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Today’s Transformation Lesson???. http:// www.youtube.com/watch?v=r5hRGpIcE5M. 9.4 Performing Rotations. A rotation is an isometry Center of rotation- a fixed point in which a figure is turned about

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  1. Today’s Transformation Lesson??? http://www.youtube.com/watch?v=r5hRGpIcE5M

  2. 9.4 Performing Rotations • A rotation is an isometry • Center of rotation- a fixed point in which a figure is turned about • Angle of Rotation- the angle formed from rays drawn from the center of rotation to a point and its image. Assume counterclockwise. • If Q is not the center of rotation , then QP=Q΄P and m∠QPQ΄= x˚ If Q is the center of rotation, then Q= Q’ R • R΄ S΄ Q Q΄ X˚ S P

  3. Draw a ray to form a 120˚angle with PA. STEP 3 DrawA′ so thatPA′= PA. • Draw a 120˚rotation of ABCabout P. STEP 2

  4. Draw a rotation STEP 4 Repeat Steps 1– 3 for each vertex. DrawA′B′C′.

  5. Preimage A(1,2), B(3,5), C (5,1) • Discovery: Determine coordinate rules for the following counterclockwise rotations: • 90˚Image • A’( ),B’( ),C’( ) • Rule______________ • 180˚Image • A’( ),B’( ),C’( ) • Rule______________ • 270˚Image • A’( ),B’( ),C’( ) • Rule_______________

  6. Coordinate Rules for Rotations • These rules apply for counterclockwise rotations about the origin • a 90o rotation (a,b) (-b,a) • a 180o rotation (a,b) (-a,-b) • a 270o rotation (a,b) (b,-a)

  7. What are the Rotation Matrices? Use to find the image of a line or polygon rotation about the origin.The rotation matrix must be first when multiplying 90˚Rotation Matrix A B C A’ B’ C’ • 1 3 5 -2 -5 -1 • X = • 2 5 1 1 3 5 90˚Rotation Matrix PreImage Matrix Image Matrix

  8. A B C 1 3 5 X 2 5 1 180˚rotation matrix Pre-Image matrix Rotation Matrices? 180˚Rotation: A΄ B΄ C ΄ -1 -3 -5 = -2 -5 -1 Image Matrix

  9. A B C 1 3 5 X 2 5 1 270˚rotation matrix Pre-Image matrix Rotation Matrices? 270˚ Rotation: A΄ B΄ C΄ 2 -5 -1 = -1 3 5 Image Matrix

  10. ANSWER • Rotate a figure 90º about the origin Graph JKLwith vertices J(3, 0), K(4, 3), and L(6, 0). Rotate the triangle 90° about the origin.

  11. Graph quadrilateral RSTUwith vertices R(3, 1),S(5, 1), T(5, –3), and U(2, –1). Then rotate the quadrilateral 270 about the origin. o o Graph RSTU. Use the coordinate rule for a 270 rotation to find the images of the vertices. (a, b) (b, –a) R′(1, –3) R(3, 1) S(5, 1) S′(1, –5) T(5, –3) T′(–3, –5) U(2, –1) U′(–1, –2) Rotate a figure using the coordinate rules SOLUTION Graph the image R′S′T′U′.

  12. Trapezoid EFGH has vertices E(–3, 2), F(–3, 4),G(1, 4), and H(2, 2). Find the image matrix for a 180 rotation of EFGH about the origin. Graph EFGHand its image. o STEP 1 Write the polygon matrix: o STEP 2 Multiply by the matrix for a 180 rotation. Use matrices to rotate a figure SOLUTION

  13. Use matrices to rotate a figure STEP 3 Graph the preimageEFGH. Graph the image E′F′G′H′.

  14. 5y 3x + 1 = 5y 3(3) + 1 = y 2 = The correct answer is B. Standardized Test Practice SOLUTION By Theorem 9.3, the rotation is an isometry, so corresponding side lengths are equal. Then 2x = 6, so x = 3. Now set up an equation to solve for y. Substitute 3 for x. Corresponding lengths in an isometry are equal. Solve for y.

  15. 6. Find the value of r in the rotation of the triangle. ANSWER The correct answer is B. More Standardized Test Practice

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