Understanding Rotations

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Lesson 19 Understanding Rotations 8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G., 8.G.3 1 Getting the idea A rotation is a tpe of transformation in which ou turn a figure about a fied point. The image formed b a rotation is congruent to the pre-image, the original figure, because the figure s size and shape do not change. Each point of the image is distinguished from those of the pre-image b using a prime smbol (9). CLOCKWISE COUNTERCLOCKWISE B A Center of Rotation C C 180 B 70 Center of Rotation A C C A B A B The direction of rotation can be clockwise or counterclockwise. Each point on the figure is rotated the same degree and direction around the center of rotation, indicated b a dot in the diagram above. To rotate a figure, hold the dot fied while turning the rest of the figure. If the center of rotation is outside the figure, first draw a segment between an verte and the dot, and then turn the segment around the dot. Eample 1 Which figure shows a rotation of nklm? K M Strateg Figure A L L K M K M Figure B L K M L Figure C K M K M L L K M Figure D K L Compare each image to the pre-image. Then decide which shows a rotation. M L Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 00 Domain : Geometr

Step 1 Compare each image to the pre-image. In Figure A, a congruent image is formed b flipping nklm over a line. In Figure B, a congruent image is formed b sliding nklm to a different location. In Figure C, a congruent image is formed b turning nklm about a point. In Figure D, the image is formed b sliding nklm to a different location and b reducing its size. The shape is the same, but the image is not congruent to the pre-image. Step Identif which of the four figures shows a rotation of the pre-image. Figure C shows a rotation because the pre-image is turned about a center of rotation at point L to form a congruent image. Solution The image in Figure C is a rotation of nklm. Eample nd9e9f9 is formed b a counterclockwise 58 rotation of ndef about the point indicated b a dot. Verif that the rotation produced a congruent image b showing that (a) the sides of the image and the pre-image are congruent and (b) the angles of the image and the pre-image are congruent. F D D E Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Strateg E Use a ruler to show that the sides are congruent. Use a protractor to show that the angles are congruent. Step 1 Use a ruler to show that corresponding sides of the image and the pre-image are congruent. DE 5 1.7 cm and D9E9 5 1.7 cm, so DE > D9E9 EF 5 3.1 cm and E9F9 5 3.1 cm, so EF > E9F9 FD 5 3.8 cm and F9D9 5 3.8 cm, so FD > F9D9 Corresponding sides of ndef and nd9e9f9 are congruent. F Lesson 19: Understanding Rotations 01

Step Use a protractor to show that corresponding angles of the image and the pre-image are congruent. m/d 5 508 and m/d9 5 508, so /D > /D9 m/e 5 1058 and m/e9 5 1058, so /E > /E9 m/f 5 58 and m/f9 5 58, so /F > /F9 Corresponding angles of ndef and nd9e9f9 are congruent. Solution ndef > nd9e9f9 because corresponding sides are congruent and corresponding angles are congruent. Eample 3 Show that trapezoid ABCD and its image after a clockwise 1308 rotation about point A, trapezoid A9B9C9D9, are congruent. C D B A A B C Strateg Use a ruler to measure side lengths and compare. Use a protractor to measure angles and compare. Step 1 Use a ruler to show that corresponding sides of the image and the pre-image are congruent. AB 5.7 cm and A9B9 5.7 cm, so AB > A9B9 BC 5 1.3 cm and B9C9 5 1.3 cm, so BC > B9C9 CD 5 1.8 cm and C9D9 5 1.8 cm, so CD > C9D9 DA 5 3.7 cm and D9A9 5 3.7 cm, so DA > D9A9 D Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 0 Domain : Geometr

Step Use a protractor to show that corresponding angles of the image and the pre-image are congruent. m/a 5 08 and m/a9 5 08, so /A > /A9 m/b 5 108 and m/b9 5 108, so /B > /B9 m/c 5 1008 and m/c9 5 1008, so /C > /C9 m/d 5 808 and m/d9 5 808, so /D > /D9 Solution Trapezoid ABCD and its image, trapezoid A9B9C9D9, are congruent because their corresponding sides and angles are congruent. You can use coordinates to rotate a figure on the coordinate plane. The table below summarizes how to rotate a figure 908, 1808, or 708 around the origin. Rotation about the Origin 90 180 70 Clockwise (, ) (, ) (, ) (, ) (, ) (, ) Counterclockwise (, ) (, ) (, ) (, ) (, ) (, ) Eample Rotate rectangle QRST 708 clockwise about the origin. Label the image Q9R9S9T9. Verif that the pre-image and the image are congruent. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Strateg 0 Q T R S Determine the coordinates of the vertices of the pre-image and the image. Find the side lengths of the pre-image and the image and compare. Step 1 Determine the coordinates of each verte of the pre-image. Q(, 3) R(1, 3) S(1, ) T(, ) Step Determine the coordinates of each verte of the image using the table. Rotating a figure 708 clockwise about the origin changes (, ) to (, ). Q9(3, ) R9(3, 1) S9(, 1) T9(, ) Lesson 19: Understanding Rotations 03

Step 3 Draw the rotated figure on the coordinate grid. 0 Q R R S T S Q T Step Find the side lengths of the pre-image and the image and compare. Notice that for each horizontal side of both rectangles, the -coordinate of both vertices is the same, but the -coordinate is different. For each vertical side, the -coordinate of both vertices is the same, but the -coordinate is different. In each case, the side length is the difference in the coordinates. QR 5 Q9R9 5 5 units RS 5 R9S9 5 3 units ST 5 S9T9 5 5 units TQ 5 T9Q9 5 3 units Step 5 Find the measure of each angle and compare. All of the angles in rectangles QRST and Q9R9S9T9 are right angles. Corresponding angles are congruent. Solution The ordered pairs for the image Q9R9S9T9 are Q9(3, ), R9(3, 1), S9(, 1), and T9(, ). Corresponding sides and angles are congruent, so the pre-image and image are congruent. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 0 Domain : Geometr

COACHED EXAMPLE Rotate rectangle EFGH 908 counterclockwise about the origin to form image E9F9G9H9. Then verif that the image and the pre-image are congruent. E F H G 0 Write the coordinates of each verte of the pre-image rectangle: E (, ) F (, ) G (, ) H (, ) Rotating a figure 908 counterclockwise about the origin changes (, ) to (, ). Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Write the coordinates of each verte of the rotated figure: E9(, ) F9(, ) G9(, ) H9(, ) Plot each of these four points on the coordinate grid above. Then draw line segments between the points to form the image rectangle. Find the lengths of each side, and compare. EF 5 5 units FG 5 5 units GH 5 5 units HE 5 5 units Compare the measures of corresponding angles. All of the angles are angles, so corresponding angles are. The pre-image and the image are congruent because. Lesson 19: Understanding Rotations 05

3 LESSON PRACTICE 1 The figure shows npqr rotated to produce np9q9r9. Select True or False for each statement. 0 8 P P R Q 8 R Q A. npqr is rotated 58 about the origin. True False B. /Q > /Q9 True False C. The center of rotation is at point P. True False D. PR > P9R9 True False Anton and Kiera both tried to rotate ntuv b 1808. The drew the triangles shown on the grid. U Anton 0 Kiera T V Part A Who correctl rotated ntuv 1808? Part B What are the coordinates of the center of rotation for the 1808 rotation? Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 0 Domain : Geometr

3 The vertices of a triangle on the coordinate plane are (1, 1), (, ), and (3, 1). What are the coordinates of the image triangle produced b each of the following rotations? Part A a 908 clockwise rotation about the origin (, ), (, ), (, ) Part B a 708 clockwise rotation about the origin (, ), (, ), (, ) Part C a 1808 counterclockwise rotation about the origin (, ), (, ), (, ) The figure shows the rotation of quadrilateral CDEF to form quadrilateral C9D9E9F9. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 0 F C D E C F D E Describe two was ou could rotate quadrilateral CDEF to form quadrilateral C9D9E9F9. Lesson 19: Understanding Rotations 07

5 Draw the image of njkl b rotating it 908 clockwise about the origin. What are the coordinates of the image triangle? K 0 J L The figure shows rectangle PQRS and its image after a rotation, rectangle P9Q9R9S9. P Q Q R 0 S R P S Verif that rectangle PQRS > rectangle P9Q9R9S9. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 08 Domain : Geometr

7 Use a protractor and a ruler to draw the image formed b rotating n ABC in the figure below 908 clockwise about the center of rotation indicated b the dot. B A C 8 The figure shows trapezoid WXYZ. W X Z Y 0 Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Part A What are the coordinates of the image W9X9Y9Z9 produced if WXYZ is rotated 908 clockwise about the origin? Part B W9(, ) X9(, ) Y9(, ) Z9(, ) What are the coordinates of a second image produced b rotating W9X9Y9Z9 1808 clockwise about the origin? W 0(, ) X 0(, ) Y 0(, ) Z 0(, ) Lesson 19: Understanding Rotations 09

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