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...... 746 secant segment............. 793 arc......................... 756 external secant segment..... 793 sector of a circle............ 764 arc length.................. 766 inscribed angle............. 772 segment of a circle.......... 765 central angle............... 756 intercepted arc............. 772 semicircle.................. 756 chord...................... 746 interior of a circle........... 746 subtend.................... 772 common tangent........... 748 major arc.................. 756 tangent of a circle........... 746 concentric circles........... 747 minor arc.................. 756 tangent circles.............. 747 congruent arcs............. 757 point of tangency........... 746 tangent segment............ 794 congruent circles........... 747 secant..................... 746 Complete the sentences below with vocabulary words from the list above. 1. A(n)? is a region bounded by an arc and a chord. ̶̶̶̶ 2. An angle whose vertex is at |
the center of a circle is called a(n)?. ̶̶̶̶ 3. The measure of a(n)? is 360° minus the measure of its central angle. ̶̶̶̶? are coplanar circles with the same center. ̶̶̶̶ 4. 11-1 Lines That Intersect Circles (pp. 746–754) TEKS G.1.A, G.2.A, G.2.B, G.9.C E X A M P L E S ■ Identify each line or segment that intersects ⊙A. chord: ̶̶ DE tangent: BC ̶̶ ̶̶ AD, and AE, radii: ̶̶ AB secant: DE diameter: ̶̶ DE ̶̶ RW are tangent to ⊙T. RS = x + 5 and ■ ̶̶ RS and RW = 3x - 7. Find RS. RS = RW 2 segs. tangent to ⊙ from same ext. pt. → segs. ≅. Substitute the given values. Subtract 3x from both sides. Subtract 5 from both sides. Divide both sides by -2. Substitute 6 for y. Simplify. x + 5 = 3x - 7 -2x + 5 = -7 -2x = -12 x = 6 RS = 6 + 5 = 11 810 810 Chapter 11 Circles EXERCISES Identify each line or segment that intersects each circle. 5. 6. Given the measures of the following segments that are tangent to a circle, find each length. 7. AB = 9x - 2 and BC = 7x + 4. Find AB. 8. EF = 5y + 32 and EG = 8 - y. Find EG. 9. JK = 8m - 5 and JL = 2m + 4. Find JK. 10. WX = 0.8x + 1.2 and WY = 2.4x. Find WY. ����������������� 11-2 Arcs and Chords (pp. 756–763) TEKS G.1.A, G.2.A, G.2.B, G.8.C, G.9.C E X A M P L E S Find each measure. ■ |
m ⁀ BF ∠BAF and ∠FAE are supplementary, so m∠BAF = 180° - 62° = 118°. m ⁀ BF = m∠BAF = 118° ■ m ⁀ DF Since m∠DAE = 90°, m ⁀ DE = 90°. m∠EAF = 62°, so m ⁀ EF = 62°. By the Arc Addition Postulate, m ⁀ DF = m ⁀ DE + m ⁀ EF = 90° + 62° = 152°. EXERCISES Find each measure. 11. m ⁀ KM 12. m ⁀ HMK 13. m ⁀ JK 14. m ⁀ MJK Find each length to the nearest tenth. 15. ST 16. CD 11-3 Sector Area and Arc Length (pp. 764–769) TEKS G.1.A, G.1.B, G.8.B, G.9.C E X A M P L E S ■ Find the area of sector PQR. Give your answer in terms of π and rounded to the nearest hundredth. EXERCISES Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 17. sector DEF 18. sector JKL 360° A = π r 2 ( m° _ ) = π (4) 2 ( 135° _ ) = 16π ( 3 _ ) = 6π m 2 360 8 ≈ 18.85 m 2 ■ Find the length of ⁀ AB. Give your answer in terms of π and rounded to the nearest hundredth. 360° L = 2πr ( m° _ ) = 2π (9) ( 80° _ ) = 18π ( 4 _ ) 360° 9 Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 19. ⁀ GH 20. ⁀ MNP = 8π ft ≈ 25.13 ft Study Guide: Review 811 811 �������������������������������������������������������������������������������������������� 11-4 Inscribed Angles (pp. 772–779) TEKS G.1.A, G.2.A, G.2.B, G.5.B |
, G.9.C E X A M P L E S Find each measure. ■ m∠ABD By the Inscribed Angle Theorem, m∠ABD = 1 __ 2 m ⁀ AD, so m∠ABD = 1 __ 2 (108°) = 54°. ■ m ⁀ BE By the Inscribed Angle Theorem, m∠BAE = 1 __ 2 m ⁀ BE. So 28° = 1 __ 2 m ⁀ BE, and m ⁀ BE = 2 (28°) = 56°. EXERCISES Find each measure. 21. m ⁀JL 22. m∠MKL Find each value. 23. x 24. m∠RSP 11-5 Angle Relationships in Circles (pp. 782–789) TEKS G.1.A, G.2.B, G.5.A, G.5.B, G.9.C E X A M P L E S Find each measure. ■ m∠UWX m∠UWX = 1 _ m ⁀ UW 2 = 1 _ (160°) 2 = 80° EXERCISES Find each measure. 25. m ⁀ MR 26. m∠QMR ■ m ⁀ VW Since m∠UWX = 80°, m∠UWY = 100° and m∠VWY = 50°. m∠VWY = 1 __ 2 m ⁀ VW. So 50° = 1 __ 2 m ⁀ VW, and m ⁀ VW = 2 (50°) = 100°. 27. m∠GKH ■ m∠AED (m ⁀ AD + m ⁀ BC ) m∠AED = 1 _ 2 = 1 _ (31° + 87°) 2 = 1 _ (118°) 2 = 59° 812 812 Chapter 11 Circles 28. A piece of string art is made by placing 16 evenly spaced nails around the circumference of a circle. A piece of string is wound from A to B to C to D. What is m∠BXC? ����������������������������������������������������������������������������������������������������������� 11-6 Segment Relationships in Circles (pp. 792–798) |
TEKS G.1.A, G.2.B, G.5.A EXERCISES Find the value of the variable and the length of each chord. 29. 30. Find the value of the variable and the length of each secant segment. 31. 32. E X A M P L E S ■ Find the value of x and the length of each chord. AE ⋅ EB = DE ⋅ EC 12x = 8 (6) 12x = 48 x = 4 AB = 12 + 4 = 16 DC = 8 + 6 = 14 ■ Find the value of x and the length of each secant segment. FJ ⋅ FG = FK ⋅ FH 16 (4) = (6 + x) 6 64 = 36 + 6x 28 = 6x x = 4 2 _ 3 FJ = 12 + 4 = 16 FK = 4 2 _ + 6 = 10 2 _ 3 3 11-7 Circles in the Coordinate Plane (pp. 799–805) TEKS G.1.A, G.2.B, G.4.A, G.5.A E X A M P L E S EXERCISES ■ Write the equation of ⊙A that passes through (-1, 1) and that has center A (2, 3). The equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r 2. 2 r = √ (2 - (-1) ) + (3 - 1) 2 = √ 3 2 + 2 2 = √ 13 The equation of ⊙A is (x - 2) 2 + (y - 3) 2 = 13. ■ Graph (x - 2) 2 + (y + 1) 2 = 4. The center of the circle is (2, -1), and the radius is √ 4 = 2. Write the equation of each circle. 33. ⊙A with center (-4, -3) and radius 3 34. ⊙B that passes through (-2, -2) and that has center B (-2, 0) 35. ⊙C 36. Graph (x |
+ 2)2 + (y - 2)2 = 1. Study Guide: Review 813 813 ����������������������������������������������������������������������������������������� 1. Identify each line or segment that intersects the circle. 2. A jet is at a cruising altitude of 6.25 mi. To the nearest mile, what is the distance from the jet to a point on Earth’s horizon? (Hint: The radius of Earth is 4000 mi.) Find each measure. 3. m ⁀ JK 4. UV 5. Find the area of the sector. Give your answer in terms of π and rounded to the nearest hundredth. 6. Find the length of ⁀ BC. Give your answer in terms of π and rounded to the nearest hundredth. 7. If m∠SPR = 47° in the diagram of a logo, find m ⁀ SR. 8. A printer is making a large version of the logo for a banner. According to the specifications, m ⁀ PQ = 58°. What should the measure of ∠QTR be? Find each measure. 9. m∠ABC 10. m∠NKL 11. A surveyor S is studying the positions of four columns A, B, C, and D that lie on a circle. He finds that m∠CSD = 42° and m ⁀ CD = 124°. What is m ⁀ AB? Find the value of the variable and the length of each chord or secant segment. 12. 13. 14. The illustration shows a fragment of a circular plate. AB = 8 in., and CD = 2 in. What is the diameter of the plate? 15. Write the equation of the circle that passes through (-2, 4) and that has center (1, -2). � � � 16. An artist uses a coordinate plane to plan a mural. The mural will include portraits of civic leaders at X (2, 4), Y (-6, 0), and Z (2, -8) and a circle that passes through all three portraits. What are the coordinates of the center of the circle? � � � 814 814 Chapter 11 Circles ��������������������������������������������������������������������������������� FOCUS ON SAT MATHEMATICS SUBJECT TESTS The topics covered on |
the SAT Mathematics Subject Tests vary only slightly each time the test is administered. You can find out the general distribution of questions across topics, then determine which areas need more of your attention when you are studying for the test. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. To prepare for the SAT Mathematics Subject Tests, start reviewing course material a couple of months before your test date. Take sample tests to find the areas you might need to focus on more. Remember that you are not expected to have studied all topics on the test. 1. ̶̶ ̶̶ BD intersect at the center of the circle AC and shown. If m∠BDC = 30°, what is the measure of minor ⁀ AB? 4. Circle D has radius 6, and m∠ABC = 25°. What is the length of minor ⁀ AC? (A) 15° (B) 30° (C) 60° (D) 105° (E) 120° Note: Figure not drawn to scale. 2. Which of these is the equation of a circle that is tangent to the lines x = 1 and y = 3 and has radius 2? (A) (x + 1) 2 + (y - 1) 2 = 4 (B) (x - 1) 2 + (y + 1) 2 = 4 (C) x 2 + (y - 1) 2 = 4 (D) (x - 1) 2 + y 2 = 4 (E) x 2 + y 2 = 4 3. If LK = 6, LN = 10, and PK = 3, what is PM? (A) 7 (B) 8 (C) 9 (D) 10 (E) 11 Note: Figure not drawn to scale. (A) 5π _ 6 (B) 5π _ 4 (C) 5π _ 3 (D) 3π (E) 5π 5. A square is inscribed in a circle as shown. If the radius of the circle is 9, what is the area of the shaded region, rounded to the nearest hundredth? (A) 11.56 (B) 23.12 (C) 57.84 (D) 104.12 (E) 156.23 College Entrance Exam Practice 815 815 ������������� Multiple Choice: Recognize Distracters In a multiple choice test item, incorrect answer choices that may seem right are called distract |
ers. Test writers create distracters by using common errors that students make. Be sure you always check your answer. The answer you get when you solve the problem may be one of the answer choices, but it may not be the correct answer. ̶̶̶ CD is tangent to ⊙B at C, and m ⁀ AC = 65°. What is m∠ABC? 130° 65° 32.5° 25° Look at each answer choice carefully. A This is a distracter. The m ⁀ ACE, not m∠ABC, is 130°. Doubling the arc length is a common error. B This is the correct answer. C This is a distracter. Using the Inscribed Angle Theorem to solve this problem is an error a student might make. m∠ABC is not equal to half the m ⁀ AC. D This is a distracter. The sum of m∠ACD and m∠ABC is not 90°. In a circle, the length of an arc intercepted by a central angle is 4π, and the radius is 16 inches. What is the measure of the central angle? 5.625 22.5° 45° 90° Look at each answer choice carefully. F This is a distracter. Students who use π r 2 instead of 2πr will get this answer. G This is a distracter. Students often make errors when dividing. This distracter was created by dividing 4 by 32 and getting a quotient of 1 __. 16 H This is the correct answer. J This is a distracter. You would get this answer if you simplified the formula for arc length incorrectly. 816 816 Chapter 11 Circles �������� ���� ���� ���� When you calculate an answer to a multiplechoice test item, solve the problem again with a different method to make sure your answer is correct. Read each test item and answer the questions that follow. Item A Which point is the center of the circle defined by the equation x 2 + (y - 9) 2 = 81? (9, 9) (-9, -9) (0, 9) (9, 0) Item C A regular hexagon is inscribed in a circle with a radius of 8 inches. What is the length of one arc of the circle intercepted by one side of the hexagon? 4 _ π 3 8 _ π 3 32 _ π 3 16π 7. What is the formula for arc length? 8. |
How do you determine the measure of the central angle? 9. Describe the errors a student might make to 1. What common error do the coordinates in get each of the distracters. choice B represent? 2. The y-coordinate in choice C is correct, but the x-coordinate is not. What error was made in finding the x-coordinate? 3. Which choice is the correct answer? What alternative method can you use to see whether your answer is correct? Item B What is m∠FHG? 50° 70° 100° 200° 4. How would you describe ̶̶ EH and ̶̶ HD? 5. If a student chose choice H, what common mistake might the student have made? 6. What common error would lead to answer choice J? Item D What is the equation of a circle centered at (4, -5) with a radius of 6? (x + 5) 2 + (y - 4) 2 = 6 (x - 4) 2 + (y + 5) 2 = 6 (x + 5) 2 + (y - 4) 2 = 36 (x - 4) 2 + (y + 5) 2 = 36 10. What common error does the equation in choice G represent? 11. What common error does the equation in choice H represent? 12. Why is choice J the correct answer? Item E What is m∠ABC? 45° 90° 180° 360° 13. How does knowing what ̶̶ AC is help you determine m∠ABC? 14. What mistake would lead to choice C? TAKS Tackler 817 817 �������������������� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–11 Multiple Choice 1. The composite figure is a right prism that shares a base with the regular pentagonal pyramid on top. If the lateral area of this figure is 328 square feet, what is the slant height of the pyramid? 6. △JKL is a right triangle where m∠K = 90° and tan J = 3 __. Which of the following could be 4 the side lengths of △JKL? KL = 16, KJ = 12, and JL = 20 KL = 15, KJ = 25, and JL = 20 KL = 20, KJ = 16, and JL = 12 KL = 18, KJ = 24, and JL = 30 |
Use the diagram for Items 7 and 8. 2.5 feet 5.0 feet 8.4 feet 9.0 feet 2. What is the area of the polygon with vertices A (2, 3), B (12, 3), C (6, 0), and D (2, 0)? 12 square units 30 square units 21 square units 42 square units 7. What is m ⁀ QU? Use the diagram for Items 3–5. 25° 42° 58° 71° 8. Which expression can be used to calculate the length of ̶̶ PS? PR ⋅ PQ _ PU PR ⋅ PR _ PU PQ ⋅ QR _ PU PQ ⋅ PR _ PS 3. What is m ⁀ BC? 36° 45° 54° 72° 4. If the length of ⁀ ED is 6π, what is the area of sector EFD? 20π square centimeters 72π square centimeters 120π square centimeters 240π square centimeters 5. Which of these line segments is NOT a chord of ⊙F? ̶̶ EC ̶̶ CA ̶̶ AF ̶̶ AE 818 818 Chapter 11 Circles 9. △ABC has vertices A (0, 0), B (-1, 3), and C (2, 4). If △ABC ∼ △DEF and △DEF has vertices D (5, -3), E (4, -2), and F (3, y), what is the value of y? -7 -5 -3 -1 10. What is the equation of the circle with ̶̶̶ MN that has endpoints M (-1, 1) diameter and N (3, -5)? (x + 1) 2 + (y - 2) 2 = 13 (x - 1) 2 + (y + 2) 2 = 13 (x + 1) 2 + (y - 2) 2 = 26 (x - 1) 2 + (y + 2) 2 = 52 ������������������������������ ���� ���� ���� Remember that an important part of writing a proof is giving a justification for each step in the proof. Justifications may include theorems, postulates, definitions, properties, or the information that is given to you. STANDARDIZED TEST PREP Short Response 21. Use the diagram to find the value of x. Show your work or explain |
in words how you determined your answer. 11. Kite PQRS has diagonals ̶̶ PR and ̶̶ QS that intersect at T. Which of the following is the shortest ̶̶ PR? segment from Q to ̶̶ PT ̶̶ QP ̶̶ RQ ̶̶ TQ 12. If the perimeter of an equilateral triangle is reduced by a factor of 1 __, what is the effect on 2 the area of the triangle? The area remains constant. The area is reduced by a factor of 1 __. 2 The area is reduced by a factor of 1 __. 4 The area is reduced by a factor of 1 __. 6 13. The area of a right isosceles triangle is 36 m 2. What is the length of the hypotenuse of the triangle? 6 meters 6 √ 2 meters 12 meters 12 √ 2 meters Gridded Response 14. The ratio of the side lengths of a triangle is 4 : 5 : 8. If the perimeter is 38.25 centimeters, what is the length in centimeters of the shortest side? 22. Paul needs to rent a storage unit. He finds one that has a length of 10 feet, a width of 5 feet, and a height of 9 feet. He finds a second storage unit that has a length of 11 feet, a width of 4 feet, and a height of 8 feet. Suppose that the first storage unit costs $85.00 per month and that the second storage unit costs $70.00 per month. a. Which storage unit has a lower price per cubic foot? Show your work or explain in words how you determined your answer. b. Paul finds a third storage unit that charges $0.25 per cubic foot per month. What are possible dimensions of the storage unit if the charge is $100.00 per month? 23. The equation of ⊙C is x 2 + (y + 1) 2 = 25. a. Graph ⊙C. b. Write the equation of the line that is tangent to ⊙C at (3, 3). Show your work or explain in words how you determined your answer. 15. What is the geometric mean of 4 and 16? 24. A tangent and a secant intersect on a circle at 16. For △HGJ and △LMK suppose that ∠H ≅ ∠L, HG = 4x + 5, KL |
= 9, HJ = 5x -1, and LM = 13. What must be the value of x to prove that △HGJ and △LMK are congruent by SAS? 17. If the length of a side of a regular hexagon is 2, what is the area of the hexagon to the nearest tenth? 18. What is the arc length of a semicircle in a circle with radius 5 millimeters? Round to the nearest hundredth. 19. What is the surface area of a sphere whose volume is 288π cubic centimeters? Round to the nearest hundredth. 20. Convert (6, 60°) to rectangular coordinates. What is the value of the x-coordinate? the point of tangency and form an acute angle. Explain how you would find the range of possible measures for the intercepted arc. Extended Response 25. Let ABCD be a quadrilateral inscribed in a circle such that ̶̶ AB ǁ ̶̶ DC. a. Prove that m ⁀ AD = m ⁀ BC. b. Suppose ABCD is a trapezoid. Show that ABCD must be isosceles. Justify your answer. c. If ABCD is not a trapezoid, explain why ABCD must be a rectangle. Cumulative Assessment, Chapters 1–11 819 819 �������������������������������� Extending Transformational Geometry 12A Congruence Transformations 12-1 Reflections 12-2 Translations 12-3 Rotations Lab Explore Transformations with Matrices 12-4 Compositions of Transformations 12B Patterns 12-5 Symmetry 12-6 Tessellations Lab Use Transformations to Extend Tessellations 12-7 Dilations Ext Using Patterns to Generate Fractals KEYWORD: MG7 ChProj Each year, hundreds of millions of monarch butterflies pass through Texas during their annual migration. 820 820 Chapter 12 Vocabulary Match each term on the left with a definition on the right. 1. image A. a mapping of a figure from its original position to a new 2. preimage 3. transformation 4. vector position B. a ray that divides an angle into two congruent angles C. a shape that undergoes a transformation D. a quantity that has both a size and a direction E. the shape that results from a transformation of a figure Ordered Pairs Graph each ordered pair. 5. (0, 4) 8. (3, - |
1) Congruent Figures 6. (-3, 2) 9. (-1, -3) 7. (4, 3) 10. (-2, 0) Can you conclude that the given triangles are congruent? If so, explain why. 11. △PQS and △PRS 12. △DEG and △FGE Identify Similar Figures Can you conclude that the given figures are similar? If so, explain why. 13. △JKL and △JMN 14. rectangle PQRS and rectangle UVWX Angles in Polygons 15. Find the measure of each interior angle of a regular octagon. 16. Find the sum of the interior angle measures of a convex pentagon. 17. Find the measure of each exterior angle of a regular hexagon. 18. Find the value of x in hexagon ABCDEF. Extending Transformational Geometry 821 821 ������������������������������������������������ Key Vocabulary/Vocabulario composition of transformations composición de transformaciones glide reflection deslizamiento con inversión isometry symmetry tessellation isometría simetría teselado Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. A composition is something that has been put together. How can you use this idea to understand what is meant by a composition of transformations? 2. The prefix iso- means “equal.” The suffix -metry means “measure.” What do you think might be true about the preimage and image of a figure under a transformation that is an isometry? Geometry TEKS G.2.A Geometric structure* use constructions to explore attributes of geometric figures... Les. 12-1 Les. 12-2 Les. 12-3 ★ ★ ★ 12-3 Tech. Lab Les. 12-4 Les. 12-5 Les. 12-6 12-6 Geo. Lab Les. 12-7 Ext. ★ G.2.B Geometric structure* make conjectures … and ★ ★ ★ ★ ★ determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic G.5.B Geometric patterns* use numeric and geometric ★ patterns to make |
generalizations about geometric properties … G.5.C Geometric patterns* use properties of transformations ★ ★ ★ ★ ★ and their compositions to make connections between mathematics and the real world such as tessellations G.7.A Dimensionality and the geometry of location* use one- ★ ★ ★ ★ and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures G.10.A Congruence and the geometry of size* use congruence transformations to make conjectures and justify properties of geometric figures … G.11.A Similarity and the geometry of shape* use and extend similarity properties and transformations to explore and justify conjectures … G.11.B Similarity and the geometry of shape* uses ratios to solve problems involving similar figures G.11.D Similarity and the geometry of shape* describe the effect on perimeter, area, and volume when one or more dimensions of figure are changed and apply this idea in solving problems ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 822 822 Chapter 12 Study Strategy: Prepare for Your Final Exam Math is a cumulative subject, so your final exam will probably cover all of the material you have learned since the beginning of the course. Preparation is essential for you to be successful on your final exam. It may help you to make a study timeline like the one below. 2 weeks before the final: • Look at previous exams and homework to determine areas I need to focus on; rework problems that were incorrect or incomplete. • Make a list of all formulas, postulates, and theorems I need to know for the final. • Create a practice exam using problems from the book that are similar to problems from each exam. 1 week before the final: • Take the practice exam and check it. For each problem I miss, find two or three similar ones and work those. • Work with a friend in the class to quiz each other on formulas, postulates, and theorems from my list. 1 day before the final: • Make sure I have pencils, calculator (check batteries!), ruler, compass, and protractor. Try This 1. Create a timeline that you will use to study for your final exam. Extending Transformational Geometry 823 823 12-1 Reflections TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and |
justify properties of geometric figures.... Objective Identify and draw reflections. Vocabulary isometry Who uses this? Trail designers use reflections to find shortest paths. (See Example 3.) An isometry is a transformation that does not change the shape or size of a figure. Reflections, translations, and rotations are all isometries. Isometries are also called congruence transformations or rigid motions. The Houston skyline Also G.2.A, G.2.B, G.7.A Recall that a reflection is a transformation that moves a figure (the preimage) by flipping it across a line. The reflected figure is called the image. A reflection is an isometry, so the image is always congruent to the preimage. E X A M P L E 1 Identifying Reflections Tell whether each transformation appears to be a reflection. Explain. A B To review basic transformations, see Lesson 1-7, pages 50−55. Yes; the image appears to be flipped across a line. No; the figure does not appear to be flipped. Tell whether each transformation appears to be a reflection. 1a. 1b. Construction Reflect a Figure Using Patty Paper Draw a triangle and a line of reflection on a piece of patty paper. Fold the patty paper back along the line of reflection. Trace the triangle. Then unfold the paper. Draw a segment from each vertex of the preimage to the corresponding vertex of the image. Your construction should show that the line of reflection is the perpendicular bisector of every segment connecting a point and its image. 824 824 Chapter 12 Extending Transformational Geometry Reflections A reflection is a transformation across a line, called the line of reflection, so that the line of reflection is the perpendicular bisector of each segment joining each point and its image. E X A M P L E 2 Drawing Reflections For more on reflections, see the Transformation Builder on page xxiv. Copy the quadrilateral and the line of reflection. Draw the reflection of the quadrilateral across the line. Step 1 Through each vertex draw a line perpendicular to the line of reflection. Step 2 Measure the distance from each vertex to the line of reflection. Locate the image of each vertex on the opposite side of the line of reflection and the same distance from it. Step 3 Connect the images of the vertices. 2. Copy the quadrilateral and the line of reflection. Draw the reflection of the quadrilateral |
across the line. E X A M P L E 3 Problem-Solving Application A trail designer is planning two trails that connect campsites A and B to a point on the river. He wants the total length of the trails to be as short as possible. Where should the trail meet the river? Understand the Problem The problem asks you to locate point X on the river so that AX + XB has the least value possible. Make a Plan Let B' be the reflection of point B across the river. For any point X on the river, are collinear. ̶̶ XB, so AX + XB = AX + XB'. AX + XB' is least when A, X, and B' ̶̶̶ XB' ≅ Solve Reflect B across the river to locate B'. Draw locate X at the intersection of ̶̶̶ AB' and the river. ̶̶̶ AB' and Look Back To verify your answer, choose several possible locations for X and measure the total length of the trails for each location. 3. What if…? If A and B were the same distance from the river, ̶̶ AX and what would be true about ̶̶ BX? 12-1 Reflections 825 825 ����������������������������123�����4�������������� Reflections in the Coordinate Plane ACROSS THE x-AXIS ACROSS THE y-AXIS ACROSS THE LINE Drawing Reflections in the Coordinate Plane Reflect the figure with the given vertices across the given line. A M (1, 2), N (1, 4), P (3, 3) ; y-axis The reflection of (x, y) is (-x, y). M (1, 2) → M' (-1, 2) N (1, 4) → N' (-1, 4) P (3, 3) → P' (-3, 3) Graph the preimage and image. B D (2, 0), E (2, 2), F (5, 2), G (5, 1) ; y = x The reflection of (x, y) is (y, x). D (2, 0) → D' (0, 2) E (2, 2) → E' (2, 2) F (5, 2) → F' (2, 5) G (5, 1) → G' ( |
1, 5) Graph the preimage and image. 4. Reflect the rectangle with vertices S (3, 4), T (3, 1), U (-2, 1), and V (-2, 4) across the x-axis. THINK AND DISCUSS 1. Acute scalene △ABC is reflected across ̶̶ BC. Classify quadrilateral ABA'C. Explain your reasoning. 2. Point A' is a reflection of point A across line ℓ. What is the relationship of ℓ to ̶̶̶ AA'? 3. GET ORGANIZED Copy and complete the graphic organizer. 826 826 Chapter 12 Extending Transformational Geometry ������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� 12-1 Exercises Exercises KEYWORD: MG7 12-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary If a transformation is an isometry, how would you describe the relationship between the preimage and the image Tell whether each transformation appears to be a reflection. p. 824 2. 4. 3. 5. 825 Multi-Step Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 6. 7. 825 8. City Planning The towns of San Pablo and Tanner are located on the same side of Highway 105. Two access roads are planned that connect the towns to a point P on the highway. Draw a diagram that shows where point P should be located in order to make the total length of the access roads as short as possible Reflect the figure with the given vertices across the given line. p. 826 9. A (-2, 1), B (2, 3), C (5, 2) ; x-axis 10. R (0, -1), S (2, 2), T (3, 0) ; y-axis 11. M (2, 1), N (3, 1), P (2, -1), Q (1, -1) ; y = x 12. A (-2, 2), B (-1, 3), C (1, 2), D (-2, -2) ; y = x |
PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a reflection. 13. 15. 14. 16. Independent Practice For See Exercises Example 13–16 17–18 19 20–23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 12-1 Reflections 827 827 ������������������������� Multi-Step Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 17. 18. 19. Recreation Cara is playing pool. She wants to hit the ball at point A without hitting the ball at point B. She has to bounce the cue ball, located at point C, off the side rail and into her ball. Draw a diagram that shows the exact point along the rail that Cara should aim for. Reflect the figure with the given vertices across the given line. 20. A (-3, 2), B (0, 2), C (-2, 0) ; y-axis 21. M (-4, -1), N (-1, -1), P (-2, -2) ; y = x 22. J (1, 2), K (-2, -1), L (3, -1) ; x-axis 23. S (-1, 1), T (1, 4), U (3, 2), V (1, -3) ; y = x Copy each figure. Then complete the figure by drawing the reflection image across the line. 24. 25. 26. Chemistry Louis Pasteur (1822– 1895) is best known for the pasteurization process, which kills germs in milk. He discovered chemical chirality when he observed that two salt crystals were mirror images of each other. 27. Chemistry In chemistry, chiral molecules are mirror images of each other. Although they have similar structures, chiral molecules can have very different properties. For example, the compound R- (+) -limonene smells like oranges, while its mirror image, S- (-) -limonene, smells like lemons. Use the figure and the given line of reflection to draw S- (-) -limonene. Each figure shows a preimage and image under a reflection. Copy the figure and draw the line of reflection. 28. 29. 30. Use arrow notation to describe the mapping of each point when it is reflected across the |
ven line. 40. x-axis 41. y-axis 42. Write About It Imagine reflecting all the points in a plane across line ℓ. Which points remain fixed under this transformation? That is, for which points is the image the same as the preimage? Explain. Construction Use the construction of a line perpendicular to a given line through a given point (see page 179) and the construction of a segment congruent to a given segment (see page 14) to construct the reflection of each figure across a line. 43. a point 44. a segment 45. a triangle 46. Daryl is using a coordinate plane to plan a garden. He draws a flower bed with vertices (3, 1), (3, 4), (-2, 4), and (-2, 1). Then he creates a second flower bed by reflecting the first one across the x-axis. Which of these is a vertex of the second flower bed? (-2, -4) (-3, 1) (2, 1) (-3, -4) 12-1 Reflections 829 829 ��������������������������������������������������������������������� 47. In the reflection shown, the shaded figure is the preimage. Which of these represents the mapping? � � � MJNP → DSWG DGWS → MJNP JMPN → GWSD PMJN → SDGW 48. What is the image of the point (-3, 4) when it is reflected across the y-axis? (4, -3) (-3, -4) (3, 4) (-4, -3) � � � � � CHALLENGE AND EXTEND Find the coordinates of the image when each point is reflected across the given line. 49. (4, 2) ; y = 3 51. (3, 1) ; y = x + 2 50. (-3, 2) ; x = 1 52. Prove that the reflection image of a segment is congruent ̶̶̶ A'B' ̶̶ AB across line ℓ. to the preimage. ̶̶̶ A'B' is the reflection image of ̶̶ AB ≅ Given: Prove: Plan: Draw auxiliary lines that △ACD ≅ △A'CD. Then use CPCTC to conclude that ∠CDA ≅ ∠CDA'. Therefore ∠ADB ≅ ∠ |
A'DB', which makes it possible to prove that △ADB ≅ △A'DB'. Finally use CPCTC to conclude that ̶̶̶ AA' and ̶̶̶ BB' as shown. First prove ̶̶ AB ≅ ̶̶̶ A'B'. � � �� � � � �� Once you have proved that the reflection image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof. 53. If ̶̶̶ A'B' is the reflection of ̶̶ AB, then AB = A'B'. 54. If ∠A'B'C' is the reflection of ∠ABC, then m∠ABC = m∠A'B'C'. 55. The reflection △A'B'C' is congruent to the preimage △ABC. 56. If point C is between points A and B, then the reflection C' is between A' and B'. 57. If points A, B, and C are collinear, then the reflections A', B', and C' are collinear. SPIRAL REVIEW A jar contains 2 red marbles, 6 yellow marbles, and 4 green marbles. One marble is drawn and replaced, and then a second marble is drawn. Find the probability of each outcome. (Previous course) 58. Both marbles are green. 59. Neither marble is red. 60. The first marble is yellow, and the second is green. The width of a rectangular field is 60 m, and the length is 105 m. Use each of the following scales to find the perimeter of a scale drawing of the field. (Lesson 7-5) 61. 1 cm : 30 m 62. 1.5 cm : 15 m 63. 1 cm : 25 m Find each unknown measure. Round side lengths to the nearest hundredth and angle measures to the nearest degree. (Lesson 8-3) 64. BC 65. m∠A 66. m∠C � � � ��������� � 830 830 Chapter 12 Extending Transformational Geometry 12-2 Translations TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties... Objective Identify and draw translations. Who uses this? Marching band directors use translations to plan their bands’ field shows. |
(See Example 4.) Also G.2.A, G.2.B, G.7.A 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. E X A M P L E 1 Identifying Translations Tell whether each transformation appears to be a translation. Explain. 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. Tell whether each transformation appears to be a translation. 1a. 1b. Construction Translate a Figure Using Patty Paper Draw a triangle and a translation vector on a sheet of paper. Place a sheet of patty paper on top of the diagram. Trace the triangle and vector. Slide the bottom paper in the direction of the vector until the head of the top vector aligns with the tail of the bottom vector. Trace the triangle. To review vectors, see Lesson 8-6, pages 559−567. Draw a segment from each vertex of the preimage to the corresponding vertex of the image. Your construction should show that every segment connecting a point and its image is the same length as the translation vector. These segments are also parallel to the translation vector. 12-2 Translations 831 831 Translations A translation is a transformation along a vector such that each segment joining a point and its image has the same length as the vector and is parallel to the vector. E X A M P L E 2 Drawing Translations Copy the triangle and the translation vector. v. Draw the translation of the triangle along Step 1 Draw a line parallel to the vector through each vertex of the triangle. For more on translations, see the Transformation Builder on page xxiv. Step 2 Measure the length of the vector. Then, from each vertex mark off this distance in the same direction as the vector, on each of the parallel lines.w Step 3 Connect the images of the vertices. 2. Copy the quadrilateral and the translation vector. Draw the translation of the quadrilateral along w. Recall that a vector in the coordinate plane can be written as 〈a, b〉, where a is the horizontal change and b is the vertical change from the initial point to the terminal point. Translations in the Coordinate Plane HOR |
IZONTAL TRANSLATION ALONG VECTOR 〈a, 0〉 VERTICAL TRANSLATION ALONG VECTOR 〈0, b〉 GENERAL TRANSLATION ALONG VECTOR 〈a, b〉 832 832 Chapter 12 Extending Transformational Geometry ��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� E X A M P L E 3 Drawing Translations in the Coordinate Plane Translate the triangle with vertices A (-2, -4), B (-1, -2), and C (-3, 0) along the vector 〈2, 4〉. The image of (x, y) is (x + 2, y + 4). A (-2, -4) → A' (-2 + 2, -4 + 4) = A' (0, 0) B (-1, -2) → B' (-1 + 2, -2 + 4) = B' (1, 2) C (-3, 0) → C' (-3 + 2, 0 + 4) = C' (-1, 4) Graph the preimage and image. 3. Translate the quadrilateral with vertices R (2, 5), S (0, 2), T (1, -1), and U (3, 1) along the vector 〈-3, -3〉. E X A M P L E 4 Entertainment Application Entertainment In 1955, the University of Texas Longhorn Band, pictured above, became the proud owner of Big Bertha, the largest bass drum in the world. The drum is 54 inches wide and 8 feet in diameter. In a marching drill, it takes 8 steps to march 5 yards. A drummer starts 8 steps to the left and 8 steps up from the center of the field. She marches 16 steps to the right to her second position. Then she marches 24 steps down the field to her final position. What is the drummer’s final position? What single translation vector moves her from the starting position to her final position? The drummer’s starting coordinates are (-8, 8). Her second position is (-8 + 16, 8) = (8, 8). Her final position is (8, 8 - 24 |
) = (8, -16). The vector that moves her directly from her starting position to her final position is 〈16, 0〉 + 〈0, -24〉 = 〈16, -24〉. 4. What if…? Suppose another drummer started at the center of the field and marched along the same vectors as above. What would this drummer’s final position be? THINK AND DISCUSS 1. Point A' is a translation of point A along v to ̶̶ AA '? v. What is the 2. relationship of ̶̶̶ ̶̶ AB is translated to form A'B'. Classify quadrilateral AA'B'B. Explain your reasoning. 3. GET ORGANIZED Copy and complete the graphic organizer. 12-2 Translations 833 833 ������������������������������������������������������������������������������������������������������������ 12-2 Exercises Exercises GUIDED PRACTICE Tell whether each transformation appears to be a translation. p. 831 1. 3. 2. 4. KEYWORD: MG7 12-2 KEYWORD: MG7 Parent. 832 Multi-Step Copy each figure and the translation vector. Draw the translation of the figure along the given vector. 5. 6 Translate the figure with the given vertices along the given vector. p. 833 7. A (-4, -4), B (-2, -3), C (-1, 3) ; 〈5, 0〉 8. R (-3, 1), S (-2, 3), T (2, 3), U (3, 1) ; 〈0, -4〉 9. J (-2, 2), K (-1, 2), L (-1, -2), M (-3, -1) ; 〈3, 2〉 10. Art The Zulu people of southern Africa are known for their beadwork. To create a typical Zulu pattern, translate the polygon with vertices (1, 5), (2, 3), (1, 1), and (0, 3) along the vector 〈0, -4〉. Translate the image along the same vector. Repeat to generate a pattern. What are the vertices |
of the fourth polygon in the pattern? PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a translation. 11. 13. 13. 12. 14. 833 Independent Practice For See Exercises Example 11–14 15–16 17–19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 834 834 Chapter 12 Extending Transformational Geometry ������ Multi-Step Copy each figure and the translation vector. Draw the translation of the figure along the given vector. 15. 16. Animation Each frame of a computer-animated feature represents 1 __ 24 of a second of film. Source: www.pixar.com Translate the figure with the given vertices along the given vector. 17. P (-1, 2), Q (1, -1), R (3, 1), S (2, 3) ; 〈-3, 0〉 18. A (1, 3), B (-1, 2), C (2, 1), D (4, 2) ; 〈-3, -3〉 19. D (0, 15), E (-10, 5), F (10, -5) ; 〈5, -20〉 20. Animation An animator draws the ladybug shown and then translates it along the vector 〈1, 1〉, followed by a translation of the new image along the vector 〈2, 2〉, followed by a translation of the second image along the vector 〈3, 3〉. a. Sketch the ladybug’s final position. b. What single vector moves the ladybug from its starting position to its final position? Draw the translation of the graph of each function along the given vector. 21. 〈3, 0〉 22. 〈-1, -1〉 23. Probability The point P (3, 2) is translated along one of the following four vectors chosen at random: 〈-3, 0〉, 〈-1, -4〉, 〈3, -2〉, and 〈2, 3〉. Find the probability of each of the following. a. The image of P is in the fourth quadrant. b. The image of |
P is on an axis. c. The image of P is at the origin. 24. This problem will prepare you for the Multi-Step TAKS Prep on page 854. The figure shows one hole of a miniature golf course and the path of a ball from the tee T to the hole H. a. What translation vector represents the path of the ball from T to ̶̶ DC? b. What translation vector represents the path of the ball from ̶̶ DC to H? c. Show that the sum of these vectors is equal to the vector that represents the straight path from T to H. 12-2 Translations 835 835 �����yx0–5–5ge07sec12l02008a4th pass5/11/5cmurphy��������������������������������������������������������� Each figure shows a preimage (blue) and its image (red) under a translation. Copy the figure and draw the vector along which the polygon is translated. 25. 26. 27. Critical Thinking The points of a plane are translated along the given vector remain fixed under this transformation? That is, are there any points for which the image coincides with the preimage? Explain. AB. Do any points 28. Carpentry Carpenters use a tool called adjustable parallels to set up level work areas and to draw parallel lines. Describe how a carpenter could use this tool to translate a given point along a given vector. What additional tools, if any, would be needed? Find the vector associated with each translation. Then use arrow notation to describe the mapping of the preimage to the image. 29. the translation that maps point A to point B 30. the translation that maps point B to point A 31. the translation that maps point C to point D 32. the translation that maps point E to point B 33. the translation that maps point C to the origin 34. Multi-Step The rectangle shown is translated two-thirds of the way along one of its diagonals. Find the area of the region where the rectangle and its image overlap. 35. Write About It Point P is translated along the vector 〈a, b〉. Explain how to find the distance between point P and its image. Construction Use the construction of a line parallel to a given line through a given point (see page 163) and the construction of a segment congruent to a |
given segment (see page 13) to construct the translation of each figure along a vector. 36. a point 37. a segment 38. a triangle 39. What is the image of P (1, 3) when it is translated along the vector 〈-3, 5〉? (0, 4) (-2, 8) (0, 6) (1, 3) 40. After a translation, the image of A (-6, -2) is B (-4, -4). What is the image of the point (3, -1) after this translation? (-5, 1) (5, -3) (5, 1) (-5, -3) 836 836 Chapter 12 Extending Transformational Geometry ����������������������������������� 41. Which vector translates point Q to point P? 〈-2, -4〉 〈4, -2〉 〈-2, 4〉 〈2, -4〉 � � � � � � � � � CHALLENGE AND EXTEND 42. The point M (1, 2) is translated along a vector that is parallel to the line y = 2x + 4. The translation vector has magnitude √ 5. What are the possible images of point M? 43. A cube has edges of length 2 cm. Point P is translated along u, v, and w as shown. a. Describe a single translation vector that maps point P to point Q. b. Find the magnitude of this vector to the nearest hundredth. 44. Prove that the translation image of a segment is congruent to the preimage. ̶̶̶ A'B' is the translation image of Given: ̶̶ AB ≅ Prove: (Hint: Draw auxiliary lines What can you conclude about ̶̶̶ BB'. ̶̶̶ AA' and ̶̶̶ AA' and ̶̶̶ A'B' ̶̶ AB. ̶̶̶ BB'?) � � ��� ��� � ��� � �� �� Once you have proved that the translation image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof. 45. If ̶̶̶ A'B' is a translation of ̶̶ AB, then AB = A' |
B'. 46. If ∠A'B'C' is a translation of ∠ABC, then m∠ABC = m∠A'B'C'. 47. The translation △A'B'C' is congruent to the preimage △ABC. 48. If point C is between points A and B, then the translation C' is between A' and B'. 49. If points A, B, and C are collinear, then the translations A', B', and C' are collinear. SPIRAL REVIEW Solve each system of equations and check your solution. (Previous course) ⎧ -5x - 2y = 17 50. ⎨ 6x - 2y = -5 ⎩ ⎧ 2x - 3y = -7 51. ⎨ 6x + 5y = 49 ⎩ ⎧ 4x + 4y = -1 52. ⎨ 12x - 8y = -8 ⎩ Solve to find x and y in each diagram. (Lesson 3-4) 53. 54. ���������� ���������� �� ��� ��� △MNP has vertices M (-2, 0), N (-3, 2), and P (0, 4). Find the coordinates of the vertices of △M'N'P' after a reflection across the given line. (Lesson 12-1) 55. x-axis 56. y-axis 57. y = x 12-2 Translations 837 837 Transformations of Functions Algebra Transformations can be used to graph complicated functions by using the graphs of simpler functions called parent functions. The following are examples of parent functions and their graphs. See Skills Bank page S63 y = ⎜x Transformation of Parent Function y = f (x) Reflection Vertical Translation Horizontal Translation Across x-axis: y = -f (x) y = f (x) + k y = f (x - h) Across y-axis: y = f (-x) Up k units if k > 0 Right h units if h > 0 Down k units if k < 0 Left h units if h < 0 Example For the parent function y = x 2, write a function rule for the given transformation and graph the preimage and image. A a reflection across the x-axis B a translation up 2 units and right 3 units function rule: y |
= - x 2 graph: function rule: y = (x - 3) 2 + 2 graph: Try This TAKS Grades 9–11 Obj. 2, 5 For each parent function, write a function rule for the given transformation and graph the preimage and image. 1. parent function: y = x 2 transformation: a translation down 1 unit and right 4 units 2. parent function: y = √x transformation: a reflection across the x-axis 3. parent function: y = ⎜x⎟ transformation: a translation up 2 units and left 1 unit 838 838 Chapter 12 Extending Transformational Geometry ���������������������������������������������������������������������� 12-3 Rotations TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties of geometric figures.... Objective Identify and draw rotations. Who uses this? Astronomers can use properties of rotations to analyze photos of star trails. (See Exercise 35.) Also G.2.A, G.2.B, G.7.A 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. E X A M P L E 1 Identifying Rotations Tell whether each transformation appears to be a rotation. Explain. A B Yes; the figure appears to be turned around a point. No; the figure appears to be flipped, not turned. Tell whether each transformation appears to be a rotation. 1a. 1b. Construction Rotate a Figure Using Patty Paper On a sheet of paper, draw a triangle and a point. The point will be the center of rotation. Place a sheet of patty paper on top of the diagram. Trace the triangle and the point. Hold your pencil down on the point and rotate the bottom paper counterclockwise. Trace the triangle. Draw a segment from each vertex to the center of rotation. Your construction should show that a point’s distance to the center of rotation is equal to its image’s distance to the center of rotation. The angle formed by a point, the center of rotation, and the point’s image is the angle by which the figure was rotated. 12-3 Rotations 839 839 |
Rotations A rotation is a transformation about a point P, called the center of rotation, such that each point and its image are the same distance from P, and such that all angles with vertex P formed by a point and its image are congruent. In the figure, ∠APA' is the angle of rotation. E X A M P L E 2 Drawing Rotations Copy the figure and the angle of rotation. Draw the rotation of the triangle about point P by m∠A. Step 1 Draw a segment from each vertex to point P. Unless otherwise stated, all rotations in this book are counterclockwise. Step 2 Construct an angle congruent to ∠A onto each segment. Measure the distance from each vertex to point P and mark off this distance on the corresponding ray to locate the image of each vertex. Step 3 Connect the images of the vertices. 2. Copy the figure and the angle of rotation. Draw the rotation of the segment about point Q by m∠X. Rotations in the Coordinate Plane BY 90° ABOUT THE ORIGIN BY 180° ABOUT THE ORIGIN For more on rotations, see the Transformation Builder on page xxiv. If the angle of a rotation in the coordinate plane is not a multiple of 90°, you can use sine and cosine ratios to find the coordinates of the image. 840 840 Chapter 12 Extending Transformational Geometry ������������������������������������������������������������������������������������������������������������������������� E X A M P L E 3 Drawing Rotations in the Coordinate Plane Rotate △ABC with vertices A (2, -1), B (4, 1), and C (3, 3) by 90° about the origin. The rotation of (x, y) is (-y, x). A (2, -1) → A' (1, 2) B (4, 1) → B' (-1, 4) C (3, 3) → C' (-3, 3) Graph the preimage and image. 3. Rotate △ABC by 180° about the origin. E X A M P L E 4 Engineering Application The Texas Star Ferris wheel has a radius of 106 ft and takes 40 seconds to make a complete rotation. A car starts at position (106, 0). What are the coordinates of the car’s location after 5 seconds? Step 1 Find |
the angle of rotation. Five seconds is 5 __ 40 = 1 __ 8 of a complete revolution, or 1 __ 8 (360°) = 45°. Step 2 Draw a right triangle to represent the car’s location (x, y) after a rotation of 45° about the origin. To review the sine and cosine ratios, see Lesson 8-2, pages 525–532. Step 3 Use the cosine ratio to find the x-coordinate. cos 45° = x _ 106 cos = adj. _ hyp. Solve for x. x = 106 cos 45° ≈ 75.0 Step 4 Use the sine ratio to find the y-coordinate. sin 45° = y _ 106 y = 106 sin 45° ≈ 75.0 sin = opp. _ hyp. Solve for y. The car’s location after 5 seconds is approximately (75.0, 75.0). 4. The London Eye observation wheel has a radius of 67.5 m and takes 30 minutes to make a complete rotation. A car starts at position (67.5, 0). Find the coordinates of the car after 6 minutes. Round to the nearest tenth. THINK AND DISCUSS 1. Describe the image of a rotation of a figure by an angle of 360°. 2. Point A' is a rotation of point A about point P. What is the relationship ̶̶ AP to ̶̶ A'P? of 3. GET ORGANIZED Copy and complete the graphic organizer. 12-3 Rotations 841 841 �������������������������������������������������������������������������������������������������������� 12-3 Exercises Exercises GUIDED PRACTICE Tell whether each transformation appears to be a rotation. p. 839 1. 3. 2. 4. KEYWORD: MG7 12-3 KEYWORD: MG7 Parent. 840 Copy each figure and the angle of rotation. Draw the rotation of the figure about point P by m∠A. 5. 6. 841 Rotate the figure with the given vertices about the origin using the given angle of rotation. 7. A (1, 0), B (3, 2), C (5, 0) ; 90° 8. J (2, 1), K (4, 3), L (2, 4), M (-1, 2 |
) ; 90° 9. D (2, 3), E (-1, 2), F (2, 1) ; 180° 10. P (-1, -1), Q (-4, -2), R (0, -2) ; 180 11. Animation An artist uses a coordinate plane to plan the motion of p. 841 an animated car. To simulate the car driving around a curve, the artist places the car at the point (10, 0) and then rotates it about the origin by 30°. Give the car’s final position, rounding the coordinates to the nearest tenth. PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a rotation. 12. 14. 14. 13. 15. Independent Practice For See Exercises Example 12–15 16–17 18–21 22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 842 842 Chapter 12 Extending Transformational Geometry ���� Copy each figure and the angle of rotation. Draw the rotation of the figure about point P by m∠A. 16. 17. Rotate the figure with the given vertices about the origin using the given angle of rotation. 18. E (-1, 2), F (3, 1), G (2, 3) ; 90° 19. A (-1, 0), B (-1, -3), C (1, -3), D (1, 0) ; 90° 20. P (0, 2), Q (2, 0), R (3, -3) ; 180° 21. L (2, 0), M (-1, -2), N (2, -2) ; 180° 22. Architecture The CN Tower in Toronto, Canada, features a revolving restaurant that takes 72 minutes to complete a full rotation. A table that is 50 feet from the center of the restaurant starts at position (50, 0). What are the coordinates of the table after 6 minutes? Round coordinates to the nearest tenth. Copy each figure. Then draw the rotation of the figure about the red point using the given angle measure. 23. 90° 24. 180° 25. 180° 26. Point Q has coordinates (2, 3). After a rotation about the origin, the image of point Q lies on the y-axis. a. Find the angle of rotation |
to the nearest degree. b. Find the coordinates of the image of point Q. Round to the nearest tenth. Rectangle RSTU is the image of rectangle LMNP under a 180° rotation about point A. Name each of the following. 27. the image of point N 28. the preimage of point S 29. the image of ̶̶̶ MN 30. the preimage of ̶̶ TU 31. This problem will prepare you for the Multi-Step TAKS Prep on page 854. A miniature golf course includes a hole with a windmill. Players must hit the ball through the opening at the base of the windmill while the blades rotate. a. The blades take 20 seconds to make a complete rotation. Through what angle do the blades rotate in 4 seconds? b. Find the coordinates of point A after 4 seconds. (Hint: (4, 3) is the center of rotation.) � � � � � � 12-3 Rotations 843 843 ����������������� �� ��������������������������� Each figure shows a preimage and its image under a rotation. Copy the figure and locate the center of rotation. 32. 33. 34. 35. Astronomy The photograph was made by placing a camera on a tripod and keeping the camera’s shutter open for a long time. Because of Earth’s rotation, the stars appear to rotate around Polaris, also known as the North Star. a. Estimation Estimate the angle of rotation of the stars in the photo. b. Estimation Use your result from part a to estimate the length of time that the camera’s shutter was open. (Hint: If the shutter was open for 24 hours, the stars would appear to make one complete rotation around Polaris.) ������� 36. Estimation In the diagram, △ABC → △A'B'C' under �� a rotation about point P. a. Estimate the angle of rotation. b. Explain how you can draw two segments and can then use a protractor to measure the angle of rotation. c. Copy the figure. Use the method from part b to find the angle of rotation. How does your result compare to your estimate? �� � � 37. Critical Thinking A student wrote the following in his math journal. “Under a rotation, every point moves around the center of rotation by the same angle measure. This means that every point moves the same distance.” Do you agree? |
Explain. �� � � Use the figure for Exercises 38–40. 38. Sketch the image of pentagon ABCDE under a rotation of 90° about the origin. Give the vertices of the image. 39. Sketch the image of pentagon ABCDE under a rotation of 180° about the origin. Give the vertices of the image. 40. Write About It Is the image of ABCDE under a rotation of 180° about the origin the same as its image under a reflection across the x-axis? Explain your reasoning. 41. Construction Copy the figure. Use the construction of an angle congruent to a given angle (see page 22) to construct the image of point X under a rotation about point P by m∠A. � � � � � �� � �� �� � � � � � � � � 844 844 Chapter 12 Extending Transformational Geometry 42. What is the image of the point (-2, 5) when it is rotated about the origin by 90°? (-5, 2) (5, -2) (-5, -2) (2, -5) 43. The six cars of a Ferris wheel are located at the vertices of a regular hexagon. Which rotation about point P maps car A to car C? 60° 90° 120° 135° 44. Gridded Response Under a rotation about the origin, the point (-3, 4) is mapped to the point (3, -4). What is the measure of the angle of rotation? CHALLENGE AND EXTEND 45. Engineering Gears are used to change the speed and direction of rotating parts in pieces of machinery. In the diagram, suppose gear B makes one complete rotation in the counterclockwise direction. Give the angle of rotation and direction for the rotation of gear A. Explain how you got your answer. 46. Given: Prove: ̶̶̶ A'B' is the rotation image of ̶̶ AB ≅ ̶̶̶ A'B' ̶̶ AB about point P. (Hint: Draw auxiliary lines show that △APB ≅ △A'PB'.) ̶̶ AP, ̶̶ BP, ̶̶ A'P, and ̶̶̶ B'P and Once you have proved that the rotation image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof. ̶ |
̶ AB, then AB = A'B'. ̶̶̶ A'B' is a rotation of 47. If 48. If ∠A'B'C' is a rotation of ∠ABC, then m∠ABC = m∠A'B'C'. 49. The rotation △A'B'C' is congruent to the preimage △ABC. 50. If point C is between points A and B, then the rotation C' is between A' and B'. 51. If points A, B, and C are collinear, then the rotations A', B', and C' are collinear. SPIRAL REVIEW Find the value(s) of x when y is 3. (Previous course) 53. y = 2x 2 - 5x - 9 52. y = x 2 - 4x + 7 54. y = x 2 - 2 Find each measure. (Lesson 6-6) 55. m∠XYR 56. QR Given the points A (1, 3), B (5, 0), C (-3, -2), and D (5, -6), find the vector associated with each translation. (Lesson 12-2) 57. the translation that maps point A to point D 58. the translation that maps point D to point B 59. the translation that maps point C to the origin 12-3 Rotations 845 845 �������ge07sec12l03010aABAB�������������������� 12-3 Explore Transformations with Matrices Use with Lesson 12-3 TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties of geometric figures.... Also G.2.B, G.5.B, G.7.A, G.11.A KEYWORD: MG7 Lab12 The vertices of a polygon in the coordinate plane can be represented by a point matrix in which row 1 contains the x-values and row 2 contains the y-values. For example, the triangle with vertices (1, 2), (-2, 0), ⎡ 1 and (3, -4) can be represented by ⎢ ⎣ 2 -2 0 ⎤ 3. ⎥ ⎦ -4 On the graphing calculator, enter a matrix using the Matrix Edit menu. Enter |
the number of rows and columns and then enter the values. Matrix operations can be used to perform transformations. Activity 1 1 Graph the triangle with vertices (1, 0), (2, 4), and (5, 3) on graph paper. Enter the point matrix that represents the vertices into matrix [B] on your calculator. ⎡ 1 2 Enter the matrix ⎢ ⎣ 0 ⎤ 0 into matrix [A] on your calculator. Multiply ⎥ ⎦ -1 [A] * [B] and use the resulting matrix to graph the image of the triangle. Describe the transformation. Try This ⎡ -1 1. Enter the matrix ⎢ ⎣ 0 ⎤ 0 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ 1 matrix to graph the image of the triangle. Describe the transformation. ⎡ 0 2. Enter the matrix ⎢ ⎣ 1 ⎤ 1 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ 0 matrix to graph the image of the triangle. Describe the transformation. 846 846 Chapter 12 Extending Transformational Geometry ���������������������������������� Activity 2 1 Graph the triangle with vertices (0, 0), (3, 1), and (2, 4) on graph paper. Enter the point matrix that represents the vertices into matrix [B] on your calculator. ⎡ 0 2 Enter the matrix ⎢ ⎣ 2 0 2 ⎤ 0 into matrix [A]. Add [A] + [B] and use the resulting ⎥ ⎦ 2 matrix to graph the image of the triangle. Describe the transformation. Try This ⎡ -1 3. Enter the matrix ⎢ ⎣ 4 -1 4 ⎤ -1 into matrix [A]. Add [A] + [B] and use the resulting ⎥ ⎦ 4 matrix to graph the image of the triangle. Describe the transformation. 4. Make a Conjecture How do you think you could use matrices to translate a triangle by the vector 〈a, b〉? Choose several values for a and b and test your conjecture. Activity 3 1 Graph the triangle with vertices (1, 1), (4 |
, 1), and (1, 2) on graph paper. Enter the point matrix that represents the vertices into matrix [B] on your calculator. 2 Enter the matrix ⎡ ⎣ 0 -1 1 0 ⎤ ⎦ into matrix [A]. Multiply [A] * [B] and use the resulting matrix to graph the image of the triangle. Describe the transformation. Try This ⎡ -1 5. Enter the values ⎢ ⎣ 0 ⎤ 0 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ -1 matrix to graph the image of the triangle. Describe the transformation. ⎡ 0 6. Enter the values ⎢ ⎣ -1 ⎤ 1 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ 0 matrix to graph the image of the triangle. Describe the transformation. 12-3 Technology Lab 847 847 12-4 Compositions of Transformations TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties of geometric figures.... Also G.5.C Objectives Apply theorems about isometries. Identify and draw compositions of transformations, such as glide reflections. Vocabulary composition of transformations glide reflection Why learn this? Compositions of transformations can be used to describe chess moves. (See Exercise 11.) A composition of transformations is one transformation followed by another. For example, a glide reflection is the composition of a translation and a reflection across a line parallel to the translation vector. The glide reflection that maps △JKL to △J'K'L' is the composition of a translation along followed by a reflection across line ℓ. v A life-sized chessboard in Galveston The image after each transformation is congruent to the previous image. By the Transitive Property of Congruence, the final image is congruent to the preimage. This leads to the following theorem. Theorem 12-4-1 A composition of two isometries is an isometry. E X A M P L E 1 Drawing Compositions of Isometries Draw the result of the composition of isometries. A Reflect △ABC across line ℓ and then translate it along v |
. Step 1 Draw △A'B'C', the reflection image of △ABC. Step 2 Translate △A'B'C' along find the final image, △A''B''C''. v to 848 848 Chapter 12 Extending Transformational Geometry ���������������������������������������������������������������������������������������������������������������������������������� B △RST has vertices R (1, 2), S (1, 4), and T (-3, 4). Rotate △RST 90° about the origin and then reflect it across the y-axis. Step 1 The rotation image of (x, y) is (-y, x). R (1, 2) → R' (-2, 1), S (1, 4) → S' (-4, 1), and T (-3, 4) → T'(-4, -3). Step 2 The reflection image of (x, y) is (-x, y). R' (-2, 1) → R'' (2, 1), S' (-4, 1) → S'' (4, 1), and T'(-4, -3) → T '' (4, -3). Step 3 Graph the preimage and images. 1. △JKL has vertices J (1, -2), K (4, -2), and L (3, 0). Reflect △JKL across the x-axis and then rotate it 180° about the origin. Theorem 12-4-2 The composition of two reflections across two parallel lines is equivalent to a translation. • The translation vector is perpendicular to the lines. • The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. • The center of rotation is the intersection of the lines. • The angle of rotation is twice the measure of the angle formed by the lines. E X A M P L E 2 Art Application For more on the composition of two reflections, see the Transformation Builder on page xxiv. Tabitha is creating a design for an art project. She reflects a figure across line ℓ and then reflects the image across line m. Describe a single transformation that moves the figure from its starting position to its final position |
. By Theorem 12-4-2, the composition of two reflections across intersecting lines is equivalent to a rotation about the point of intersection. Since the lines are perpendicular, they form a 90° angle. By Theorem 12-4-2, the angle of rotation is 2 · 90° = 180°. 2. What if…? Suppose Tabitha reflects the figure across line n and then the image across line p. Describe a single transformation that is equivalent to the two reflections. 12-4 Compositions of Transformations 849 849 ���������������������������������������������������� Theorem 12-4-3 Any translation or rotation is equivalent to a composition of two reflections. E X A M P L E 3 Describing Transformations in Terms of Reflections Copy each figure and draw two lines of reflection that produce an equivalent transformation. A translation: △ABC → △A'B'C' Step 1 Draw ̶̶̶ AA' and locate the midpoint M of ̶̶̶ AA'. Step 2 Draw the perpendicular ̶̶̶ AM and bisectors of ̶̶̶ A'M. B rotation with center P : △DEF → △D'E'F' To draw the perpendicular bisector of a segment, use a ruler to locate the midpoint, and then use a right angle to draw a perpendicular line. Step 1 Draw ∠DPD'. Draw the angle bisector PX. Step 2 Draw the bisectors of ∠DPX and ∠D'PX. 3. Copy the figure showing the translation that maps LMNP → L'M'N'P'. Draw the lines of reflection that produce an equivalent transformation. THINK AND DISCUSS 1. Which theorem explains why the image of a rectangle that is translated and then rotated is congruent to the preimage? 2. Point A' is a glide reflection of point A along What is the relationship between would use to draw a glide reflection. v and across line ℓ. v and ℓ? Explain the steps you 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe an equivalent transformation and sketch an example. 850 850 Chapter 12 Extending Transformational Geometry ���������������������������������������������������������������������������������������������������������������� |
��������������� 12-4 Exercises Exercises KEYWORD: MG7 12-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain the steps you would use to draw a glide reflection Draw the result of each composition of isometries. p. 848 2. Translate △DEF along u and then reflect it across line ℓ. 3. Reflect rectangle PQRS across line m and then translate it along v. 4. △ABC has vertices A (1, -1), B (4, -1), and C (3, 2). Reflect △ABC across the y-axis and then translate it along the vector 〈0, -2〉.. Sports To create the opening graphics p. 849 for a televised football game, an animator reflects a picture of a football helmet across line ℓ. She then reflects its image across line m, which intersects line ℓ at a 50° angle. Describe a single transformation that moves the helmet from its starting position to its final position. 850 Copy each figure and draw two lines of reflection that produce an equivalent transformation. 6. translation: △EFG → △E'F'G' 7. rotation with center P: △ABC → △A'B'C' Independent Practice Draw the result of each composition of isometries. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–10 11 12–13 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 8. Translate △RST along translate it along v. u and then 9. Rotate △ABC 90° about point P and then reflect it across line ℓ. 10. △GHJ has vertices G (1, -1), H (3, 1), and J (3, -2). Reflect △GHJ across the line y = x and then reflect it across the x-axis. 12-4 Compositions of Transformations 851 851 ������������������������������������������������� 11. Games In chess, a knight moves in the shape of the letter L. The piece moves two spaces horizontally or vertically. Then it turns 90° in either direction and moves |
one more space. a. Describe a knight’s move as a composition of transformations. b. Copy the chessboard with the knight. Label all the positions the knight can reach in one move. c. Label all the positions the knight can reach in two moves. Copy each figure and draw two lines of reflection that produce an equivalent transformation. 12. translation: ABCD → A'B'C'D' 13. rotation with center Q: �� �� △JKL → △J'K'L' �� �� �� �� � � � � �� � � � � 14. /////ERROR ANALYSIS///// The segment with endpoints A (4, 2) and B (2, 1) is reflected across the y-axis. The image is reflected across the x-axis. What transformation is equivalent to the composition of these two reflections? Which solution is incorrect? Explain the error. � � � ��������������������������������� ��������������������������������� ������������������������������� �������������������������������� ��������������������������������� ���������������������������������� ������������������������������� ��������� � ����������������������������� ��������������������������������� ��������������������������� ������������������������������� �������������������������������� ������������������������������� ����� � � 15. Equilateral △ABC is reflected across ̶̶ AB. Then its image is translated along BC. Copy △ABC and draw its final image. � Tell whether each statement is sometimes, always, or never true. 16. The composition of two reflections is equivalent to a rotation. � � 17. An isometry changes the size of a figure. 18. The composition of two isometries is an isometry. 19. A rotation is equivalent to a composition of two reflections. 20. Critical Thinking Given a composition of reflections across two parallel lines, does the order of the reflections matter? For example, does reflecting △ABC across m and then its image across n give the same result as reflecting △ABC across n and then its image across m? Explain. � � � � � 21. Write About It Under a glide reflection, △RST → △R'S'T '. The vertices of △RST are R |
(-3, -2), S (-1, -2), and T (-1, 0). The vertices of △R'S'T'are R' (2, 2), S' (4, 2), and T'(4, 0). Describe the reflection and translation that make up the glide reflection. 852 852 Chapter 12 Extending Transformational Geometry 22. This problem will prepare you for the Multi-Step TAKS Prep on page 854. The figure shows one hole of a miniature golf course where T is the tee and H is the hole. a. Yuriko makes a hole in one as shown by the red arrows. Write the ball’s path as a composition of translations. b. Find a different way to make a hole in one, and write the ball’s path as a composition of translations. 23. △ABC is reflected across the y-axis. Then its image is rotated 90° about the origin. What are the coordinates of the final image of point A under this composition of transformations? (-1, -2) (-2, 1) (1, 2) (-2, -1) 24. Which composition of transformations maps △ABC into the fourth quadrant? Reflect across the x-axis and then reflect across the y-axis. Rotate about the origin by 180° and then reflect across the y-axis. Translate along the vector 〈-5, 0〉 and then rotate about the origin by 90°. Rotate about the origin by 90° and then translate along the vector 〈1, -2〉. 25. Which is equivalent to the composition of two translations? Reflection Rotation Translation Glide reflection CHALLENGE AND EXTEND 26. The point A (3, 1) is rotated 90° about the point P (-1, 2) and then reflected across the line y = 5. Find the coordinates of the image A'. 27. For any two congruent figures in a plane, one can be transformed to the other by a composition of no more than three reflections. Copy the figure. Show how to find a composition of three reflections that maps △MNP to △M'N'P'. 28. A figure in the coordinate plane is reflected across the line y = x + 1 and then across the line y = x + 3. Find a translation vector that is equivalent to the composition of |
the reflections. Write the vector in component form. SPIRAL REVIEW Determine whether the set of ordered pairs represents a function. (Previous course) (-3, -1), (1, 2), (-3, 1), (5, 10) 30. ⎬ ⎨ (-6, -5), (-1, 0), (0, -5), (1, 0) 29. ⎬ ⎨ Find the length of each segment. (Lesson 11-6) 31. ̶̶ EJ 32. ̶̶ CD 33. ̶̶ FH Determine the coordinates of each point after a rotation about the origin by the given angle of rotation. (Lesson 12-3) 35. N (-1, -3) ; 180° 34. F (2, 3) ; 90° 36. Q (-2, 0) ; 90° 12-4 Compositions of Transformations 853 853 ��������������������������������������������� SECTION 12A Congruence Transformations A Hole in One The figure shows a plan for one hole of a miniature golf course. The tee is at point T and the hole is at point H. Each unit of the coordinate plane represents one meter. 1. When a player hits the ball in a straight line from T to H, the path of the ball can be represented by a translation. What is the translation vector? How far does the ball travel? Round to the nearest tenth. 2. The designer of the golf course decides to make the hole more difficult by placing a barrier between the tee and the hole, as shown. To make a hole in one, a player must hit the ball so that ̶̶ DC. What point it bounces off wall along the wall should a player aim for? Explain. 3. Write the path of the ball in Problem 2 as a composition of two translations. What is the total distance that the ball travels in this case? Round to the nearest tenth. 4. The designer decides to remove the barrier and put a revolving obstacle between the tee and the hole. The obstacle consists of a turntable with four equally spaced pillars, as shown. The designer wants the turntable to make one complete rotation in 16 seconds. What should be the coordinates of the pillar at (4, 2) after 2 seconds? |
854 854 Chapter 12 Extending Transformational Geometry ������������������������������������������ SECTION 12A Quiz for Lessons 12-1 Through 12-4 12-1 Reflections Tell whether each transformation appears to be a reflection. 1. 2. Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 3. 4. 12-2 Translations Tell whether each transformation appears to be a translation. 5. 6. 7. A landscape architect represents a flower bed by a polygon with vertices (1, 0), (4, 0), (4, 2), and (1, 2). She decides to move the flower bed to a new location by translating it along the vector 〈-4, -3〉. Draw the flower bed in its final position. 12-3 Rotations Tell whether each transformation appears to be a rotation. 8. 9. Rotate the figure with the given vertices about the origin using the given angle of rotation. 10. A (1, 0), B (4, 1), C (3, 2) ; 180° 11. R (-2, 0), S (-2, 4), T (-3, 4), U (-3, 0) ; 90° 12-4 Compositions of Transformations 12. Draw the result of the following composition of transformations. Translate GHJK along and then reflect it across line m. v 13. △ABC with vertices A (1, 0), B (1, 3), and C (2, 3) is reflected across the y-axis, and then its image is reflected across the x-axis. Describe a single transformation that moves the triangle from its starting position to its final position. Ready to Go On? 855 855 ������� 12-5 Symmetry TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties.... Objective Identify and describe symmetry in geometric figures. Who uses this? Marine biologists use symmetry to classify diatoms. Vocabulary symmetry line symmetry line of symmetry rotational symmetry Also G.2.B, G.5.C Diatoms are microscopic algae that are found in aquatic environments. Scientists use a system that was developed in the 1970s to classify diatoms based on their symmetry. A figure has symmetry if there is a transformation |
of the figure such that the image coincides with the preimage. Line Symmetry A figure has line symmetry (or reflection symmetry) if it can be reflected across a line so that the image coincides with the preimage. The line of symmetry (also called the axis of symmetry) divides the figure into two congruent halves. E X A M P L E 1 Identifying Line Symmetry Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. A B C yes; one line of symmetry no line symmetry yes; five lines of symmetry Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. 1a. 1b. 1c. 856 856 Chapter 12 Extending Transformational Geometry ge07se_c12l05005aABeckmann Rotational Symmetry A figure has rotational symmetry (or radial symmetry) if it can be rotated about a point by an angle greater than 0° and less than 360° so that the image coincides with the preimage. The angle of rotational symmetry is the smallest angle through which a figure can be rotated to coincide with itself. The number of times the figure coincides with itself as it rotates through 360° is called the order of the rotational symmetry. Angle of rotational symmetry: 90° Order Identifying Rotational Symmetry Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. A B C yes; 180°; order: 2 no rotational symmetry yes; 60°; order: 6 Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 2a. 2b. 2c. E X A M P L E 3 Biology Application Describe the symmetry of each diatom. Copy the shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order. A B line symmetry and rotational symmetry; angle of rotational symmetry: 180°; order: 2 line symmetry and rotational symmetry; angle of rotational symmetry: 120°; order: 3 Describe the symmetry of each diatom. Copy the shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order. 3a. 3b. 12-5 Symmetry 857 857 ������������ A three-dimensional figure has plane symmetry if a |
plane can divide the figure into two congruent reflected halves. A three-dimensional figure has symmetry about an axis if there is a line about which the figure can be rotated (by an angle greater than 0° and less than 360°) so that the image coincides with the preimage. E X A M P L E 4 Identifying Symmetry in Three Dimensions Tell whether each figure has plane symmetry, symmetry about an axis, or neither. A trapezoidal prism B equilateral triangular prism plane symmetry plane symmetry and symmetry about an axis Tell whether each figure has plane symmetry, symmetry about an axis, or no symmetry. 4a. cone 4b. pyramid THINK AND DISCUSS 1. Explain how you could use scissors and paper to cut out a shape that has line symmetry. 2. Describe how you can find the angle of rotational symmetry for a regular polygon with n sides. 3. GET ORGANIZED Copy and complete the graphic organizer. In each region, draw a figure with the given type of symmetry. 858 858 Chapter 12 Extending Transformational Geometry �������������������������������������������������������������������������������� 12-5 Exercises Exercises KEYWORD: MG7 12-5 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Describe the line of symmetry of an isosceles triangle. 2. The capital letter T has?. (line symmetry or rotational symmetry) ̶̶̶̶. 856 Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. 3. 4. 5. 857 Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 6. 7. 8. 857 9. Architecture The Pentagon in Alexandria, Virginia, is the world’s largest office building. Copy the shape of the building and draw all lines of symmetry. Give the angle and order of rotational symmetry Tell whether each figure has plane symmetry, symmetry about an axis, or neither. p. 858 10. prism 11. cylinder 12. rectangular prism PRACTICE AND PROBLEM SOLVING Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. 13. 14. 15. Independent Practice For See Exercises Example 13– |
15 16–18 19 20–22 1 2 3 4 TEKS TEKS TAKS TAKS Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. Skills Practice p. S27 Application Practice p. S39 16. 17. 18. 12-5 Symmetry 859 859 ����������������� 19. Art Op art is a style of art that uses optical effects to create an impression of movement in a painting or sculpture. The painting at right, Vega-Tek, by Victor Vasarely, is an example of op art. Sketch the shape in the painting and draw any lines of symmetry. If there is rotational symmetry, give the angle and order. Tell whether each figure has plane symmetry, symmetry about an axis, or neither. 20. sphere 21. triangular pyramid 22. torus Draw a triangle with the following number of lines of symmetry. Then classify the triangle. 23. exactly one line of symmetry 24. three lines of symmetry 25. no lines of symmetry Data Analysis The graph shown, called the standard normal curve, is used in statistical analysis. The area under the curve is 1 square unit. There is a vertical line of symmetry at x = 0. The areas of the shaded regions are indicated on the graph. 26. Find the area under the curve for x > 0. � ���� ����� � 27. Find the area under the curve for x > 3. �� �� �� � � � � 28. If a point under the curve is selected at random, what is the probability that the x-value of the point will be between -1 and 1? Tell whether the figure with the given vertices has line symmetry and/or rotational symmetry. Give the angle and order if there is rotational symmetry. Draw the figure and any lines of symmetry. 29. A (-2, 2), B (2, 2), C (1, -2), D (-1, -2) 30. R (-3, 3), S (3, 3), T (3, -3), U (-3, -3) 31. J (4, 4), K (-2, 2), L (2, -2) 32. A (3, 1), B (0, 2), C (-3, 1), D (-3, -1), E (0, -2), F |
(3, -1) 33. Art The Chokwe people of Angola are known for their traditional sand designs. These complex drawings are traced out to illustrate stories that are told at evening gatherings. Classify the symmetry of the Chokwe design shown. Algebra Graph each function. Tell whether the graph has line symmetry and/or rotational symmetry. If there is rotational symmetry, give the angle and order. Write the equations of any lines of symmetry. 35. y = (x - 2) 2 34. y = x 2 36. y = x 3 860 860 Chapter 12 Extending Transformational Geometry 37. This problem will prepare you for the Multi-Step TAKS Prep on page 880. This woodcut, entitled Circle Limit III, was made by Dutch artist M. C. Escher. a. Does the woodcut have line symmetry? If so, describe the lines of symmetry. If not, explain why not. b. Does the woodcut have rotational symmetry? If so, give the angle and order of the symmetry. If not, explain why not. c. Does your answer to part b change if color is not taken into account? Explain. Classify the quadrilateral that meets the given conditions. First make a conjecture and then verify your conjecture by drawing a figure. 38. two lines of symmetry perpendicular to the sides and order-2 rotational symmetry 39. no line symmetry and order-2 rotational symmetry 40. two lines of symmetry through opposite vertices and order-2 rotational symmetry 41. four lines of symmetry and order-4 rotational symmetry 42. one line of symmetry through a pair of opposite vertices and no rotational symmetry 43. Physics High-speed photography makes it possible to analyze the physics behind a water splash. When a drop lands in a bowl of liquid, the splash forms a crown of evenly spaced points. What is the angle of rotational symmetry for a crown with 24 points? 44. Critical Thinking What can you conclude about a rectangle that has four lines of symmetry? Explain. 45. Geography The Isle of Man is an island in the Irish Sea. The island’s symbol is a triskelion that consists of three running legs radiating from the center. Describe the symmetry of the triskelion. 46. Critical Thinking Draw several examples of figures that have two perpendicular lines of symmetry. What other type of symmetry do these figures have? Make a conjecture based on your observation. Each figure shows |
part of a shape with a center of rotation and a given rotational symmetry. Copy and complete each figure. 47. order 4 48. order 6 49. order 2 50. Write About It Explain the connection between the angle of rotational symmetry and the order of the rotational symmetry. That is, if you know one of these, explain how you can find the other. 12-5 Symmetry 861 861 51. What is the order of rotational symmetry for the hexagon shown? 2 3 4 6 52. Which of these figures has exactly four lines of symmetry? Regular octagon Equilateral triangle Isosceles triangle Square 53. Consider the graphs of the following equations. Which graph has the y-axis as a line of symmetry? y = (x - 3x + 3⎟ 54. Donnell designed a garden plot that has rotational symmetry, but not line symmetry. Which of these could be the shape of the plot? CHALLENGE AND EXTEND 55. A regular polygon has an angle of rotational symmetry of 5°. How many sides does the polygon have? 56. How many lines of symmetry does a regular n-gon have if n is even? if n is odd? Explain your reasoning. Find the equation of the line of symmetry for the graph of each function. 57. y = (x + 4) 2 58. y = ⎜x - 2⎟ 59. y = 3x 2 + 5 Give the number of axes of symmetry for each regular polyhedron. Describe all axes of symmetry. 60. cube 61. tetrahedron 62. octahedron SPIRAL REVIEW 63. Shari worked 16 hours last week and earned $197.12. The amount she earns in one week is directly proportional to the number of hours she works in that week. If Shari works 20 hours one week, how much does she earn? (Previous course) Find the slant height of each figure. (Lesson 10-5) 64. a right cone with radius 5 in. and surface area 61π in 2 65. a square pyramid with lateral area 45 cm 2 and surface area 65.25 cm 2 66. a regular triangular pyramid with base perimeter 24 √ 3 m and surface area 120 √ 3 m 2 Determine the coordinates of the final image of the point P (-1, 4) under each composition of isometries. (Lesson 12 |
-4) 67. Reflect point P across the line y = x and then translate it along the vector 〈2, -4〉. 68. Rotate point P by 90° about the origin and then reflect it across the y-axis. 69. Translate point P along the vector 〈1, 0〉 and then rotate it 180° about the origin. 862 862 Chapter 12 Extending Transformational Geometry 12-6 Tessellations TEKS G.5.C Geometric patterns: use... compositions [of transformations] to make connections between mathematics and the real world.... Objectives Use transformations to draw tessellations. Identify regular and semiregular tessellations and figures that will tessellate. Vocabulary translation symmetry frieze pattern glide reflection symmetry tessellation regular tessellation semiregular tessellation Who uses this? Repeating patterns play an important role in traditional Native American art. A pattern has translation symmetry if it can be translated along a vector so that the image coincides with the preimage. A frieze pattern is a pattern that has translation symmetry along a line. Both of the frieze patterns shown below have translation symmetry. The pattern on the right also has glide reflection symmetry. A pattern with glide reflection symmetry coincides with its image after a glide reflection. E X A M P L E 1 Art Application Identify the symmetry in each frieze pattern. A B When you are given a frieze pattern, you may assume that the pattern continues forever in both directions. translation symmetry and glide reflection symmetry translation symmetry Identify the symmetry in each frieze pattern. 1a. 1b. A tessellation, or tiling, is a repeating pattern that completely covers a plane with no gaps or overlaps. The measures of the angles that meet at each vertex must add up to 360°. In the tessellation shown, each angle of the quadrilateral occurs once at each vertex. Because the angle measures of any quadrilateral add to 360°, any quadrilateral can be used to tessellate the plane. Four copies of the quadrilateral meet at each vertex. 12-6 Tessellations 863 863 The angle measures of any triangle add up to 180°. This means that any triangle can be used to tessellate a plane. Six copies of the triangle meet at each vertex, as shown. m∠1 |
+ m∠2 + m∠3 = 180° m∠1 + m∠2 + m∠3 + m∠1 + m∠2 + m∠3 = 360° E X A M P L E 2 Using Transformations to Create Tessellations Copy the given figure and use it to create a tessellation. Step 1 Rotate the triangle 180° about the midpoint of one side. Step 2 Translate the resulting pair of triangles to make a row of triangles. Step 3 Translate the row of triangles to make a tessellation. A B Step 1 Rotate the quadrilateral 180° about the midpoint of one side. Step 2 Translate the resulting pair of quadrilaterals to make a row of quadrilaterals. Step 3 Translate the row of quadrilaterals to make a tessellation. 2. Copy the given figure and use it to create a tessellation. A regular tessellation is formed by congruent regular polygons. A semiregular tessellation is formed by two or more different regular polygons, with the same number of each polygon occurring in the same order at every vertex. Every vertex has two squares and three triangles in this order: square, triangle, square, triangle, triangle. 864 864 Chapter 12 Extending Transformational Geometry �������������������������������������������������������������������������������� Tessellations When I need to decide if given figures can be used to tessellate a plane, I look at angle measures. To form a regular tessellation, the angle measures of a regular polygon must be a divisor of 360°. To form a semiregular tessellation, the angle measures around a vertex must add up to 360°. For example, regular octagons and equilateral triangles cannot be used to make a semiregular tessellation because no combination of 135° and 60° adds up to exactly 360°. Ryan Gray Sunset High School E X A M P L E 3 Classifying Tessellations Classify each tessellation as regular, semiregular, or neither. A B C Two regular octagons and one square meet at each vertex. The tessellation is semiregular. Only squares are used. The tessellation is regular. Irregular hexagons are used in the |
tessellation. It is neither regular nor semiregular. Classify each tessellation as regular, semiregular, or neither. 3a. 3b. 3c. E X A M P L E 4 Determining Whether Polygons Will Tessellate Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. A B No; each angle of the pentagon measures 108°, and 108 is not a divisor of 360. Yes; two octagons and one square meet at each vertex. 135° + 135° + 90° = 360° Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. 4a. 4b. 12-6 Tessellations 865 865 ������� THINK AND DISCUSS 1. Explain how you can identify a frieze pattern that has glide reflection symmetry. 2. Is it possible to tessellate a plane using circles? Why or why not? 3. GET ORGANIZED Copy and complete the graphic organizer. 12-6 Exercises Exercises KEYWORD: MG7 12-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Sketch a pattern that has glide reflection symmetry. 2. Describe a real-world example of a regular tessellation. 863 Transportation The tread of a tire is the part that makes contact with the ground. Various tread patterns help improve traction and increase durability. Identify the symmetry in each tread pattern. 3. 4 Copy the given figure and use it to create a tessellation. p. 864 6. 7. 5. 8 Classify each tessellation as regular, semiregular, or neither. p. 865 9. 10. 11. 865 Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. 12. 13. 14. 866 866 Chapter 12 Extending Transformational Geometry ������������������������������������������������������������������������������������ Independent Practice For See Exercises Example 15–17 18–20 21–23 24–26 1 2 |
3 4 TEKS TEKS TAKS TAKS Skills Practice p. S27 Application Practice p. S39 PRACTICE AND PROBLEM SOLVING Interior Decorating Identify the symmetry in each wallpaper border. 15. 16. Copy the given figure and use it to create a tessellation. 18. 18. 19. 17. 20. Classify each tessellation as regular, semiregular, or neither. 21. 22. 23. Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. 24. 25. 26. 27. Physics A truck moving down a road creates whirling pockets of air called a vortex train. Use the figure to classify the symmetry of a vortex train. Identify all of the types of symmetry (translation, reflection, and/or rotation) in each tessellation. 28. 29. 30. Tell whether each statement is sometimes, always, or never true. 31. A triangle can be used to tessellate a plane. 32. A frieze pattern has glide reflection symmetry. 33. The angles at a vertex of a tessellation add up to 360°. 34. It is possible to use a regular pentagon to make a regular tessellation. 35. A semiregular tessellation includes scalene triangles. 12-6 Tessellations 867 867 36. This problem will prepare you for the Multi-Step TAKS Prep on page 880. Many of the patterns in M. C. Escher’s works are based on simple tessellations. For example, the pattern at right is based on a tessellation of equilateral triangles. Identify the figure upon which each pattern is based. a. a. b. Use the given figure to draw a frieze pattern with the given symmetry. 37. translation symmetry 38. glide reflection symmetry 39. translation symmetry 40. glide reflection symmetry 41. Optics A kaleidoscope is formed by three mirrors joined to form the lateral surface of a triangular prism. Small objects are reflected in the mirrors to form a tessellation. Copy the triangle and reflect the triangle over each side. Repeat to form a tessellation. Describe the symmetry of the tessellation. 42. Critical Thinking The pattern on a soccer ball is a tessellation of a |
sphere using regular hexagons and regular pentagons. Can these two shapes be used to tessellate a plane? Explain your reasoning. 43. Chemistry A polymer is a substance made of repeating chemical units or molecules. The repeat unit is the smallest structure that can be repeated to create the chain. Draw the repeat unit for polypropylene, the polymer shown below. 44. The dual of a tessellation is formed by connecting the centers of adjacent polygons with segments. Copy or trace the semiregular tessellation shown and draw its dual. What type of polygon makes up the dual tessellation? 45. Write About It You can make a regular tessellation from an equilateral triangle, a square, or a regular hexagon. Explain why these are the only three regular tessellations that are possible. 868 868 Chapter 12 Extending Transformational Geometry �������������������������������� 46. Which frieze pattern has glide reflection symmetry? 47. Which shape CANNOT be used to make a regular tessellation? Equilateral triangle Square Regular pentagon Regular hexagon 48. Which pair of regular polygons can be used to make a semiregular tessellation? CHALLENGE AND EXTEND 49. Some shapes can be used to tessellate a plane in more than one way. Three tessellations that use the same rectangle are shown. Draw a parallelogram and draw at least three tessellations using that parallelogram. Determine whether each figure can be used to tessellate three-dimensional space. 50. 51. 52. SPIRAL REVIEW 53. A book is on sale for 15% off the regular price of $8.00. If Harold pays with a $10 bill and receives $2.69 in change, what is the sales tax rate on the book? (Previous course) 54. Louis lives 5 miles from school and jogs at a rate of 6 mph. Andrea lives 3.9 miles from school and jogs at a rate of 6.5 mph. Andrea leaves her house at 7:00 A.M. When should Louis leave his house to arrive at school at the same time as Andrea? (Previous course) Write the equation of each circle. (Lesson 11-7) 55. ⊙P with center (-2, 3) and radius √ 5 56. ⊙Q that passes through ( |
3, 4) and has center (0, 0) 57. ⊙T that passes through (1, -1) and has center (5, -3) Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. (Lesson 12-5) 58. 59. 60. 12-6 Tessellations 869 869 12-6 Use Transformations to Extend Tessellations In Lesson 12-6, you saw that you can use any triangle or quadrilateral to tessellate a plane. In this lab, you will learn how to use transformations to turn these basic patterns into more-complex tessellations. Use with Lesson 12-6 TEKS G.5.C Geometric patterns: use properties of transformations and their compositions to make connections between mathematics and the real world.... Activity 1 1 Cut a rectangle out of heavy paper. 2 Cut a piece from one side of the rectangle and translate it to the opposite side. Tape it into place. 3 Repeat the process with the other pair of sides. 4 The resulting shape will tessellate the plane. Trace around the shape to create a tessellation. Try This 1. Repeat Activity 1, starting with a parallelogram. 2. Repeat Activity 1, starting with a hexagon whose opposite sides are congruent and parallel. 3. Add details to one of your tessellations to make it look like a pattern of people, animals, flowers, or other objects. 870 870 Chapter 12 Extending Transformational Geometry Activity 2 1 Cut a triangle out of heavy paper. 2 Find the midpoint of one side. Cut a piece from one half of this side of the triangle and rotate the piece 180°. Tape it to the other half of this side. 3 Repeat the process with the other two sides. 4 The resulting shape will tessellate the plane. Trace around the shape to create a tessellation. Try This 4. Repeat Activity 2, starting with a quadrilateral. 5. How is this tessellation different from the ones you created in Activity 1? 6. Add details to one of your tessellations to make it look like a pattern of people, animals, flowers, or other objects. 12-6 Geometry Lab 871 871 12-7 Dilations TEKS G.11.A Similarity and the geometry of shape: use and extend similarity |
properties and transformations.... Objective Identify and draw dilations. Vocabulary center of dilation enlargement reduction Also G.2.A, G.11.B, G.11.D Who uses this? Artists use dilations to turn sketches into large-scale paintings. (See Example 2.) Recall that a dilation is a transformation that changes the size of a figure but not the shape. The image and the preimage of a figure under a dilation are similar. E X A M P L E 1 Identifying Dilations For a dilation with scale factor k, if k > 0, the figure is not turned or flipped. If k < 0, the figure is rotated by 180°. Tell whether each transformation appears to be a dilation. Explain. A B Yes; the figures are similar,and the image is not turned or flipped. No; the figures are not similar. Tell whether each transformation appears to be a dilation. 1a. 1b. Construction Dilate a Figure by a Scale Factor of 2 Draw a triangle and a point outside the triangle. The point is the center of dilation. Use a straightedge to draw a line through the center of dilation and each vertex of the triangle. Set the compass to the distance from the center of dilation to a vertex. Mark this distance along the line for each vertex as shown. Connect the vertices of the image. In the construction, the lines connecting points of the image with the corresponding points of the preimage all intersect at the center of dilation. Also, the distance from the center to each point of the image is twice the distance to the corresponding point of the preimage. 872 872 Chapter 12 Extending Transformational Geometry Dilations is the same for every point P. A dilation, or similarity transformation, is a transformation in which the lines connecting every point P with its image P' all intersect at a point C, called the center of dilation. CP' ___ CP The scale factor k of a dilation is the ratio of a linear measurement of the image to a corresponding measurement of the preimage. In the figure, k = P'Q' ___. PQ A dilation enlarges or reduces all dimensions proportionally. A dilation with a scale factor greater than 1 is an enlargement, or expansion. A dilation with a scale factor greater than 0 but less than 1 is a reduction, or contraction. E X |
A M P L E 2 Drawing Dilations Copy the triangle and the center of dilation P. Draw the image of △ABC under a dilation with a scale factor of 1 __. 2 Step 1 Draw a line through P and each vertex. Step 2 On each line, mark half the distance from P to the vertex. Step 3 Connect the vertices of the image. 2. Copy the figure and the center of dilation. Draw the dilation of RSTU using center Q and a scale factor of 3. E X A M P L E 3 Art Application An artist is creating a large painting from a photograph by dividing the photograph into squares and dilating each square by a scale factor of 4. If the photograph is 20 cm by 25 cm, what is the perimeter of the painting? The scale factor of the dilation is 4, so a 1 cm by 1 cm square on the photograph represents a 4 cm by 4 cm square on the painting. Find the dimensions of the painting. b = 4 (25) = 100 cm h = 4 (20) = 80 cm Multiply each dimension by the scale factor, 4. Find the perimeter of the painting. P = 2 (100 + 80) = 360 cm P = 2 (b + h) 3. What if…? In Example 3, suppose the photograph is a square with sides of length 10 in. Find the area of the painting. 12-7 Dilations 873 873 �������������������������� Dilations in the Coordinate Plane If P (x, y) is the preimage of a point under a dilation centered at the origin with scale factor k, then the image of the point is P' (kx, ky). If the scale factor of a dilation is negative, the preimage is rotated by 180°. For k > 0, a dilation with a scale factor of -k is equivalent to the composition of a dilation with a scale factor of k that is rotated 180° about the center of dilation. E X A M P L E 4 Drawing Dilations in the Coordinate Plane Draw the image of a triangle with vertices A (-1, 1), B (-2, -1), and C (-1, -2) under a dilation with a scale factor of -2 centered at the origin. The dilation of (x, y) is (-2x, -2y). A (-1, 1) → A' |
(-2 (-1), -2 (1) ) = A' (2, -2) B (-2, -1) → B' (-2 (-2), -2 (-1) ) = B' (4, 2) C (-1, -2) → C' (-2 (-1), -2 (-2) ) = C' (2, 4) Graph the preimage and image. 4. Draw the image of a parallelogram with vertices R (0, 0), S (4, 0), T (2, -2), and U (-2, -2) under a dilation centered at the origin with a scale factor of - 1 __ 2. THINK AND DISCUSS 1. Given a triangle and its image under a dilation, explain how you could use a ruler to find the scale factor of the dilation. 2. A figure is dilated by a scale factor of k, and then the image is rotated 180° about the center of dilation. What single transformation would produce the same image? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the dilation with the given scale factor. 874 874 Chapter 12 Extending Transformational Geometry ���������������������������������������������������������������������������������������������������������������������������������������������������������� 12-7 Exercises Exercises KEYWORD: MG7 12-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary What are the center of dilation and scale factor for the transformation (x, y) → (3x, 3y Tell whether each transformation appears to be a dilation. p. 872 2. 4. 3. 5. 873 Copy each triangle and center of dilation P. Draw the image of the triangle under a dilation with the given scale factor. 6. Scale factor: 2 7. Scale factor: 1 __. 873 8. Architecture A blueprint shows a reduction of a room using a scale factor of 1 __ 50. In the blueprint, the room’s length is 8 in., and its width is 6 in. Find the perimeter of the room. 874 Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 9. A (1, 0) |
, B (2, 2), C (4, 0) ; scale factor: 2 10. J (-2, 2), K (4, 2), L (4, -2), M (-2, -2) ; scale factor: 1 _ 2 11. D (-3, 3), E (3, 6), F (3, 0) ; scale factor: - 1 _ 3 12. P (-2, 0), Q (-1, 0), R (0, -1), S (-3, -1) ; scale factor: -2 PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a dilation. 13. Independent Practice For See Exercises Example 13–16 17–18 19 20–23 1 2 3 4 TEKS TEKS TAKS TAKS 15. Skills Practice p. S27 Application Practice p. S39 14. 16. 12-7 Dilations 875 875 �� Copy each rectangle and the center of dilation P. Draw the image of the rectangle Copy each rectangle and the center of dilation under a dilation with the given scale factor. under a dilation with the given scale factor. 17. 17. scale factor: 3 18. scale factor: 1 __ 2 Mosaics 19. Art Jeff is making a mosaic by gluing 1 cm square tiles onto a photograph. He starts with a 6 cm by 8 cm rectangular photo and enlarges it by a scale factor of 1.5. How many tiles will Jeff need in order to cover the enlarged photo? This mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001. Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 20. M (0, 3), N (6, 0), P (0, -3) ; scale factor: - 1_ 3 21. A (-1, 3), B (1, 1), C (-4, 1) ; scale factor: -1 22. R (1, 0), S (2, 0), T (2, -2), U (-1, -2) ; scale factor: -2 23. D (4, 0), E (2, -4 |
), F (-2, -4), G (-4, 0), H (-2, 4), J (2, 4) ; scale factor: - 1 _ 2 Each figure shows the preimage (blue) and image (red) under a dilation. Write a similarity statement based on the figure. 24. 25. 26. The rectangular prism shown is enlarged by a dilation with scale factor 4. Find the surface area and volume of the image. Copy each figure and locate the center of dilation. 27. 28. 29. 30. This problem will prepare you for the Multi-Step TAKS Prep on page 880. This lithograph, Drawing Hands, was made by M. C. Escher in 1948. a. In the original drawing, the rectangular piece of paper from which the hands emerge measures 27.6 cm by 19.9 cm. On a poster of the drawing, the paper is 82.8 cm long. What is the scale factor of the dilation that was used to make the poster? b. What is the area of the paper on the poster? 876 876 Chapter 12 Extending Transformational Geometry ������������������������������ 31. /////ERROR ANALYSIS///// Rectangle A'B'C'D' is the image of rectangle ABCD under a dilation. Which calculation of the area of rectangle A'B'C'D' is incorrect? Explain the error. 32. Optometry The pupil is the circular opening that allows light into the eye. a. An optometrist dilates a patient’s pupil from 6 mm to 8 mm. What is the scale factor for this dilation? b. To the nearest tenth, find the area of the pupil before and after the dilation. c. As a percentage, how much more light is admitted to the eye after the dilation? 33. Estimation In the diagram, △ABC → △A'B'C' under a dilation with center P. a. Estimate the scale factor of the dilation. b. Explain how you can use a ruler to make measurements and to calculate the scale factor. c. Use the method from part b to calculate the scale factor. How does your result compare to your estimate? 34. △ABC has vertices A (-1, 1), B (2, 1), and C (2, 2). a. Draw the image of △ABC |
under a dilation centered at the origin with scale factor 2 followed by a reflection across the x-axis. b. Draw the image of △ABC under a reflection across the x-axis followed by a dilation centered at the origin with scale factor 2. c. Compare the results of parts a and b. Does the order of the transformations matter? 35. Astronomy The image of the sun projected through the hole of a pinhole camera (the center of dilation) has a diameter of 1 __ 4 in. The diameter of the sun is 870,000 mi. What is the scale factor of the dilation? 12-7 Dilations 877 877 ��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� Multi-Step △ABC with vertices A (-2, 2), B (1, 3), and C (1, -1) is transformed by a dilation centered at the origin. For each given image point, find the scale factor of the dilation and the coordinates of the remaining image points. Graph the preimage and image on a coordinate plane. 36. A' (-4, 4) 38. B' (-1, -3) 37. C' (-2, 2) 39. Critical Thinking For what values of the scale factor is the image of a dilation congruent to the preimage? Explain. 40. Write About It When is a dilation equivalent to a rotation by 180°? Why? 41. Write About It Is the composition of a dilation with scale factor m followed by a dilation with scale factor n equivalent to a single dilation with scale factor mn? Explain your reasoning. Construction Copy each figure. Then use a compass and straightedge to construct the dilation of the figure with the given scale factor and point P as the center of dilation. 42. scale factor: 1 _ 2 43. scale factor: 2 44. scale factor: -1 45. scale factor: -2 46. Rectangle ABCD is transformed by a dilation centered at the origin. Which scale factor produces an image |
that has a vertex at (0, -2)? - 1 _ 2 -1 -2 -4 47. Rectangle ABCD is enlarged under a dilation centered at the origin with scale factor 2.5. What is the perimeter of the image? 15 24 30 50 48. Gridded Response What is the scale factor of a dilation centered at the origin that maps the point (-2, 3) to the point (-8.4, 12.6)? 49. Short Response The rules for a photo contest state that entries must have an area no greater than 100 cm 2. Amber has a 6 cm by 8 cm digital photo, and she uses software to enlarge it by a scale factor of 1.5. Does the enlargement meet the requirements of the contest? Show the steps you used to decide your answer. 878 878 Chapter 12 Extending Transformational Geometry ������������ CHALLENGE AND EXTEND 50. Rectangle ABCD has vertices A (0, 2), B (1, 2), C (1, 0), and D (0, 0). a. Draw the image of ABCD under a dilation centered at point P with scale factor 2. b. Describe the dilation in part a as a composition of a dilation centered at the origin followed by a translation. c. Explain how a dilation with scale factor k and center of dilation (a, b) can be written as a composition of a dilation centered at the origin and a translation. 51. The equation of line ℓ is y = -x + 2. Find the equation of the image of line ℓ after a dilation centered at the origin with scale factor 3. SPIRAL REVIEW 52. Jerry has a part-time job waiting tables. He kept records comparing the number of customers served to his total amount of tips for the day. If this trend continues, how many customers would he need to serve in order to make $68.00 in tips for the day? (Previous course) Customers per Day Tips per Day ($) 15 20 20 28 25 36 30 44 Find the perimeter and area of each polygon with the given vertices. (Lesson 9-4) 53. J (-3, -2), K (0, 2), L (7, 2), and M (4, -2) 54. D (-3, 0), E (1, 2), and F |
(-1, -4) Determine whether the polygons can be used to tessellate a plane. (Lesson 12-6) 55. a right triangle and a square 56. a regular nonagon and an equilateral triangle Using Technology Use a graphing calculator to complete the following. 1. △ABC with vertices A (3, 4), B (5, 2), and C (1, 1) can be represented ⎡ 3 ⎢ by the point matrix ⎣ 4 5 2 ⎤ 1. Enter these values into matrix [B] on ⎥ ⎦ 1 your calculator. (See page 846.) ⎤ 0 can be used to perform a dilation with scale factor ⎥ ⎦ 2 ⎡ 2 2. The matrix ⎢ ⎣ 0 2. Enter these values into matrix [A] on your calculator and find [A] * [B]. Graph the triangle represented by the resulting point matrix. 3. Make a conjecture about the matrix that could be used to perform a dilation with scale factor − 1 __ 2. Enter the values into matrix [A] on your calculator. 4. Test your conjecture by finding [A] * [B] and graphing the triangle represented by the resulting point matrix. 12-7 Dilations 879 879 ��������� SECTION 12B Patterns Tessellation Fascination A museum is planning an exhibition of works by the Dutch artist M. C. Escher (1898– 1972). The exhibit will include the five drawings shown here. 1. Tell whether each drawing has parallel lines of symmetry, intersecting lines of symmetry, or no lines of symmetry. 2. Tell whether each drawing has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 3. Tell whether each drawing is a tessellation. If so, identify the basic figure upon which the tessellation is based. Drawing A Drawing B Drawing C Drawing D 4. The entrance to the exhibit will include a large mural based on drawing E. In the original drawing, the cover of the book measures 13.2 cm by 11.1 cm. In the mural, the book cover will have an area of 21,098.88 cm 2. What is the scale factor of the dilation that will be used to make the mural? Drawing E 880 880 Chapter 12 Extending Transformational Geometry SECTION 12B Qu |
iz for Lessons 12-5 Through 12-7 12-5 Symmetry Explain whether each figure has line symmetry. If so, copy the figure and draw all lines of symmetry. 1. 2. 3. Explain whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 4. 5. 6. 12-6 Tessellations Copy the given figure and use it to create a tessellation. 7. 8. Classify each tessellation as regular, semiregular, or neither. 10. 11. 9. 12. 13. Determine whether it is possible to tessellate a plane with regular octagons. If so, draw the tessellation. If not, explain why. 12-7 Dilations Tell whether each transformation appears to be a dilation. 14. 15. 16. Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 17. A (0, 2), B (-1, 0), C (0, -1), D (1, 0) ; scale factor: 2 18. P (-4, -2), Q (0, -2), R (0, 0), S (-4, 0) ; scale factor: - 1 _ 2 Ready to Go On? 881 881 EXTENSION EXTENSION Using Patterns to Generate Fractals TEKS G.5.C Geometric patterns: use properties of transformations and their compositions to make connections between mathematics and the real world.... Objective Describe iterative patterns that generate fractals. Vocabulary self-similar iteration fractal Look closely at one of the large spirals in the Romanesco broccoli. You will notice that it is composed of many smaller spirals, each of which has the same shape as the large one. This is an example of self-similarity. A figure is self-similar if it can be divided into parts that are similar to the entire figure. You can draw self-similar figures by iteration, the repeated application of a rule. To create a self-similar tree, start with the figure shown in stage 0. Replace each of its branches with the original figure to get the figure in stage 1. Again replace the branches with the original figure to get the figure in stage 2. Continue the pattern to generate the tree. Stage 0 Stage 1 Stage 2 Stage 3 Stage 8 A figure |
that is generated by iteration is called a fractal. E X A M P L E 1 Creating Fractals Continue the pattern to draw stages 3 and 4 of this fractal, which is called the Sierpinski triangle. To go from one stage to the next, remove an equilateral triangle from each remaining black triangle. Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 1. Explain how to go from one stage to the next to create the Koch snowflake fractal. 882 882 Chapter 12 Extending Transformational Geometry Stage 0 Stage 1 Stage 2 Stage 3 EXTENSION Exercises Exercises Explain how to go from one stage to the next to generate each fractal. 1. Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 Stage 10 2. Stage 0 Stage 1 Stage 2 Stage 3 3. The three-dimensional figure in the photo is called a Sierpinski tetrahedron. a. Describe stage 0 for this fractal. b. Explain how to go from one stage to the next to generate the Sierpinski tetrahedron. 4. A fractal is generated according to the following rules. Stage 0 is a segment. To go from one stage to the next, replace each segment with the figure at right. Draw Stage 2 of this fractal. 5. The first four rows of Pascal’s triangle are shown in the hexagonal tessellation at right. The beginning and end of each row is a 1. To find each remaining number, add the two numbers to the left and right from the row above. a. Continue the pattern to write the first eight rows of Pascal’s triangle. b. Shade all the hexagons that contain an odd number. c. What fractal does the resulting pattern resemble? 6. Write About It Explain why the fern leaf at right is an example of self-similarity. Chapter 12 Extension 883 883 ���������� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary center of dilation......................... 873 line of symmetry.......................... 856 composition of transformations........... 848 reduction........... |
..................... 873 enlargement............................. 873 regular tessellation....................... 864 frieze pattern............................. 863 rotational symmetry...................... 857 glide reflection........................... 848 semiregular tessellation................... 864 glide reflection symmetry................. 863 symmetry................................ 856 isometry................................. 824 tessellation............................... 863 line symmetry............................ 856 translation symmetry..................... 863 Complete the sentences below with vocabulary words from the list above. 1. A(n)? is a pattern formed by congruent regular polygons. ̶̶̶̶ 2. A pattern that has translation symmetry along a line is called a(n)?. ̶̶̶̶ 3. A transformation that does not change the size or shape of a figure is a(n)?. ̶̶̶̶ 4. One transformation followed by another is called a(n)?. ̶̶̶̶ 12-1 Reflections (pp. |
824–830) TEKS G.2.A, G.2.B, G.7.A, G.10.A E X A M P L E EXERCISES ■ Reflect the figure with the given vertices across the given line. A (1, -2), B (4, -3), C (3, 0) ; y = x To reflect across the line y = x, interchange the x- and y-coordinates of each point. The images of the vertices are A' (-2, 1), B' (-3, 4), and C' (0, 3). Tell whether each transformation appears to be a reflection. 5. 6. 7. 8. Reflect the figure with the given vertices across the given line. 9. E (-3, 2), F (0, 2), G (-2, 5) ; x-axis 10. J (2, -1), K (4, -2), L (4, -3), M (2, -3) ; y-axis 11. P (2, -2), Q (4, -2), R (3, -4) ; y = x 12. A (2, 2), B (-2, 2), C (-1, 4) ; y = x 884 884 Chapter 12 Extending Transformational Geometry �������������������������� 12-2 Translations (pp. 831–837) TEKS G.2.A, G.2.B, G.7.A, G.10.A E X A M P L E EXERCISES ■ Translate the figure with the given vertices along the given vector. D (-4, 4), E (-4, 2), F (-1, 1), G (-2, 3) ; 〈5, -5〉 To translate along 〈5, -5〉, add 5 to the x-coordinate of each point and add -5 to the y-coordinate of each point. The vertices of the image are D' (1, -1), E' (1, -3), F' (4, -4), and G' (3, -2). Tell whether each transformation appears to be a translation. 13. 14. 15. 16. Translate the figure with the given |
vertices along the given vector. 17. R (1, -1), S (1, -3), T (4, -3), U (4, -1) ; 〈-5, 2〉 18. A (-4, -1), B (-3, 2), C (-1, -2) ; 〈6, 0〉 19. M (1, 4), N (4, 4), P (3, 1) ; 〈-3, -3〉 20. D (3, 1), E (2, -2), F (3, -4), G (4, -2) ; 〈-6, 2〉 12-3 Rotations (pp. 839–845) TEKS G.2.A, G.2.B, G.7.A, G.10.A E X A M P L E EXERCISES ■ Rotate the figure with the given vertices about the origin using the given angle of rotation. A (-2, 0), B (-1, 3), C (-4, 3) ; 180° To rotate by 180°, find the opposite of the x- and y-coordinate of each point. The vertices of the image are A' (2, 0), B' (1, -3), and C' (4, -3). Tell whether each transformation appears to be a rotation. 21. 22. 23. 24. Rotate the figure with the given vertices about the origin using the given angle of rotation. 25. A (1, 3), B (4, 1), C (4, 4) ; 90° 26. A (1, 3), B (4, 1), C (4, 4) ; 180° 27. M (2, 2), N (5, 2), P (3, -2), Q (0, -2) ; 90° 28. G (-2, 1), H (-3, -2), J (-1, -4) ; 180° Study Guide: Review 885 885 ��������������������������������������������� 12-4 Compositions of Transformations (pp. 848–853) TEKS G.5.C, G.10.A E X A M P L E |
EXERCISES ■ Draw the result of the composition of isometries. Translate △MNP along across line ℓ. v and then reflect it Draw the result of the composition of isometries. 29. Translate ABCD along v and then reflect it across line m. First draw △M'N'P', the translation image of △MNP. Then reflect △M'N'P' across line ℓ to find the final image, △M''N''P''. 30. Reflect △JKL across line m and then rotate it 90° about point P. 12-5 Symmetry (pp. 856–862) TEKS G.2.B, G.5.C, G.10.A E X A M P L E S EXERCISES Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. Tell whether each figure has line symmetry. If so, copy the figure and draw all lines of symmetry. 31. 32. ■ no rotational symmetry ■ Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of symmetry. 33. 34. The figure coincides with itself when it is rotated by 90°. Therefore the angle of rotational symmetry is 90°. The order of symmetry is 4. 35. 36. 886 886 Chapter 12 Extending Transformational Geometry ��������������������������������������������� 12-6 Tessellations (pp. 863–869) TEKS G.5.C E X A M P L E S EXERCISES ■ Copy the given figure and use it to create a tessellation. Rotate the quadrilateral 180° about the midpoint of one side. Copy the given figure and use it to create a tessellation. 37. 38. Translate the resulting pair of quadrilaterals to make a row. 39. 40. Translate the row to make a tessellation. Classify each tessellation as regular, semiregular, or neither. 41. ■ Classify the tessellation as regular, semiregular, or neither. The tessellation is made of two different regular polygons, and each vertex has the same polygons in the same order. Therefore the t |
essellation is semiregular. 42. 12-7 Dilations (pp. 872–879) TEKS G.2.A, G.11.A, G.11.B, G.11.D E X A M P L E EXERCISES ■ Draw the image of the figure with the given vertices under a dilation centered at the origin using the given scale factor. A (0, -2), B (2, -2), C (2, 0) ; scale factor: 2 Tell whether each transformation appears to be a dilation. 43. 44. Multiply the x- and y-coordinates of each point by 2. The vertices of the image are A' (0, -4), B' (4, -4), and C' (4, 0). Draw the image of the figure with the given vertices under a dilation centered at the origin using the given scale factor. 45. R (0, 0), S (4, 4), T (4, -4) ; scale factor: - 1_ 2 46. D (0, 2), E (-2, 2), F (-2, 0) ; scale factor: -2 Study Guide: Review 887 887 ���������������������� Tell whether each transformation appears to be a reflection. 1. 2. Tell whether each transformation appears to be a translation. 3. 4. 5. An interior designer is using a coordinate grid to place furniture in a room. The position of a sofa is represented by a rectangle with vertices (1, 3), (2, 2), (5, 5), and (4, 6). He decides to move the sofa by translating it along the vector 〈-1, -1〉. Draw the sofa in its final position. Tell whether each transformation appears to be a rotation. 6. 7. 8. Rotate rectangle DEFG with vertices D (1, -1), E (4, -1), F (4, -3), and G (1, -3) about the origin by 180°. 9. Rectangle ABCD with vertices A (3, -1), B (3, -2), C (1, -2), and D (1, -1) is reflected across the x-axis, and then its image is reflected across the |
line y = 4. Describe a single transformation that moves the rectangle from its starting position to its final position. 10. Tell whether the “no entry” sign has line symmetry. If so, copy the sign and draw all lines of symmetry. 11. Tell whether the “no entry” sign has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. Copy the given figure and use it to create a tessellation. 12. 13. 14. 15. Classify the tessellation shown as regular, semiregular, or neither. Tell whether each transformation appears to be a dilation. 16. 17. 18. Draw the image of △ABC with vertices A (2, -1), B (1, -4), and C (4, -4) under a dilation centered at the origin with scale factor - 1 __ 2. 888 888 Chapter 12 Extending Transformational Geometry FOCUS ON ACT No question on the ACT Mathematics Test requires the use of a calculator, but you may bring certain types of calculators to the test. Check www.actstudent.org for a descriptive list of calculators that are prohibited or allowed with slight modifications. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. If you are not sure how to solve a problem, looking through the answer choices may provide you with a clue to the solution method. It may take longer to work backward from the answers provided, so make sure you are monitoring your time. 1. Which of the following functions has a graph that is symmetric with respect to the y-axis? (A) f (x) = x 4 - 2 (B) f (x) = (x + 2) 4 (C) f (x) = 2x - 4 (D) f (x) = x 2 + 4x (E) f (x) = (x - 4) 2 2. What is the image of the point (-4, 5) after the translation that maps the point (1, -3) to the point (-1, -7)? (F) (4, 1) (G) (-6, 1) (H) (-8, 3) (-2, 9) (J) (K) (0, 7) 3. When the point (-2, -5) is reflected across |
the x-axis, what is the resulting image? (A) (-5, -2) (B) (2, 5) (C) (2, -5) (D) (-2, 5) (E) (5, 2) 4. After a composition of transformations, the line segment from A (1, 4) to B (4, 2) maps to the line segment from C (-1, -2) to D (-4, -4). Which of the following describes the composition that is applied to ̶̶ AB to obtain ̶̶ CD? (F) Translate 5 units to the left and then reflect across the y-axis. (G) Reflect across the y-axis and then reflect across the x-axis. (H) Reflect across the y-axis and then translate 6 units down. (J) Reflect across the x-axis and then reflect across the y-axis. (K) Translate 6 units down and then reflect across the x-axis. 5. What is the image of the following figure after rotating it counterclockwise by 270°? (A) (B) (C) (D) (E) College Entrance Exam Practice 889 889 Any Question Type: Highlight Main Ideas Before answering a test item, identify the important information given in the problem and make sure you clearly identify the question being asked. Outlining the question or breaking a problem into parts can help you to understand the main idea. A common error in answering multi-step questions is to complete only the first step. In multiple-choice questions, partial answers are often used as the incorrect answer choices. If you start by outlining all steps needed to solve the problem, you are less likely to choose one of these incorrect answers. Gridded Response A blueprint shows a rectangular building’s layout reduced using a scale factor of 1 __ 30. On the blueprint, the building’s width is 15 in. and its length is 7 in. Find the area of the actual building in square feet. What are you asked to find? the area of the actual building in square feet List the given information you need to solve the problem. The scale factor is 1 _. 30 On the blueprint, the width is 15 in. and the length is 7 in. The area of the building is 656.25 square feet. Multiple Choice An animator uses a coordinate plane to show the motion of a flying bird. The bird begins at the point ( |
12, 0) and is then rotated about the origin by 15° every 0.005 second. Give the bird’s position after 0.015 second. Round the coordinates to the nearest tenth. (8.49, 8.49) (-12, 0) (0, 12) (-8.49, 8.49) What are you asked to find? the coordinates of the bird’s position after 0.015 seconds, to the nearest tenth What information are you given? the initial position of the bird and the angle of rotation for every 0.005 second The correct answer is A. 890 890 Chapter 12 Extending Transformational Geometry ���� ���� ���� Sometimes important information is given in a diagram. Read each test item and answer the questions that follow. Item A Multiple Choice Jonas is using a coordinate plane to plan an archaeological dig. He outlines a rectangle with vertices at (5, 2), (5, 9), (10, 9), and (10, 2). Then he outlines a second rectangle by reflecting the first area across the x-axis and then across the y-axis. Which is a vertex of the second outlined rectangle? (-5, 2) (-5, -9) (-2, -10) (10, -9) 1. Identify the sentence that gives the information regarding the coordinates of the initial rectangle. 2. What are you being asked to do? 3. How many transformations does Jonas perform before he sketches the second rectangle? Which sentence leads you to this answer? 4. A student incorrectly marked choice A as her response. What part of the test item did she fail to complete? Item B Gridded Response Gabby has a digital photo with dimensions 3.5 in. by 5 in., and she uses software to enlarge it by a scale factor of 5. How large must a frame be, in square inches, in order for the enlarged photo to fit? 5. Make a list stating the information given and what you are being asked to do. 6. Are there any intermediate steps you have to take to obtain a solution for the problem? If so, describe the steps. Item C Gridded Response Rectangle A'B'C'D' is the image of rectangle ABCD under a dilation. Once you have identified the scale factor, determine the area of rectangle A'B'C'D'. 7. What do you need to find before you can find the area of the rectangle? 8. Where |
in the test item can you find the important information (data) needed to solve the problem? Make a list of this information. Item D Multiple Choice △ABC is reflected across the x-axis. Then its image is rotated 180° about the origin. What are the coordinates of the image of point B after the reflection? (-4, -1) (-1, 4) (1, -4) (4, -1) 9. Identify the transformations described in the problem statement. 10. What are you being asked to do? 11. Identify any part of the problem statement that you will not use to answer the question. 12. There are only two pieces of information given in this test item that are important to answering this question. What are they? TAKS Tackler 891 891 �������������������������� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–12 Multiple Choice 1. Which of the following best represents the area of the shaded figure if each square in the grid has a side length of 1 centimeter? 5. Marty conjectures that the sum of any two prime numbers is even. Which of the following is a counterexample that shows Marty’s conjecture is false = 11 3 + 5 = 8 Use the graph for Items 6–8. 17 square centimeters 21 square centimeters 25 square centimeters 29 square centimeters 2. Which of the following expressions represents the number of edges of a polyhedron with n vertices and n faces? n - 2 2n - 1 2 (n - 1) 2 (n + 1) 3. The image of point A under a 90° rotation about the origin is A' (10, -4). What are the coordinates of point A? 6. What are the coordinates of the image of point C under the same translation that maps point D to point B? (4, 4) (0, 4) (0, 8) (4, -8) 7. △PQR is the image of a triangle under a dilation centered at the origin with scale factor - 1 __. Which 2 point is a vertex of the preimage of △PQR under this dilation? (-10, -4) (-10, 4) (-4, -10) (4, 10) A B C D 4. A cylinder has a volume of 24 cubic centimeters. The height of a cone with the same radius is |
two times the height of the cylinder. What is the volume of the cone? 8 cubic centimeters 12 cubic centimeters 16 cubic centimeters 48 cubic centimeters 8. What is the measure of ∠PRQ? Round to the nearest degree. 63° 127° 117° 45° 9. Which mapping represents a rotation of 270° about the origin? (x, y) → (-x, -y) (x, y) → (x, -y) (x, y) → (-y, -x) (x, y) → (y, -x) 892 892 Chapter 12 Extending Transformational Geometry ���������������������� ���� ���� ���� When problems involve geometric figures in the coordinate plane, it may be useful to describe properties of the figures algebraically. For example, you can use slope to verify that sides of a figure are parallel or perpendicular, or you can use the Distance Formula to find side lengths of the figure. 10. What are the coordinates of the center of the circle (x + 1) 2 + (y + 4) 2 = 4? (-1, -4) (-1, -2) (1, 2) (1, 4) STANDARDIZED TEST PREP Short Response 18. A (-4, -2), B (-2, -3), and C (-3, -5) are three of the vertices of rhombus ABCD. Show that ABCD is a square. Justify your answer. 19. ABCD is a square with vertices A (3, -1), B (3, -3), C (1, -3), and D (1, -1). ⊙P is a circle with equation (x - 2) 2 + (y - 2) 2 = 4. a. What is the center and radius of ⊙P? b. Describe a reflection and dilation of ABCD so that ⊙P is inscribed in the image of ABCD. Justify your answer. 11. Which regular polygon can be used with an equilateral triangle to tessellate a plane? 20. Determine the value of x if △ABC ≅ △BDC. Justify your answer. Heptagon Octagon Nonagon Dodecagon 12. What is the measure of ∠TSV in ⊙P? 24° 42° 45° 48° 13. Given the points B (-1, 2), |
C (-7, y), D (1, -3), and E (-3, -2), what is the value of y if ̶̶ BD ǁ ̶̶ CE? -12 -8 3.5 8 Gridded Response 14. △ABC is a right triangle such that m∠B = 90°. If AC = 12 and BC = 9, what is the perimeter of △ABC? Round to the nearest tenth. 15. A blueprint for an office space uses a scale of 3 inches: 20 feet. What is the area in square inches of the office space on the blueprint if the actual office space has area 1300 square feet? 16. How many lines of symmetry does a regular hexagon have? 17. What is the x-coordinate of the image of the point A (12, -7) if A is reflected across the x-axis? 21. △ABC is reflected across line m. a. What observations can be made about △ABC and its reflected image △A'B'C' regarding the following properties: collinearity, betweenness, angle measure, triangle congruence, and orientation? b. Explain. 22. Given the coordinates of points A, B, and C, explain how you could demonstrate that the three points are collinear. 23. Proving that the diagonals of rectangle KLMN are equal using a coordinate proof involves placement of the rectangle and selection of coordinates. a. Is it possible to always position rectangle KLMN so that one vertex coincides with the origin? b. Why is it convenient to place rectangle KLMN so that one vertex is at the origin? Extended Response 24. ̶̶ AB has endpoints A (0, 3) and B (2, 5). ̶̶ a. Draw AB and its image, ̶̶̶ A'B', under the translation 〈0, -8〉. b. Find the equations of two lines such that the composition of the two reflections across the lines will also map work or explain in words how you found your answer. ̶̶̶ A'B'. Show your ̶̶ AB to c. Show that any glide reflection is equivalent to a composition of three reflections. Cumulative Assessment, Chapters 1–12 893 893 ��������������������� T E X A S TAKS Grades 9–11 Obj. 10 Point Isabel Lighthouse The Point |
Isabel Lighthouse was built in 1853 on a prominent bluff on the mainland. Today, the fully restored lighthouse is the only one in Texas that is open for climbing and viewing. Port Isabel Port Isabel ge07ts_c12psl001a Choose one or more strategies to solve each problem. 1st pass 9/06/05 dtrevino 15 miles at sea. To the nearest square mile, what is the area of water covered by the beam as it rotates by an angle of 60°? 1. Suppose the beam from the lighthouse is visible for up to 2. Given that Earth’s radius is approximately 4000 miles and that the top of the tower of a lighthouse is 65 ft above sea level, find the distance from the top of the tower to the horizon. Round your answer to the nearest mile. (Hint: 65 feet = 0.01 miles) For 3, use the table. 3. Most lighthouses use Fresnel lenses, named after their inventor, Augustine Fresnel. The table shows the sizes, or orders, of the circular lenses. The diagram shows some measurements of the Fresnel lens used in the Point Isabel Lighthouse. What is the order of the lens? Fresnel Lenses Order Lens Diameter First Second Third Fourth Fifth Sixth 6 ft 1 in. 4 ft 7 in. 3 ft 3 in. 1 ft 8 in. 1 ft 3 in. 1 ft 0 in. 894 894 Chapter 12 Extending Transformational Geometry ������������ Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Tule Lake Lift Bridge Moveable Bridges Texas is home to many moveable bridges. A moveable bridge has a section that can be lifted, tilted, or swung out of the way so that tall boats can pass. 1. The Quintana Swing Bridge is a swing bridge. Part of the roadbed can pivot horizontally to let boats pass. What transformation describes the motion of the bridge? The pivoting section moves through an angle of 90°. How far does a point 10 ft from the pivot travel as the bridge opens? Round to the nearest tenth of a foot. A lift bridge contains a section that can be translated vertically. For 2–4, use the table. Lift Bridges Name Vertical Clearance (Lowered Position) Vertical Clearance (Raised Position) Tule Lake Lift Bridge Rio |
Hondo Lift Bridge 10 ft 27 ft 138 ft 73 ft 2. It takes about 7 min to completely lift the roadbed of the Tule Lake Lift Bridge. At what speed, in feet per minute, does the lifting mechanism translate the roadbed? Round your answer to the nearest foot per minute. 3. To the nearest second, how long does it take the Tule Lake Lift Bridge’s lifting mechanism to translate the roadbed 10 ft? 4. The Rio Hondo Lift Bridge can be raised at 10.2 feet per minute. To the nearest second, how long does it take to completely lift its roadbed? 5. Corpus Christi once had a bascule bridge. Weights were used to raise the bridge at an angle so that boats could pass through the channel. The moveable section of the Bascule Bridge in Corpus Christi was 121 ft long. Find the height of the bridge above the roadway after it had been rotated by an angle of 20°. Round to the nearest inch. Problem Solving on Location 895 895 ���������� Student Handbook TEKS TAKS Practice.............................. S4 Skills Practice.................................................. S4 Application Practice.......................................... S28 Problem-Solving Handbook....................... S40 Draw a Diagram................................................ S40 Make a Model.................................................. S41 Guess and Test............ |
..................................... S42 Work Backward................................................. S43 Find a Pattern.................................................. S44 Make a Table................................................... S45 Solve a Simpler Problem........................................ S46 Use Logical Reasoning.......................................... S47 Use a Venn Diagram............................................ S48 Make an Organized List......................................... S49 Skills Bank............................................ S50 Number and Operations Operations with Real Numbers................................... S50 Order of Operations............... |
............................. S50 Properties..................................................... S51 Estimation, Rounding, and Reasonableness...................... S52 Classify Real Numbers........................................... S53 Exponents...................................................... S53 Properties of Exponents......................................... S54 Powers of 10 and Scientific Notation.............................. S54 Square Roots................................................... S55 Simplifying Square Roots....................................... S55 Algebra The Coordinate Plane........................................... S56 Connecting Words with Algebra.................................. S57 Variables |
and Expressions....................................... S57 Solving Linear Equations....................................... S58 Solving Equations for a Variable.................................. S59 Writing and Graphing Inequalities............................... S59 S2S2 S2 Student Handbook Solving Linear Inequalities...................................... S60 Absolute Value................................................. S61 Relations and Functions........................................ S61 Inverse Functions.............................................. S62 Direct Variation................................................ S62 Functional Relationships in Formulas............................ S63 Transformations of Functions.................................... S63 Polynomials |
................................................... S64 Quadratic Functions............................................ S65 Factoring to Solve Quadratic Equations........................... S66 The Quadratic Formula.......................................... S66 Solving Systems of Equations.................................... S67 Solving Systems of Linear Inequalities........................... S68 Solving Radical Equations....................................... S68 Matrix Operations.............................................. S69 Measurement Structure of Measurement Systems............................... S70 Rates and Derived Measurements................................ S70 Unit Conversions............................................... S71 Accuracy, Precision, and Tolerance.... |
........................... S72 Relative and Absolute Error..................................... S73 Significant Digits............................................... S73 Choose Appropriate Units....................................... S74 Nonstandard Units.............................................. S74 Use Tools for Measurement...................................... S75 Choose Appropriate Measuring Tools............................ S75 Data Analysis and Probability Measures of Central Tendency................................... S76 Probability..................................................... S77 Organizing and Describing Data................................. S78 Displaying Data................................................ S78 Scatter Plots and Trend Lines..... |
............................... S79 Quartiles and Box-and-Whisker Plots............................. S80 Circle Graphs.................................................. S80 Misleading Graphs and Statistics................................ S81 Venn Diagrams................................................. S81 Postulates, Theorems, and Corollaries..... S82 Constructions.......................................... S87 Selected Answers............................... S88 Glossary......................................... S115 Index............................................. S161 Symbols and Formulas.......... inside back cover S3S3 TEKS TAKS Practice Chapter 1 Skills Practice Lesson 1-1 Name each of the following. 1. two points 2. two lines 3. two planes 4. a point on IH 5. a line that contains L and J 6. a plane that contains L, K, and H |
Draw and label each of the following. 7. a ray with endpoint A that passes through B 8. a line PQ that intersects plane D Lesson 1-2 Find each length. 9. MN 10. MO 11. Segments that have the same length are?. ̶̶̶̶ 12. Construct a segment congruent to AB. Then construct the midpoint M. 13. M is the midpoint of ̶̶ PR, PM = 2x + 5, and MR = 4x - 7. Solve for x and find PR. Lesson 1-3 Z is in the interior of ∠WXY. Find each of the following. 14. m∠WXY if ∠WXZ = 23° and m∠ZXY = 51° 15. m∠WXZ if m∠WXY = 44° and m∠ZXY = 20° EH bisects ∠DEF. Find each of the following. 16. m∠DEH if m∠DEH = (10z - 2) ° and m∠HEF = (6z + 10) ° 17. m∠DEF if m∠DEH = (9x + 3) ° and m∠HEF = (5x + 11) ° 18. A? is formed by two opposite rays and measures ̶̶̶̶? °. ̶̶̶̶ 19. There are? ° in a circle. ̶̶̶̶ Lesson 1-4 Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 20. ∠AOB and ∠DOE 21. ∠AOE and ∠DOE 22. ∠COE and ∠EOA 23. ∠AOB and ∠BOD 24. Name a pair of vertical angles. Given m∠A = 41.7° and m∠B = (24.2 - x) °, find the measure of each of the following. 25. complement of ∠A 26. supplement of ∠A 27. supplement of ∠B S4 S4 TEKS TAKS Practice ��������������������������������� Lesson 1-5 Find the perimeter and area of each figure. 28. 29. 30. Find the circumference and area of each circle. Give |
your answer to the nearest hundredth. 31. 33. 32. Lesson 1-6 34. The formula to find the midpoint M of ̶̶ AB with endpoints A ( x 1, y 1 ) and B ( x 2, y 2 ) is?. ̶̶̶̶ Find the coordinates of the midpoint of each segment. 35. ̶̶̶ WX with endpoints W (-4, 1) and X (2, 9) ̶̶ YZ with midpoints Y (4, 8) and Z (-1, -4) 36. 37. M is the midpoint of ̶̶ RS. R has coordinates (-7, -3), and M has coordinates (1, 1). Find the coordinates of S. Find the length of the given segments and determine if they are congruent. 38. ̶̶̶ VW and ̶̶ RS and ̶̶ PQ ̶̶ TU 39. Lesson 1-7 Identify each transformation. Then use arrow notation to describe the transformation. 40. 41. 42. A figure has vertices at (1, 1), (2, 4), and (5, 3). After a transformation, the image of the figure has vertices at (-3, -2), (-2, 1), and (1, 0). Draw the preimage and image. Then describe the transformation. 43. A figure has vertices at (5, 5), (2, 6), (1, 5), and (2, 4). After a transformation, the image of the figure has vertices at (5, 5), (6, 8), (5, 9), and (4, 8). Draw the preimage and image. Then describe the transformation. 44. The coordinates of the vertices of quadrilateral DEFG are (3, 0), (2, 3), (-3, 2), and (-2, -1). Find the coordinates for the image of rectangle DEFG after the translation (x, y) → (x, -y). Draw the preimage and image. Then describe the transformation. TEKS TAKS Practice S5S5 ����������������������������������������������������������������������������������� Chapter 2 Skills Practice Lesson Lesson Lesson 2-1 2-5 2-5 Lesson 2- |
2 Find the next item in each pattern. 1. 3, 7, 11, 15, … 2. -3, 6, -12, 24, … 3. Complete the conjecture “The product of two negative numbers is?.” ̶̶̶̶ 4. Show that the conjecture “The quotient of two integers is an integer” is false by finding a counterexample. Identify the hypothesis and conclusion of each conditional. 5. A number is divisible by 10 if it ends in zero. 6. If the temperature reaches 100° F, it will rain. Write a conditional statement from each of the following. 7. Perpendicular lines intersect to form 90° angles. 8. 9. The sum of two supplementary angles is 180°. Lesson 2-3 Determine if each conditional is true. If false, give a counterexample. 10. If a figure has four sides, then it is a square. 11. If x = 3, then 5x = 15. 12. Does the conclusion use inductive or deductive reasoning? To rent a boat, you must take a boating safety course. Jason rented a boat, so Jessica concludes that he has taken a boating safety course. 13. Determine if the conjecture is valid by the Law of Detachment. Given: If a student is in tenth grade, then the student may participate in student council. Eric is a tenth-grader. Conjecture: Eric may participate in student council. 14. Determine if the conjecture is valid by the Law of Syllogism. Given: If a triangle is isosceles, then it has two congruent sides. If a triangle has two congruent angles, then it has two congruent sides. Conjecture: If a triangle is isosceles, then it has two congruent angles. 15. Draw a conclusion from the given information. Given: If the sum of the angles of a polygon is 360°, then it is a quadrilateral. If a polygon is a quadrilateral, then it has four sides. The sum of the angles of polygon R is 360°. Lesson 2-4 16. Write the conditional statement and converse within the biconditional “A triangle is equilateral if and only if it has three congruent sides.” 17. For the conditional “If a triangle is scalene, then its |
sides have different lengths,” write the converse and a biconditional statement. 18. Determine if the biconditional “n + 3 = -1 ↔ n = -4” is true. If false, give a counterexample. Write each definition as a biconditional. 19. A parallelogram is a quadrilateral with two pairs of parallel sides. 20. Congruent angles have equal measures. S6 S6 TEKS TAKS Practice �������������� Lesson 2-5 Lesson 2-6 Lesson 2-7 Solve each equation. Write a justification for each step. x + 2 _ 5 21. 2x + 3 = 9 22. = 3 Write a justification for each step. 23. AC = AB + BC 9x - 5 = (3x + 6) + (5x + 2) 9x - 5 = 8x + 8 x - 5 = 8 x = 13 24. Fill in the blanks to complete the two-column proof. Given: ∠HMK and ∠JML are right angles. Prove: ∠1 ≅ ∠3 Proof: Statements Reasons 1. a. 2. b. c.? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶ 1. Given 2. Adjacent angles that form a right angle are complementary. 3. ∠1 ≅ ∠3 3. d.? ̶̶̶̶̶ 25. Use the given plan to write a two-column proof of the Transitive Property of Congruence. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ CD, ̶̶ EF ̶̶ CD ≅ ̶̶ EF Given: Prove: Plan: Use the definition of congruent segments to write the given congruence statements as statements of equality. Then use the Transitive Property of Equality to show that AB = EF. So definition of congruent segments. ̶̶ EF by the ̶̶ AB ≅ 26. Use the given two-column proof to write a flowchart proof. Given: ∠2 ≅ ∠3 Prove: m∠1 = m∠4 Proof: Statements Reasons 1. ∠2 ≅ ∠3 1. Given 2. ∠1 and ∠2 are supplementary. 2. Lin. Pair Thm. ∠3 |
and ∠4 are supplementary. 3. ∠1 ≅ ∠4 4. m∠1 = m∠4 3. ≅ Supps. Thm. 4. Def. of ≅ 27. Use the given two-column proof to write a paragraph proof. Given: ∠1 ≅ ∠3 Prove: ∠4 ≅ ∠5 Proof: Statements Reasons 1. ∠1 ≅ ∠3 1. Given 2. ∠1 ≅ ∠4, ∠3 ≅ ∠5 2. Vert. Thm. 3. ∠1 ≅ ∠5 4. ∠4 ≅ ∠5 3. Trans. Prop. of ≅ 4. Trans. Prop. of ≅ TEKS TAKS Practice S7S7 �������������������������������������������� Chapter 3 Skills Practice Lesson 3-1 Identify each of the following. 1. a pair of parallel segments 2. a pair of perpendicular segments 3. a pair of skew segments Identify the transversal and classify each angle pair. 4. ∠5 and ∠3 5. ∠2 and ∠4 6. ∠5 and ∠1 Lesson 3-2 Find each angle measure. 7. m∠XYZ 8. m∠KJH 9. m∠ABC 10. m∠LMN 11. m∠PQR 12. m∠TUV Lesson 3-3 Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 13. ∠2 ≅ ∠4 14. m∠8 = 5x + 36, m∠6 = 11x + 12, x = 4 Use the theorems and given information to show that p ǁ q. 15. ∠1 ≅ ∠8 16. m∠2 = 9x + 31, m∠3 = 6x + 14, x = 9 17. Write a two-column proof. Given: ∠1 and ∠5 are supplementary. Prove: ℓ ǁ m S8 S8 TEKS TAKS Practice �������������������������������������������������������������������������������������������������������������������������������������������� |
������������������������� Lesson 3-4 18. Name the shortest segment from point A to BE. 19. Write and solve an inequality for x. Solve for x and y in each diagram. 20. 21. 22. Write a two-column proof. Given: ℓ ⊥ p, m ⊥ p Prove: ℓ ǁ m Lesson 3-5 Use the slope formula to determine the slope of each line. 23. FG 24. HJ Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 25. AB and CD for A (4, 7), B (3, 2), C (-3, 4), D (2, 3) 26. EF and GH for E (-2, 4), F (3, 1), G (-1, -2), H (4, -5) 27. JK and LM for J (-3, 3), K (4, -2), L (4, 2), M (0, -4) Lesson 3-6 Write the equation of each line in the given form. 28. the line with slope - 2 _ through (3, -1) in point-slope form 3 29. the line through (-2, 2) and (4, -1) in slope-intercept form 30. the line with x-intercept -3 and y-intercept 4 in slope-intercept form Graph each line. 31 33. y = 2 32. y + 4 = -3 (x + 2) 34. x = -1 Determine whether the lines are parallel, intersect, or coincide. 35. y = 4x + 2, 4x - y = 1 37. 2x + 5y = 1, 5x + 2y = 1 36. y =- 1 _ 2 x + 3, 2y + x = 6 38. 2x - y = 5, 2x - y = -5 TEKS TAKS Practice |
S9S9 ���������������������������������������������������������������������� Chapter 4 Skills Practice Lesson Lesson Lesson 4-1 2-5 2-5 Lesson 4-2 Lesson 4-3 Classify each triangle by its angle measures. 1. △ABC 2. △BCD Classify each triangle by its side lengths. 3. △EFG 4. △FGH 5. △EFH 6. Find the side lengths of △JKL. The measure of one of the acute angles of a right triangle is given. What is the measure of the other acute angle? 7. 38° 8. 27.6° Find each angle measure. 9. m∠A 10. m∠J and m∠P Given: △GHI ≅ △JKL. Identify the congruent corresponding parts. 11. ̶̶̶ GH ≅? ̶̶̶̶ 12. ̶̶ JL ≅? ̶̶̶̶ 13. ∠K ≅? ̶̶̶̶ Given: △LMN ≅ △PQN. Find each value. 14. x 15. m∠LMN 16. Given: ̶̶ AD is the perpendicular bisector of ̶̶ AD is the bisector of ∠BAC. ̶̶ AB ≅ ̶̶ AC, ∠B ≅ ∠C ̶̶ BC. Prove: △BAD ≅ △CAD Lesson 4-4 Use SSS to explain why the triangles in each pair are congruent. 17. △QRS ≅ △QRT 18. △UVW ≅ △WXU Show that the triangles are congruent for the given value of the variable. 19. △XYZ ≅ △ABC, 20. △DEF ≅ △GFE, x = 4 y = 8 21. Given: K is the midpoint of Prove: △HJK ≅ △LMK ̶̶ HL and ̶̶ MJ. S10 S10 TEKS TAKS Practice �������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� Lesson 4 |
-5 Determine if you can use ASA to prove the triangles congruent. Explain. 22. △ACB and △ACD 23. △EFG and △HGF Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 24. △ABC ≅ △EDC 25. △FGH ≅ △FJH Lesson 4-6 Lesson 4-7 26. Given: ̶̶̶ ̶̶ LP, MN ǁ ∠N ≅ ∠L ̶̶ ̶̶̶ PN ML ≅ Prove: 27. Given: ∠1 ≅ ∠6, ∠4 ≅ ∠6 ̶̶ AE ∠1 ≅ ∠3, ̶̶ AB ≅ Prove: △ACD is isosceles. 28. Given: △ABC with vertices A (2, 4), B (3, 1), C (5, 2) and △DEF with vertices D (-4, -2), E (-1, -3), F (-2, -5) Prove: ∠BAC ≅ ∠EDF Position each figure in the coordinate plane. 29. a rectangle with length 7 units and width 3 units 30. a square with side length 3a Write a coordinate proof. 31. Given: Right △GHI has coordinates G (0, 0), H (0, 4), and I (6, 0). ̶̶̶ GH, and K is the midpoint of J is the midpoint of ̶̶ GI. Prove: The area of △GJK is 1 __ 4 the area of △GHI. Assign coordinates to each vertex and write a coordinate proof. 32. Given: A is the midpoint of B is the midpoint of ̶̶̶ XW in rectangle WXYZ. ̶̶ YZ. Prove: AB = XY Lesson 4-8 Find each angle measure. 33. m∠X Find each value. 35. x 37. Given: △XYZ is isosceles. A is the midpoint of ̶̶ XZ. ̶̶ XY ≅ ̶̶ YZ 34. m∠A 36. y Prove: |
△YAZ is isosceles. TEKS TAKS Practice S11 S11 ������������������������������������������������������������������������������������������������������� Chapter 5 Skills Practice Lesson 5-1 Find each measure. 1. CD 2. HG 3. JM 4. m∠SRT, given m∠SRU = 126° 5. PQ 6. m∠WXV 7. Write an equation in point-slope form for the perpendicular bisector of the segment Lesson 5-2 with endpoints A (1, 4) and B (-5, -2). ̶̶ ̶̶ DG, EG, and Find each length. ̶̶ FG are the perpendicular bisectors of △ABC. 8. BG 9. AG Find the circumcenter of a triangle with the given vertices. 10. H (5, 0), J (0, 3), K (0, 0) ̶̶ QS and 12. the distance from S to ̶̶ RS are angle bisectors of △QPR. Find each measure. 13. m∠SQP ̶̶ PR 11. L (0, 0), M (-2, 0), N (0, -4) Lesson 5-3 In △DEF, DJ = 30, and FM = 12. Find each length. 15. MJ 14. DM 16. GF 17. GM Find the orthocenter of a triangle with the given vertices. 18. N (-2, 2), P (4, 2), Q (0, -2) 19. R (-2, 1), S (2, 5), T (4, 1) Lesson 5-4 20. The vertices of △WXY are W (-3, 2), X (5, 2), and Y (1, -4). A is the midpoint of and B is the midpoint of ̶̶ XY. Show that ̶̶ AB ǁ ̶̶̶ WX and AB = 1 __ 2 WX. ̶̶̶ WY, Find each measure. 21. DE 23. DG 25. m∠FHE 22. FG 24. m∠CHF 26. m∠CED S12S12 TEKS TAKS Practice ������������������������������������������������������������������������������������ |
��������������������������������������� Lesson 5-5 Write an indirect proof of each statement. 27. An isosceles triangle cannot have an obtuse base angle. 28. A right triangle cannot have three congruent sides. 29. Write the angles in order from smallest to largest. 30. Write the sides in order from shortest to longest. Tell whether a triangle can have sides with the given lengths. Explain. 32. 7, 9, 18 31. 4, 7, 8 33. 2x + 5, 4x, 3 x 2, when x = 3 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. 34. 4 in., 10 in. 36. 6.2 cm, 12 cm 35. 8 ft, 8 ft Lesson 5-6 Compare the given measures. 37. Compare RS and UV. 38. Compare m∠XWY 39. Find the range of and m∠ZWY. values for x. 40. Write a two-column proof. Given: m∠X > m∠Y, m∠B > m∠A Prove: AY > XB Lesson 5-7 Lesson 5-8 Find the value of x. Give your answer in simplest radical form. 43. 41. 42. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 44. 46. 45. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 47. 4, 7.5, 8.5 48. 6, 10, 11 49. 9, 21, 25 Find the value of x. Give your answer in simplest radical form. 52. 50. 51. Find the values of x and y. Give your answers in simplest radical form. 53. 55. 54. TEKS TAKS Practice S13 S13 ����������������������������������������������������������������������������������������������������������������������������������������������� Chapter 6 Skills Practice Lesson Lesson Lesson 6-1 2-5 2-5 Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. Tell whether each polygon is regular or irregular. Tell |
whether it is concave or convex. 4. 5. 6. 7. Find the measure of each interior angle of pentagon ABCDE. 8. Find the sum of the interior angle measures of a convex heptagon. 9. Find the measure of each interior angle of a regular 15-gon. 10. Find the value of x in polygon FGHJKL. 11. Find the measure of each exterior angle of a regular dodecagon. Lesson 6-2 MNOP is a parallelogram. Find each measure. 12. MP 13. m∠M 14. m∠N Three vertices of QRST are given. Find the coordinates of T. 15. Q (-5, 3), R (3, 6), S (6, 4) 16. Q (-1, 7), R (3, 3), S (-2, 3) Write a two-column proof. 17. Given: ABFG and HDEG are parallelograms. Prove: ∠B ≅ ∠D Lesson 6-3 18. Show that RSTU is a parallelogram 19. Show that WXYZ is a parallelogram for x = 2 and y = 3. for a = 6 and b = 11. Determine if each quadrilateral must be a parallelogram. Justify your answer. 20. 22. 21. Show that the quadrilateral with the given vertices is a parallelogram. 23. W (0, 0), X (-3, 3), Y (5, 5), Z (8, 2) 24. A (-3, 1), B (-2, 4), C (1, 2), D (0, -1) S14S14 TEKS TAKS Practice ��������������������������������������������������������������������������������������������������������������������������������������������������������������������������� Lesson 6-4 EFGH is a rectangle. Find each measure. 25. EH 26. HF JKLM is a rhombus. Find each measure. 27. JK 28. m∠NKL Show that the diagonals of a square with the given vertices are congruent perpendicular bisectors of each other. 29. N (1, 4), P (4, 1) |
, Q (1, -2), R (-2, 1) 30. S (-2, 7), T (2, 8), U (3, 4), V (-1, 3) 31. Given: WXYZ is a rectangle. Prove: ̶̶̶ WB ≅ ̶̶ YA ̶̶ XB ≅ ̶̶ AZ Lesson 6-5 Lesson 6-6 Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ ̶̶ 32. Given: XY ≅ XY ǁ ̶̶ XZ ⊥ Conclusion: WXYZ is a rhombus. ̶̶̶ WZ, ̶̶̶ WZ, ̶̶̶ WY 33. Given: ̶̶̶ WX ≅ ̶̶ XY Conclusion: WXYZ is a square. ̶̶ XY, Conclusion: WXYZ is a rectangle. ̶̶̶ WX ⊥ ̶̶̶ WX ⊥ ̶̶̶ WZ 34. Given: Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 35. A (1, 0), B (2, -4), C (6, -3), D (5, 1) 36. E (-3, -1), F (-4, -4), G (2, -6), H (3, -3) In kite TUVW, m∠TUX = 65°, and m∠UVT = 32°. Find each measure. 37. m∠TUX 39. m∠TWX 38. m∠XUV Find each measure. 40. m∠C 41. HJ, given that EG = 32.8 and FJ = 24.3 42. Find the value of x so that JKLM is isosceles. 43. Given RP = 8y - 7 and NQ = 10y - 12, find the value of y so that NPQR is isosceles. 44. Find RS. 45. Find XY. TEKS TAKS Practice S15 S15 ���������������������������������������������������������������������������������������������������� |
���������������������������������������������������������� Chapter 7 Skills Practice Lesson 7-1 Write a ratio expressing the slope of each line. 1. line ℓ 2. line m 3. line n 4. The ratio of the side lengths of a quadrilateral is 2 : 4 : 5 : 6, and its perimeter is 85 inches. What is the length of the shortest side? 5. The ratio of angle measures in a triangle is 3 : 10 : 12. What is the measure of each angle? Solve each proportion. = 6 _ 6. x _ 5 20 7. 21 _ 6 9 9. Given that 3x = 12y, find the ratio of x to y in simplest form. Lesson 7-2 Identify the pairs of congruent angles and corresponding sides. 10. 11. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 12. rectangles ABCD and EFGH 13. △JKL and △MNO Lesson 7-3 Explain why the triangles are similar and write a similarity statement. 14. 15. Verify that the triangles are similar. 16. △FGH ∼ △JKH 17. △ACE ∼ △BCD Explain why the triangles are similar and then find each length. 18. △XYZ and △ABC, BC 19. △RSV and △UST, TU S16S16 TEKS TAKS Practice ����������������������������������������������������������������������������������������������������������������������������������������� Lesson 7-4 Lesson 7-5 Find the length of each segment. 20. ̶̶ AE Verify that the given segments are parallel. 22. ̶̶ EF and ̶̶ JG Find the length of each segment. 24. ̶̶ RS and ̶̶ ST 21. ̶̶ KJ 23. ̶̶ LP and ̶̶̶ MN 25. ̶̶̶ XW and ̶̶̶ WZ The scale drawing of the playhouse is 1 in. : 10 ft. Find the actual lengths of the following walls. 26. ̶̶̶ GH ̶̶ EF ̶̶ DC 27. 28. The school courtyard is 25 ft by 40 ft. Make a scale drawing of the courtyard using the following scales. 29. 1 cm |
: 1 ft 31. 1 cm : 10 ft 30. 1 cm : 5 ft 32. Given that △ABC ∼ △DEF, find the perimeter P and area A of △DEF. Lesson 7-6 33. Given that △RSV ∼ △RTU, find the coordinates of S and the scale factor. 34. Given: A (-3, 3), B (1, 7), C (5, 5), D (-1, 5), E (1, 4) Prove: △ABC ∼ △ADE TEKS TAKS Practice S17 S17 ������������������������������������������������������������������������������������������������������������������������������������������������������������������������������ Chapter 8 Skills Practice Lesson Lesson Lesson 8-1 2-5 2-5 Write a similarity statement comparing the three triangles in each diagram. 1. 2. 3. Find the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form. 4. 3 and 9 5. 4 and 7 6. 1 _ 2 and 5 Find x, y, and z. 7. 8. 9. Lesson 8-2 Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 10. sin A 11. cos A 12. tan A Use a special right triangle to write each trigonometric ratio as a fraction. 13. cos 30° 14. sin 45° 15. tan 60° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 16. sin 38° 17. cos 47° 18. tan 21° Find each length. Round to the nearest hundredth. 19. DE 20. GH 21. KL Lesson 8-3 22. tan -1 (3.5) Use your calculator to find each angle measure to the nearest degree. 23. sin -1 ( 1 _ ) 5 Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 25. 27. 26. 24. cos -1 (0.05) For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 28. A (1, 4), B (1, 1), C (4, 1) 29. D (-3, 5), E (-3 |
, 1), F (2, 5) S18S18 TEKS TAKS Practice ��������������������������������������������������������������������������������������������������������� Lesson 8-4 Lesson 8-5 Classify each angle as an angle of elevation or angle of depression. 30. ∠1 31. ∠2 32. ∠3 33. ∠4 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 34. cos 127° 37. tan 158° 35. tan 131° 38. sin 85° 36. sin 114° 39. cos 161° Find each measure. Round lengths to the nearest tenth and angle measure to the nearest degree. 40. AC 41. m∠E 42. m∠G 43. m∠T 44. VX 45. BC Lesson 8-6 Write each vector in component form. 46. AB with A (2, 3) and B (5, 6) 47. the vector with initial point C (3, 6) and terminal point D (2, 4) 48. EF 49. GH Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 50. 〈-3, 2〉 52. 〈2, -5〉 51. 〈4, 3〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 53. A wind velocity is given by the vector 〈3, 4〉. 54. The velocity of a rocket is given by the vector 〈8, 1〉. Identify each of the following in the diagram. 55. equal vectors 56. parallel vectors Find each vector sum. 57. 〈5, 0〉 + 〈-3, 6〉 58. 〈-3, -1〉 + 〈0, -7〉 59. 〈1, 8〉 + 〈2, 3〉 60. 〈-2, -1〉 + 〈-7, 9〉 TEKS TAKS Practice S19 S19 ������������������������������������ |
�������������������������������������������������������������� Chapter 9 Skills Practice Lesson 9-1 Find each measurement. 1. the area of the parallelogram 2. the perimeter of the rectangle in which A = 15 x 2 ft 2 3. b 2 of the trapezoid in which A = 35 ft 2 4. the area of the kite 5. the base of a triangle in which h = 9 and A = 135 in 2 6. the area of a rhombus in which d 1 = (3x + 5) cm and d 2 = (7x + 4) cm Lesson 9-2 Find each measurement. 7. the circumference of ⊙C in terms of π 8. the area of ⊙D in terms of π 9. the circumference of ⊙F in which A = 49 x 2 π cm 2 10. the radius of ⊙E in which C = 36π in. Find the area of each regular polygon. Round to the nearest tenth. 11. a regular hexagon with a side length of 8 in. 12. an equilateral triangle with an apothem of 5 √ 3 _ 3 cm Lesson 9-3 Find the shaded area. Round to the nearest tenth, if necessary. 15. 13. 14. Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 in. 16. 17. S20S20 TEKS TAKS Practice ������������������������������������������������������������������������������������� Lesson 9-4 Estimate the area of each irregular shape. 18. 19. Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 20. A (-2, 3), B (0, 6), C (6, 2), D (4, -1) 21. E (-1, 3), F (2, 3), G (2, -1) Find the area of each polygon with the given vertices. 22. R (-2, 3), S (1, 5), T (3, 1), U (0, -2) 23. W (-4, 0), X (4, 3), Y (6, 1), Z (2, -1) Lesson 9-5 Describe the effect |
of each change on the area of the given figure. 24. The height of the rectangle with height 10 ft and width 12 ft is multiplied by 1 _. 2 25. The base of the parallelogram with vertices A (-2, 3), B (3, 3), C (0, -1), D (-5, -1) is doubled. Describe the effect of each change on the perimeter or circumference and the area of the given figure. 26. The radius of ⊙E is multiplied by 1 _. 4 27. The base and height of a rectangle with base 6 in. and height 5 in. are multiplied by 3. 28. A square has a side length of 7 ft. If the area is tripled, what happens to the side length? 29. A circle has a diameter of 20 m. If the area is doubled, what happens to the circumference? Lesson 9-6 A point is chosen randomly on ̶̶ 30. The point is on AC. 32. The point is not on ̶̶ BC. ̶̶ AD. Find the probability of each event. ̶̶ AB or ̶̶ BD. 31. The point is on 33. The point is on ̶̶ CD. Use the spinner to find the probability of each event. 34. the pointer landing on green 35. the pointer landing on blue or red 36. the pointer not landing on orange 37. the pointer not landing on red or yellow Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth. 38. the equilateral triangle 39. the parallelogram 40. the circle 41. the part of the rectangle that does not include the circle, triangle, or parallelogram TEKS TAKS Practice S21 S21 �������������������������������������������������������������������������� Chapter 10 Skills Practice Lesson 10-1 Classify each figure. Name the vertices, edges, and bases. Classify each figure. Name the vertices, edges, and bases. 1. 1. 2. 3. Describe the three-dimensional figure that can be made from the given net. 4. 5. 6. Use the figure made of unit cubes for Exercises 7–11. Assume there are no hidden cubes. 7. Draw all six orthographic views. 8. Draw an isometric view. 9. Draw a one-point perspective view. |
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