN

- Translate figures on the coordinate plane- Translate figures on the coordinate plane-Can convert between vector notation andcoordinate notation

A translation is a transformation where all the points of a figure are moved the same distance in the same direction. A translation is an isometry, so the image of a translated figure is congruent to the preimage.

Example 1: Identifying Translations

Tell whether each transformation appears to be a translation. Explain.

No; the figure appears to be flipped.

Yes; the figure appears to slide.

A. B.

Check It Out! Example 1

Tell whether each transformation appears to be a translation.a. b.

No; not all of the points have moved the same distance.

Yes; all of the points have moved the same distance in the samedirection.

Translations using vector notation

A vector is a set of directions telling a point howto move. - Denoted with angle brackets < x, y > -Tells movement in x direction, then y direction

Example: < -2, 4 > means to move 2 units to the leftand 4 units up.

Example: Sketch the segment AB with A(1,-3) andB(-2,0). Translate this segment along the vector< -3, 5 >

A

B

x movement is 3 units to the lefty movement is 5 units up

A'

B'

Translations using arrow notation<-3, 5> in arrow notation would say(x, y) (x – 3, y + 5)

Examples:Turn < 2 , -1> into arrow notation1.(x ,y) (x + 2, y – 1)

2. Turn (x, y) (x + 3, y) into vector notation< 3, 0 >

Example: Drawing Translations in the Coordinate PlaneTranslate the triangle with vertices D(–3, –1), E(5, –3), and F(–2, –2) along the vector <3, –1>.

The image of (x, y) is (x + 3, y – 1).D(–3, –1) D’(–3 + 3, –1 – 1)

= D’(0, –2)E(5, –3) E’(5 + 3, –3 – 1)

= E’(8, –4)F(–2, –2) F’(–2 + 3, –2 – 1)

= F’(1, –3)Graph the preimage and the image.

Check It Out! ExampleTranslate the quadrilateral with vertices R(2, 5), S(0, 2), T(1,–1), and U(3, 1) along the vector <–3, –3>. The image of (x, y) is (x – 3, y – 3).R(2, 5) R’(2 – 3, 5 – 3)

= R’(–1, 2)S(0, 2) S’(0 – 3, 2 – 3)

= S’(–3, –1)T(1, –1) T’(1 – 3, –1 – 3)

= T’(–2, –4)U(3, 1) U’(3 – 3, 1 – 3)

= U’(0, –2)Graph the preimage and the image.

R

S

T

UR’

S’

T’

U’

12-3 Rotations

Holt Geometry

I CANI CAN

Rotate 90º, 180º, and 270º around the originRotate 90º, 180º, and 270º around the originDetermine the angle of rotationDetermine the angle of rotation

Remember that a rotation is a transformation that turns a figure around a fixed point, called the center of rotation. A rotation is an isometry, so the image of a rotated figure is congruent to the preimage.

Example 1: Identifying Rotations

Tell whether each transformation appears to be a rotation. Explain.

No; the figure appearsto be flipped.

Yes; the figure appearsto be turned around a point.

A. B.

Check It Out! Example 1

Tell whether each transformation appears to be a rotation.

No, the figure appears to be a translation.

Yes, the figure appears to be turned around a point.

a. b.

We will only be dealing with right-angle rotations.(90º, 180º, 270º, and 360º)

Unless otherwise stated, all rotations in this book are counterclockwise.

Helpful Hint

Easy way to do a rotation on a coordinate plane:

- Actually TURN the paper the number of degrees you require

-IGNORE the old numbering of the axes. Countout to your new coordinates, and write them downsomewhere.

-Return paper to original orientation and plot thosenew points.

Do the following example on your white board

Rotate triangle ABC with verticesA(2,-1), B(4,1), and C(3,3)by 90º about the origin.

A(2,-1) A'( , )

B(4, 1) B'( , )

C(3, 3) C'( , )

Do the following example on your white board

Rotate triangle DEF with verticesD(2,3), E(-1,2), and F(2,1)by 180º about the origin.

D(2,3) D'( , )

E(-1,2) E'( , )

F(2, 1) F'( , )

Do the following example on your white boardGraph the triangle A(2,-4), B(3,5),C(6,1); Then rotate it 270º.

A(2,-4) A'( , )

B(3, 5) B'( , )

C(6, 1) C'( , )

By 270º, (x, y) (y, –x)

Example : Drawing Rotations in the Coordinate Plane

Rotate ΔJKL with vertices J(2, 2), K(4, –5), and L(–1, 6) by 180° about the origin.

The rotation of (x, y) is (–x, –y).

Graph the preimage and image.

J(2, 2) J’(–2, –2)

K(4, –5) K’(–4, 5) L(–1, 6) L’(1, –6)

Check It Out! Example

Rotate ∆ABC by 180° about the origin.

The rotation of (x, y) is (–x, –y).A(2, –1) A’(–2, 1)B(4, 1) B’(–4, –1)C(3, 3) C’(–3, –3)

Graph the preimage and image.