# Rotations 4.3 Essential Question MAKING

4.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.3.A G.3.B G.3.C G.3.D Rotations Essential Question How can you rotate a figure in a coordinate plane? Rotating a Triangle in a Coordinate Plane Work with a partner. a. Use dynamic geometry software to draw any triangle and label it △ABC. b. Rotate the triangle 90° counterclockwise about the origin to form △A′B′C′. c. What is the relationship between the coordinates of the vertices of △ABC and those of △A′B′C′? d. What do you observe about the side lengths Sample and angle measures of the two triangles? B′ C′ Points A(1, 3) B(4, 3) C(4, 1) D(0, 0) Segments AB = 3 BC = 2 AC = 3.61 Angles m∠A = 33.69° m∠B = 90° m∠C = 56.31° 4 A 3 B 2 1 A′ 0 MAKING MATHEMATICAL ARGUMENTS −3 y B A −3 −1 −3 −5 D 0 1 2 3 4 Rotating a Triangle in a Coordinate Plane Work with a partner. a. The point (x, y) is rotated 90° counterclockwise about the origin. Write a rule to determine the coordinates of the image of (x, y). b. Use the rule you wrote in part (a) to rotate △ABC 90° counterclockwise about the origin. What are the coordinates of the vertices of the image, △A′B′C′? c. Draw △A′B′C′. Are its side lengths the same as those of △ABC? Justify your answer. Rotating a Triangle in a Coordinate Plane 1 −5 −1 −1 To be proficient in math, you need to use previously established results in constructing arguments. 5 −2 C 5x 1 C Work with a partner. a. The point (x, y) is rotated 180° counterclockwise about the origin. Write a rule to determine the coordinates of the image of (x, y). Explain how you found the rule. b. Use the rule you wrote in part (a) to rotate △ABC (from Exploration 2) 180° counterclockwise about the origin. What are the coordinates of the vertices of the image, △A′B′C′? Communicate Your Answer 4. How can you rotate a figure in a coordinate plane? 5. In Exploration 3, rotate △A′B′C′ 180° counterclockwise about the origin. What are the coordinates of the vertices of the image, △A″B″C″? How are these coordinates related to the coordinates of the vertices of the original triangle, △ABC? Section 4.3 HSTX_GEOM_PE_04.03.indd 193 Rotations 193 6/19/14 6:04 PM 4.3 Lesson What You Will Learn Perform rotations. Perform compositions with rotations. Core Vocabul Vocabulary larry Identify rotational symmetry. rotation, p. 194 center of rotation, p. 194 angle of rotation, p. 194 rotational symmetry, p. 197 center of symmetry, p. 197 Performing Rotations Core Concept Rotations A rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form the angle of rotation. Q 40° Q′ • If Q is not the center of rotation P, then QP = Q′P and m∠QPQ′ = x°, or angle of rotation center of rotation • If Q is the center of rotation P, then Q = Q′. Direction of rotation R R′ A rotation about a point P through an angle of x° maps every point Q in the plane to a point Q′ so that one of the following properties is true. P The figure above shows a 40° counterclockwise rotation. Rotations can be clockwise or counterclockwise. In this chapter, all rotations are counterclockwise unless otherwise noted. Drawing a Rotation clockwise Draw a 120° rotation of △ABC about point P. A C counterclockwise B P SOLUTION Step 1 Draw a segment from P to A. Step 2 Draw a ray to form a 120° angle —. with PA A B Step 3 Draw A′ so that PA′ = PA. 50 60 30 40 20 150 140 130 120 70 11 0 0 1 80 10 0 16 00 0 0 17 18 C 0 18 0 P A 140 15 120 130 0 110 60 50 40 30 160 0 20 17 10 0 70 10 0 8 90 0 9 P C B Step 4 Repeat Steps 1–3 for each vertex. Draw △A′B′C′. B′ A A′ 194 Chapter 4 HSTX_GEOM_PE_04.03.indd 194 120° P C A B A′ C′ C B P Transformations 6/19/14 6:04 PM USING ROTATIONS You can rotate a figure more than 360°. The effect, however, is the same as rotating the figure by the angle minus 360°. y You can rotate a figure more than 180°. The diagram shows rotations of point A 130°, 220°, and 310° about the origin. Notice that point A and its images all lie on the same circle. A rotation of 360° maps a figure onto itself. A A′ 130° You can use coordinate rules to find the coordinates of a point after a rotation of 90°, 180°, or 270° about the origin. x 220° A‴ 310° A″ Core Concept Coordinate Rules for Rotations about the Origin When a point (a, b) is rotated counterclockwise about the origin, the following are true. • For a rotation of 90°, (a, b) → (−b, a). y (−b, a) (a, b) 90° 180° • For a rotation of 180°, (a, b) → (−a, −b). x • For a rotation of 270°, (a, b) → (b, −a). (−a, −b) 270° (b, −a) Rotating a Figure in the Coordinate Plane Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, −3), and U(2, −1) and its image after a 270° rotation about the origin. SOLUTION Use the coordinate rule for a 270° rotation to find the coordinates of the vertices of the image. Then graph quadrilateral RSTU and its image. 2 −4 (a, b) → (b, −a) −2 U′ R(3, 1) → R′(1, −3) y R S(5, 1) → S′(1, −5) T′ −6 U(2, −1) → U′(−1, −2) Monitoring Progress 6 x U R′ T(5, −3) → T′(−3, −5) S T S′ Help in English and Spanish at BigIdeasMath.com 1. Trace △DEF and point P. Then draw a 50° rotation of △DEF about point P. E D F P 2. Graph △JKL with vertices J(3, 0), K(4, 3), and L(6, 0) and its image after a 90° rotation about the origin. Section 4.3 HSTX_GEOM_PE_04.03.indd 195 Rotations 195 6/19/14 6:04 PM Performing Compositions with Rotations Postulate Postulate 4.3 Rotation Postulate A rotation is a rigid motion. D E F′ D′ E′ F Because a rotation is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the rotation shown. • DE = D′E′, EF = E′F′, FD = F′D′ • m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′ Because a rotation is a rigid motion, the Composition Theorem (Theorem 4.1) guarantees that compositions of rotations and other rigid motions, such as translations and reflections, are rigid motions. Performing a Composition — with endpoints R(1, −3) and S(2, −6) and its image after the composition. Graph RS Reflection: in the y-axis Rotation: 90° about the origin COMMON ERROR Unless you are told otherwise, perform the transformations in the order given. SOLUTION —. Step 1 Graph RS — in the y-axis. Step 2 Reflect RS — R′S′ has endpoints R′(−1, −3) and S′(−2, −6). — 90° about the Step 3 Rotate R′S′ — has endpoints origin. R″S″ R″(3, −1) and S″(6, −2). Monitoring Progress 3. y −4 R″(3, −1) −2 R′(−1, −3) S′(−2, −6) R(1, −3) −6 8 x S″(6, −2) S(2, −6) Help in English and Spanish at BigIdeasMath.com — from Example 3. Perform the rotation first, followed by the reflection. Graph RS Does the order of the transformations matter? Explain. — — the origin. Graph RS and its image after the composition. — with endpoints A(−4, 4) and B(−1, 7) and its image after Graph AB 4. WHAT IF? In Example 3, RS is reflected in the x-axis and rotated 180° about 5. the composition. Translation: (x, y) → (x − 2, y − 1) Rotation: 90° about the origin 6. Graph △TUV with vertices T(1, 2), U(3, 5), and V(6, 3) and its image after the composition. Rotation: 180° about the origin Reflection: in the x-axis 196 Chapter 4 HSTX_GEOM_PE_04.03.indd 196 Transformations 6/19/14 6:04 PM Identifying Rotational Symmetry A figure in the plane has rotational symmetry when the figure can be mapped onto itself by a rotation of 180° or less about the center of the figure. This point is the center of symmetry. Note that the rotation can be either clockwise or counterclockwise. A regular octagon has rotational symmetry. For example, the regular octagon at the left has rotational symmetry. The center is the intersection of the diagonals. Rotations of 45°, 90°, 135°, or 180° about the center all map the octagon onto itself. The regular octagon also has point symmetry, which is 180° rotational symmetry. Identifying Rotational Symmetry Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself. a. parallelogram b. trapezoid SOLUTION a. The parallelogram has rotational symmetry. The center is the intersection of the diagonals. A 180° rotation about the center maps the parallelogram onto itself. b. The trapezoid does not have rotational symmetry because no rotation of 180° or less maps the trapezoid onto itself. A parallelogram has rotational symmetry, but a trapezoid does not. Distinguishing Between Types of Symmetry Identify the line symmetry and rotational symmetry of the equilateral triangle. SOLUTION The triangle has line symmetry. Three lines of symmetry can be drawn for the figure. For a figure with s lines of symmetry, the smallest rotation that maps the figure onto itself has the 360° 360° measure —. So, the equilateral triangle has —, s 3 or 120° rotational symmetry. Monitoring Progress 120° Help in English and Spanish at BigIdeasMath.com Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. 7. rhombus 8. octagon 9. right triangle 10. Identify the line symmetry and rotational symmetry of a non-equilateral isosceles triangle. Section 4.3 HSTX_GEOM_PE_04.03.indd 197 Rotations 197 6/19/14 6:04 PM Exercises 4.3 Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE When a point (a, b) is rotated counterclockwise about the origin, (a, b) → (b, −a) is the result of a rotation of ______. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. What are the coordinates of the vertices of the image after a 90° counterclockwise rotation about the origin? y 4 2 What are the coordinates of the vertices of the image after a 270° clockwise rotation about the origin? −4 B A C −2 What are the coordinates of the vertices of the image after turning the figure 90° to the left about the origin? 4 x 2 −2 −4 What are the coordinates of the vertices of the image after a 270° counterclockwise rotation about the origin? Monitoring Progress and Modeling with Mathematics In Exercises 3–6, trace the polygon and point P. Then draw a rotation of the polygon about point P using the given number of degrees. (See Example 1.) 3. 30° E 4. 80° B D E −2 F 9. 180° G 5. 150° 4x D P C A P F 10. 270° y y K J 4 −6 6. 130° G 2 R P F Q In Exercises 7–10, graph the polygon and its image after a rotation of the given number of degrees about the origin. (See Example 2.) 7. 90° B 6 S x −2 Q T x — with endpoints X(−3, 1) In Exercises 11–14, graph XY and Y(4, −5) and its image after the composition. (See Example 3.) 11. Translation: (x, y) → (x, y + 2) Rotation: 90° about the origin 12. Rotation: 180° about the origin 13. Rotation: 270° about the origin Reflection: in the y-axis A −4 4 R Translation: (x, y) → (x − 1, y + 1) y 4 L M 2 P J 198 y 8. 180° −2 Chapter 4 HSTX_GEOM_PE_04.03.indd 198 C 2 4x 14. Reflection: in the line y = x Rotation: 180° about the origin Transformations 6/19/14 6:04 PM In Exercises 15 and 16, graph △LMN with vertices L(1, 6), M(−2, 4), and N(3, 2) and its image after the composition. (See Example 3.) 27. CONSTRUCTION Follow these steps to construct a rotation of △ABC by angle D around a point O. Use a compass and straightedge. 15. Rotation: 90° about the origin A′ Translation: (x, y) → (x − 3, y + 2) B A 16. Reflection: in the x-axis C Rotation: 270° about the origin D O In Exercises 17–20, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. (See Example 4.) 17. 18. Step 1 Draw △ABC, ∠D, and O, the center of rotation. —. Use the construction for copying Step 2 Draw OA an angle to copy ∠D at O, as shown. Then use distance OA and center O to find A′. Step 3 Repeat Step 2 to find points B′ and C′. Draw △A′B′C′. 19. 20. 28. REASONING You enter the revolving door at a hotel. a. You rotate the door 180°. What does this mean in the context of the situation? Explain. In Exercises 21–24, identify the line symmetry and rotational symmetry of the figure. (See Example 5.) 21. b. You rotate the door 360°. What does this mean in the context of the situation? Explain. 22. 29. MATHEMATICAL CONNECTIONS Use the graph of y = 2x − 3. y 23. 24. −2 2 x −2 ERROR ANALYSIS In Exercises 25 and 26, the endpoints — are C(−1, 1) and D(2, 3). Describe and correct of CD the error in finding the coordinates of the vertices of the image after a rotation of 270° about the origin. 25. 26. ✗ C (−1, 1) → C ′ (−1, −1) D (2, 3) → D ′ (2, −3) ✗ C (−1, 1) → C ′ (1, −1) D (2, 3) → D ′ (3, 2) a. Rotate the line 90°, 180°, 270°, and 360° about the origin. Write the equation of the line for each image. Describe the relationship between the equation of the preimage and the equation of each image. b. Do you think that the relationships you described in part (a) are true for any line? Explain your reasoning. 30. MAKING AN ARGUMENT Your friend claims that rotating a figure by 180° is the same as reflecting a figure in the y-axis and then reflecting it in the x-axis. Is your friend correct? Explain your reasoning. Section 4.3 HSTX_GEOM_PE_04.03.indd 199 Rotations 199 6/19/14 6:04 PM 31. DRAWING CONCLUSIONS A figure only has point 38. HOW DO YOU SEE IT? You are finishing the puzzle. symmetry. How many times can you rotate the figure before it is back where it started? The remaining two pieces both have rotational symmetry. 32. ANALYZING RELATIONSHIPS Is it possible for a figure to have 90° rotational symmetry but not 180° rotational symmetry? Explain your reasoning. 1 33. ANALYZING RELATIONSHIPS Is it possible for a 2 figure to have 180° rotational symmetry but not 90° rotational symmetry? Explain your reasoning. 34. THOUGHT PROVOKING Can rotations of 90°, 180°, a. Describe the rotational symmetry of Piece 1 and of Piece 2. 270°, and 360° be written as the composition of two reflections? Justify your answer. b. You pick up Piece 1. How many different ways can it fit in the puzzle? 35. USING AN EQUATION Inside a kaleidoscope, two mirrors are placed next to each other to form a V. The angle between the mirrors determines the number of lines of symmetry in the mirror image. Use the formula 1 n(m∠1) = 180° to find the measure of ∠1, the angle black glass between the mirrors, for the number n of lines of symmetry. c. Before putting Piece 1 into the puzzle, you connect it to Piece 2. Now how many ways can it fit in the puzzle? Explain. 39. USING STRUCTURE A polar coordinate system locates a point in a plane by its distance from the origin O and by the measure of an angle with its vertex at the origin. For example, the point A(2, 30°) is 2 units from the origin and m∠XOA = 30°. What are the polar coordinates of the image of point A after a 90° rotation? a 180° rotation? a 270° rotation? Explain. b. a. 90° 60° 120° 150° 30° A 36. REASONING Use the coordinate rules for 180° counterclockwise rotations about the origin to write coordinate rules for clockwise rotations of 90°, 180°, or 270° about the origin. O 1 2 210° 37. USING STRUCTURE △XYZ has vertices X(2, 5), 330° 240° Y(3, 1), and Z(0, 2). Rotate △XYZ 90° about the point P(−2, −1). Maintaining Mathematical Proficiency X 0° 3 300° 270° Reviewing what you learned in previous grades and lessons The figures are congruent. Name the corresponding angles and the corresponding sides. (Skills Review Handbook) 40. P Q W V 41. A B J X T S 200 Chapter 4 HSTX_GEOM_PE_04.03.indd 200 R Z Y D K C M L Transformations 6/19/14 6:04 PM

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