12-2 Translations Holt Geometry I CAN I CAN - Translate figures on the coordinate plane - Translate figures on the coordinate plane -Can convert between vector notation and coordinate 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 same direction.

Translations using vector notation A vector is a set of directions telling a point how to move. - Denoted with angle brackets -Tells movement in x direction, then y direction Example: means to move 2 units to the left and 4 units up.

Example: Sketch the segment AB with A(1,-3) and B(-2,0). Translate this segment along the vector A B x movement is 3 units to the left y movement is 5 units up A' B'

Translations using arrow notation in arrow notation would say (x, y) (x – 3, y + 5) Examples: Turn into arrow notation1. (x,y) (x + 2, y – 1) 2. Turn (x, y) (x + 3, y) into vector notation

Example: Drawing Translations in the Coordinate Plane Translate the triangle with vertices D(–3, –1), E(5, –3), and F(–2, –2) along the vector. 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! Example Translate the quadrilateral with vertices R(2, 5), S(0, 2), T(1,–1), and U(3, 1) along the vector. 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 U R’ S’ T’ U’

12-3 Rotations Holt Geometry I CAN I CAN Rotate 90º, 180º, and 270º around the origin Rotate 90º, 180º, and 270º around the origin Determine the angle of rotation Determine 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 appears to be flipped. Yes; the figure appears to 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. Count out to your new coordinates, and write them down somewhere. -Return paper to original orientation and plot those new points.

Do the following example on your white board Rotate triangle ABC with vertices A(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 vertices D(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 board Graph 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.