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�re congruent, so each angle must be 90°, or a right angle. Investigation 3 Do Rectangle Diagonals Have Special Properties? Now let’s look at the diagonals of rectangles. You will need ● graph paper ● a compass Step 1 Step 2 Step 1 Step 2 Draw a large rectangle using the lines on a piece of graph paper as a guide. Draw in both diagonals. With your compass, compare the lengths of the two diagonals. Compare results with your group. In addition, recall that a rectangle is also a parallelogram. So its diagonals also have the properties of a parallelogram’s diagonals. Combine these ideas to complete the conjecture. Rectangle Diagonals Conjecture The diagonals of a rectangle are? and?. C-52 Career A tailor uses a button spacer to mark the locations of the buttons. The tool opens and closes, but the tips always remain equally spaced. What quadrilateral properties make this tool work correctly? LESSON 5.6 Properties of Special Parallelograms 289 What happens if you combine the properties of a rectangle and a rhombus? We call the shape a square, and you can think of it as a regular quadrilateral. So you can define it in two different ways. A square is an equiangular rhombus. Or A square is an equilateral rectangle. A square is a parallelogram, as well as both a rectangle and a rhombus. Use what you know about the properties of these three quadrilaterals to copy and complete this conjecture. Square Diagonals Conjecture The diagonals of a square are?,?, and?. C-53 EXERCISES For Exercises 1–10 identify each statement as true or false. For each false statement, sketch a counterexample or explain why it is false. You will need Construction tools for Exercises 17–19, 23, 24, and 30 1. The diagonals of a parallelogram are congruent. 2. The consecutive angles of a rectangle are congruent and supplementary. 3. The diagonals of a rectangle bisect each other. 4. The diagonals of a rectangle bisect the angles. 5. The diagonals of a square are perpendicular bisectors of each other. 6. Every rhombus is a square. 7. Every square is a rectangle. |
8. A diagonal divides a square into two isosceles right triangles. 9. Opposite angles in a parallelogram are always congruent. 10. Consecutive angles in a parallelogram are always congruent. 11. WREK is a rectangle. CR 10 WE? 12. PARL is a parallelogram. y? K W C 10 E R L R 48° P y 95° A 290 CHAPTER 5 Discovering and Proving Polygon Properties 13. SQRE is a square 16. Is TILE a parallelogram? Why? y T (0, 18) X I (–10, 0) E (10, 0) x x L (0, –18) 14. Is DIAM a rhombus? Why? 15. Is BOXY a rectangle? Why? y x A –9 I M D –9 y 9 B Y O 9 17. Construction Given the diagonal LV, construct square LOVE. L V 18. Construction Given diagonal BK and B, construct rhombus BAKE. B B K 19. Construction Given side PS and diagonal PE, construct rectangle PIES. P P S E 20. To make sure that a room is rectangular, builders check the two diagonals of the room. Explain what they must check, and why this works. 21. The platforms shown at the beginning of this lesson lift objects straight up. The platform also stays parallel to the floor. You can clearly see rhombuses in the picture, but you can also visualize the frame as the diagonals of three rectangles. Explain why the diagonals of a rectangle guarantee this vertical movement. LESSON 5.6 Properties of Special Parallelograms 291 22. At the street intersection shown at right, one of the streets is wider than the other. Do the crosswalks form a rhombus or a parallelogram? Explain. What would have to be true about the streets if the crosswalks formed a rectangle? A square? In Exercises 23 and 24, use only the two parallel edges of your double-edged straightedge. You may not fold the paper or use any marks on the straightedge. 23. Construction Draw an angle on your paper. Use your double-edged straightedge to construct the bisector of the angle. 24. Construction Draw a segment on your paper. Use your double-edged straightedge to construct the perpendicular bisector of the segment. Review 25. Trace the figure below |
. Calculate the measure of each lettered angle. 1 2 and 54° a 1 2 26. Complete the flowchart proof below to demonstrate logically that if a quadrilateral has four congruent sides then it is a rhombus. One possible proof for this argument has been started for you. Given: Quadrilateral QUAD has QU UA AD DQ with diagonal DU Show: QUAD is a rhombus Flowchart Proof QU UA AD DQ Given QU DA QD UA? 4 QUD ADU? 5 1 2 3 4? 3 DU DU Same segment 6 QU AD QD UA If alternate interior angles are congruent, then lines are parallel 7 8 QUAD is a parallelogram Definition of parallelogram QUAD is a rhombus? 292 CHAPTER 5 Discovering and Proving Polygon Properties 27. Find the coordinates of three more points that lie on the line passing through the points (2, 1) and (3, 4). 28. Find the coordinates of the circumcenter and the orthocenter for RGT with vertices R(2, 1), G(5, 2), and T(3, 4). 29. Draw a counterexample to show that this statement is false: If a triangle is isosceles, then its base angles are not complementary. 30. Construction Oran Boatwright is rowing at a 60° angle from the upstream direction as shown. Use a ruler and a protractor to draw the vector diagram. Draw the resultant vector and measure it to find his actual velocity and direction. 31. In Exercise 26, you proved that if the four sides of a quadrilateral are congruent, then the quadrilateral is a rhombus. So, when we defined rhombus, we did not need the added condition of it being a parallelogram. We only needed to say that it is a quadrilateral with all four sides congruent. Is this true for rectangles? Your conjecture would be, “If a quadrilateral has all four angles congruent, it must be a rectangle.” Can you find a counterexample that proves it false? If you cannot, try to create a proof showing that it is true. 2 mi/hr 60° 1.5 mi/hr IMPROVING YOUR REASONING SKILLS How Did the Farmer Get to the Other Side? A farmer was taking her pet rabbit, a basket of prize-winning |
baby carrots, and her small—but hungry—rabbit-chasing dog to town. She came to a river and realized she had a problem. The little boat she found tied to the pier was big enough to carry only herself and one of the three possessions. She couldn’t leave her dog on the bank with the little rabbit (the dog would frighten the poor rabbit), and she couldn’t leave the rabbit alone with the carrots (the rabbit would eat all the carrots). But she still had to figure out how to cross the river safely with one possession at a time. How could she move back and forth across the river to get the three possessions safely to the other side? LESSON 5.6 Properties of Special Parallelograms 293 L E S S O N 5.7 “For instance” is not a “proof.” JEWISH SAYING Proving Quadrilateral Properties Most of the paragraph proofs and flowchart proofs you have done so far have been set up for you to complete. Creating your own proofs requires a great deal of planning. One excellent planning strategy is “thinking backward.” If you know where you are headed but are unsure where to start, start at the end of the problem and work your way back to the beginning one step at a time. The firefighter below asks another firefighter to turn on one of the water hydrants. But which one? A mistake could mean disaster—a nozzle flying around loose under all that pressure. Which hydrant should the firefighter turn on? Did you “think backward” to solve the puzzle? You’ll find it a useful strategy as you write proofs. To help plan a proof and visualize the flow of reasoning, you can make a flowchart. As you think backward through a proof, you draw a flowchart backward to show the steps in your thinking. Work with a partner when you first try planning your geometry proof. Think backward to make your plan: start with the conclusion and reason back to the given. Let’s look at an example. A concave kite is sometimes called a dart. EXAMPLE Given: Dart ADBC with AC BC, AD BD Show: CD bisects ACB C D A B Solution Plan: Begin by drawing a diagram and marking the given information on it. Next, construct your proof by reasoning backward. Then convert this reasoning into a flowchart. Your flowchart should start with boxes containing the given information and end with |
what you are trying to demonstrate. The arrows indicate the flow of your logical argument. Your thinking might go something like this: 294 CHAPTER 5 Discovering and Proving Polygon Properties “I can show CD is the bisector of ACB if I can show ACD BCD.” ACD BCD CD is the bisector of ACB “I can show ACD BCD if they are corresponding angles in congruent triangles.” ADC BDC ACD BCD CD is the bisector of ACB “Can I show ADC BDC? Yes, I can, by SSS, because it is given that AC BC and AD BD, and CD CD because it is the same segment in both triangles.” AD BD AC BC ADC BDC ACD BCD CD is the bisector of ACB CD CD AD BD Given AC BC Given CD CD Same segment 1 2 3 By adding the reason for each statement below each box in your flowchart, you can make the flowchart into a complete flowchart proof. 4 ADC BDC 5 ACD BCD 6 SSS CPCTC CD is the bisector of ACB Definition of angle bisector Some students prefer to write their proofs in a flowchart format, and others prefer to write out their proof as an explanation in paragraph form. By reversing the reasoning in your plan, you can make the plan into a complete paragraph proof. “It is given that AC BC and AD BD. CD CD because it is the same segment in both triangles. So, ADC BDC by the SSS Congruence Conjecture. So, ACD BCD by the definition of congruent triangles (CPCTC). Therefore, by the definition of angle bisectors, CD is the bisector of ACB. Q.E.D.” The abbreviation Q.E.D. at the end of a proof stands for the Latin phrase quod erat demonstrandum, meaning “which was to be demonstrated.” You can also think of Q.E.D. as a short way of saying “Quite Elegantly Done” at the conclusion of your proof. In the exercises you will prove some of the special properties of quadrilaterals discovered in this chapter. LESSON 5.7 Proving Quadrilateral Properties 295 EXERCISES 1. Let’s start with a puzzle. Copy the 5-by-5 puzzle grid at right. Start |
at square 1 and end at square 100. You can move to an adjacent square horizontally, vertically, or diagonally if you can add, subtract, multiply, or divide the number in the square you occupy by 2 or 5 to get the number in that square. For example, if you happen to be in square 11, you could move to square 9 by subtracting 2 or to square 55 by multiplying by 5. When you find the path from 1 to 100, show it with arrows. Notice that in this puzzle you may start with different moves. You could start with 1 and go to 5. From 5 you could go to 10 or 3. Or you could start with 1 and go to 2. From 2 you could go to 4. Which route should you take? In Exercises 2–10, each conjecture has also been stated as a “given” and a “show.” Any necessary auxiliary lines have been included. Complete a flowchart proof or write a paragraph proof. 2. Prove the conjecture: The diagonal of a parallelogram divides the parallelogram into two congruent triangles. Given: Parallelogram SOAK with diagonal SA Show: SOA AKS Flowchart Proof K 1 3 S A 4 2 O 2 SO KA 4 3 4 Definition of parallelogram AIA 1 SOAK is a parallelogram Given 3 OA?? 7? SOA? 5 6 1 2? SA?? 3. Prove the conjecture: The opposite angles of a parallelogram are congruent. H T Given: Parallelogram BATH with diagonals BT and HA Show: HBA ATH and BAT THB Flowchart Proof 1 Parallelogram BATH with diagonal BT Given 4 Parallelogram BATH with diagonal HA? 2 BAT THB 3 BAT THB Conjecture from Exercise 2 CPCTC 5 BAH =? 6 HBA??? 296 CHAPTER 5 Discovering and Proving Polygon Properties B A 4. Prove the conjecture: If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Given: Quadrilateral WATR, with WA RT and WR AT, and diagonal WT R T 2 3 Show: WATR is a parallelogram Flowchart Proof WATR is a parallelogram?. Write a flowchart proof to demonstrate that quadrilateral SOAP is a parallelogram. |
Given: Quadrilateral SOAP with SP OA and SP OA Show: SOAP is a parallelogram P 1 3 A S 4 2 O 6. The results of the proof in Exercise 5 can now be stated as a proved conjecture. Complete this statement beneath your proof: “If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a?.” 7. Prove the conjecture: The diagonals of a rectangle are congruent. Given: Rectangle YOGI with diagonals YG and OI Show: YG OI 8. Prove the conjecture: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Given: Parallelogram BEAR, with diagonals BA ER Show: BEAR is a rectangle 9. Prove the Isosceles Trapezoid Conjecture: The base angles of an isosceles trapezoid are congruent. Given: Isosceles trapezoid PART with PA TR, PT AR, and TZ constructed parallel to RA Show: TPA RAP LESSON 5.7 Proving Quadrilateral Properties 297 10. Prove the Isosceles Trapezoid Diagonals Conjecture: The diagonals of an isosceles trapezoid are congruent. Given: Isosceles trapezoid GTHR with GR TH and diagonals GH and TR R H G T Show: GH TR 1? Given 2 3 4? Given GT GT?? Isosceles Trapezoid Conjecture 5??? 6?? 11. If an adjustable desk lamp, like the one at right, is adjusted by bending or straightening the metal arm, it will continue to shine straight down onto the desk. What property that you proved in the previous exercises explains why? 12. You have discovered that triangles are rigid but parallelograms are not. This property shows up in the making of fabric, which has warp threads and weft threads. Fabric is constructed by weaving thread at right angles, creating a grid of rectangles. What happens when you pull the fabric along the warp or weft? What happens when you pull the fabric along a diagonal (the bias)? Warp threads (vertical) Bias Weft threads (horizontal) Review 13. Find the measure of the acute angles in the |
4-pointed star in the Islamic tiling shown at right. The polygons are squares and regular hexagons. Find the measure of the acute angles in the 6-pointed star in the Islamic tiling on the far right. The 6-pointed star design is created by arranging six squares. Are the angles in both stars the same? 14. A contractor tacked one end of a string to each vertical edge of a window. He then handed a protractor to his apprentice and said, “Here, find out if the vertical edges are parallel.” What should the apprentice do? No, he can’t quit, he wants this job! Help him. 298 CHAPTER 5 Discovering and Proving Polygon Properties 15. The last bus stops at the school some time between 4:45 and 5:00. What is the probability that you will miss the bus if you arrive at the bus stop at 4:50? 16. The 3-by-9-by-12-inch clear plastic sealed container shown is resting on its smallest face. It is partially filled with a liquid to a height of 8 inches. Sketch the container resting on its middle-sized face. What will be the height of the liquid in the container in this position? 12 9 3 JAPANESE PUZZLE QUILTS When experienced quilters first see Japanese puzzle quilts, they are often amazed (or puzzled?) because the straight rows of blocks so common to block quilts do not seem to exist. The sewing lines between apparent blocks seem jagged. At first glance, Japanese puzzle quilts look like American crazy quilts that must be handsewn and that take forever to make! However, Japanese puzzle quilts do contain straight sewing lines. Study the Japanese puzzle quilt at right. Can you find the basic quilt block? What shape is it? Mabry Benson designed this puzzle quilt, Red and Blue Puzzle (1994). Can you find any rhombic blocks that are the same? How many different types of fabric were used? The puzzle quilt shown above is made of four different-color kites sewn into rhombuses. The rhombic blocks are sewn together with straight sewing lines as shown in the diagram at left. Look closely again at the puzzle quilt. Now for your project. You will need copies of the Japanese puzzle quilt grid, color pencils or markers, and color paper or fabrics. 1. To produce the z |
igzag effect of a Japanese puzzle quilt, you need to avoid pseudoblocks of the same color sharing an edge. How many different colors or fabrics do you need in order to make a puzzle quilt? 2. How many different types of rhombic blocks do you need for a four-color Japanese puzzle quilt? What if you want no two pseudoblocks of the same color to touch at either an edge or a vertex? 3. Can you create a four-color Japanese puzzle quilt that requires more than four different color combinations in the rhombic blocks? 4. Plan, design, and create a Japanese puzzle quilt out of paper or fabric, using the Japanese puzzle quilt technique. Detail of a pseudoblock Detail of an actual block LESSON 5.7 Proving Quadrilateral Properties 299 ● CHAPTER 11 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER CHAPTER 5 R E V I E W In this chapter you extended your knowledge of triangles to other polygons. You discovered the interior and exterior angle sums for all polygons. You investigated the midsegments of triangles and trapezoids and the properties of parallelograms. You learned what distinguishes various quadrilaterals and what properties apply to each class of quadrilaterals. Along the way you practiced proving conjectures with flowcharts and paragraph proofs. Be sure you’ve added the new conjectures to your list. Include diagrams for clarity. How has your knowledge of triangles helped you make discoveries about other polygons? EXERCISES 1. How do you find the measure of one exterior angle of a regular polygon? You will need Construction tools for Exercises 19–24 2. How can you find the number of sides of an equiangular polygon by measuring one of its interior angles? By measuring one of its exterior angles? 3. How do you construct a rhombus by using only a ruler or double-edged straightedge? 4. How do you bisect an angle by using only a ruler or double-edged straightedge? 5. How can you use the Rectangle Diagonals Conjecture to determine if the corners of a room are right angles? 6. How can you use the Triangle Midsegment Conjecture to find a distance between two points that you can’t measure directly? 7. Find x and y. 8. Perimeter 266 cm. 9. Find a and |
c. Find x. x 50° 80° y 94 cm 52 cm x 116° c a 10. MS is a midsegment. Find the perimeter of MOIS. I 11. Find x. 12. Find y and z. G S 20 18 x – 12 32 cm x O M 26 T D 17 cm y z 300 CHAPTER 5 Discovering and Proving Polygon Properties ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 13. Copy and complete the table below by placing a yes (to mean always) or a no (to mean not always) in each empty space. Use what you know about special quadrilaterals. Kite Isosceles trapezoid Parallelogram Rhombus Rectangle Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent Diagonals bisect each other Diagonals are perpendicular Diagonals are congruent Exactly one line of symmetry Exactly two lines of symmetry Yes 14. APPLICATION A 2-inch-wide frame is to be built around the regular decagonal window shown. At what angles a and b should the corners of each piece be cut? 15. Find the measure of each lettered angle 16. Archaeologist Ertha Diggs has uncovered one stone that appears to be a voussoir from a semicircular stone arch. On each isosceles trapezoidal face, the obtuse angles measure 96°. Assuming all the stones were identical, how many stones were in the original arch? No 2 in. b a CHAPTER 5 REVIEW 301 EW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTE 17. Kite ABCD has vertices A(3, 2), B(2, 2), C(3, 1), and D(0, 2). Find the coordinates of the point of intersection of the diagonals. 18. When you swing left to right on a swing, the seat stays parallel to the ground. Explain why. 19. Construction The tiling of congruent pentagons shown below is created from a honeycomb grid (tiling of regular hexagons). What is the measure of each lettered angle? Re-create the design with compass and straightedge. a b 20. Construction An airplane is heading north at 900 km/hr. However, a |
50 km/hr wind is blowing from the east. Use a ruler and a protractor to make a scale drawing of these vectors. Measure to find the approximate resultant velocity, both speed and direction (measured from north). Construction In Exercises 21–24, use the given segments and angles to construct each figure. Use either patty paper or a compass and a straightedge. The small letter above each segment represents the length of the segment. x y 21. Construct rhombus SQRE with SR y and QE x. 22. Construct kite FLYR given F, L, and FL x. 23. Given bases LP with length z z F L and EN with length y, nonparallel side LN with length x, and L, construct trapezoid PENL. 24. Given F, FR x, and YD z, construct two trapezoids FRYD that are not congruent to each other. 302 CHAPTER 5 Discovering and Proving Polygon Properties ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 25. Three regular polygons meet at point B. Only four sides of the third polygon are visible. How many sides does this polygon have? B 26. Find x. x 48 cm 27. Prove the conjecture: The diagonals of a rhombus bisect the angles. Given: Rhombus DENI, with diagonal DN Show: Diagonal DN bisects D and N Flowchart Proof DN bisects IDE and INE? 1 DENI is a rhombus? 2 DE?? 3 NE?? 4 DN?? TAKE ANOTHER LOOK 1. Draw several polygons that have four or more sides. In each, draw all the diagonals from one vertex. Explain how the Polygon Sum Conjecture follows logically from the Triangle Sum Conjecture. Does the Polygon Sum Conjecture apply to concave polygons? 2. A triangle on a sphere can have three right angles. Can you find a “rectangle” with four right angles on a sphere? Investigate the Polygon Sum Conjecture on a sphere. Explain how it is related to the Triangle Sum Conjecture on a sphere. Be sure to test your conjecture on polygons with the smallest and largest possible angle measures. The small, precise polygons in the painting, Boy With Birds (1953, oil on |
canvas), by American artist David C. Driskell (b 1931), give it a look of stained glass. CHAPTER 5 REVIEW 303 EW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTER 5 REVIEW ● CHAPTE 3. Draw a polygon and one set of its exterior angles. Label the exterior angles. Cut out the exterior angles and arrange them all about a point. Explain how this activity demonstrates the Exterior Angle Sum Conjecture. 4. Is the Exterior Angle Sum Conjecture also true for concave polygons? Are the kite conjectures also true for darts (concave kites)? Choose your tools and investigate. 5. Investigate exterior angle sums for polygons on a sphere. Be sure to test polygons with the smallest and largest angle measures. Assessing What You’ve Learned GIVING A PRESENTATION Giving a presentation is a powerful way to demonstrate your understanding of a topic. Presentation skills are also among the most useful skills you can develop in preparation for almost any career. The more practice you can get in school, the better. Choose a topic to present to your class. There are a number of things you can do to make your presentation go smoothly. Work with a group. Make sure your group presentation involves all group members so that it’s clear everyone contributed equally. Choose a topic that will be interesting to your audience. Prepare thoroughly. Make an outline of important points you plan to cover. Prepare visual aids—like posters, models, handouts, and overhead transparencies—ahead of time. Rehearse your presentation. Communicate clearly. Speak up loud and clear, and show your enthusiasm about your topic. ORGANIZE YOUR NOTEBOOK Your conjecture list should be growing fast! Review your notebook to be sure it’s complete and well organized. Write a one-page chapter summary. WRITE IN YOUR JOURNAL Write an imaginary dialogue between your teacher and a parent or guardian about your performance and progress in geometry. UPDATE YOUR PORTFOLIO Choose a piece that represents your best work from this chapter to add to your portfolio. Explain what it is and why you chose it. PERFORMANCE ASSESSMENT While a classmate, a friend, a family member, or a teacher observes, carry out one of the investigations from this chapter. Explain what you’re doing at each step, including how you arrived at the conjecture. 304 CH |
APTER 5 Discovering and Proving Polygon Properties CHAPTER 6 Discovering and Proving Circle Properties I am the only one who can judge how far I constantly remain below the quality I would like to attain. M. C. ESCHER Curl-Up, M. C. Escher, 1951 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● learn relationships among chords, arcs, and angles ● discover properties of tangent lines ● learn how to calculate the length of an arc ● prove circle conjectures Chord Properties Let’s review some basic terms before you begin discovering the properties of circles. You should be able to identify the terms below. Match the figures at the right with the terms at the left. 1. Congruent circles 2. Concentric circles 3. Radius 4. Chord 5. Diameter 6. Tangent 7. Minor arc 8. Major arc 9. Semicircle A. DC B. TG C. OE D. AB E. F. 3 in. G. RQ H. PRQ I. PQR D A O C B T E G 3 in. P T R Q Check your answers: ( 1.F,2.E,3.C,4.A,5.D,6.B,7.G,8.I,9.H ) In addition to these terms, you will become familiar with two more, central angle and inscribed angle, in the next investigation. Can you find parts of the water wheel that match the circle terms above? L E S S O N 6.1 Learning by experience is good, but in the case of mushrooms and toadstools, hearsay evidence is better. ANONYMOUS Double Splash Evidence, part of modern California artist Gerrit Greve’s Water Series, uses brushstrokes to produce an impression of concentric ripples in water. 306 CHAPTER 6 Discovering and Proving Circle Properties Investigation 1 How Do We Define Angles in a Circle? Look at the examples and non-examples for each term. Then write a definition for each. Discuss your definitions with others in your class. Agree on a common set of definitions and add them to your definition list. In your notebook, draw and label a picture to illustrate each definition. Step 1 Define central angle. C D O B A DOC intercepts arc DC BOC, COD, DOA, |
and DOB are central angles of circle O.. AOB, Step 2 Define inscribed angle PQR, PQS, RST, QST, and QSR are not central angles of circle P ABC, BCD, and CDE are inscribed angles. ABC is inscribed in. and intercepts (or determines) AC ABC PQR, STU, and VWX are not inscribed angles. You will need ● a compass ● a straightedge ● a protractor Step 1 Step 2 Investigation 2 Chords and Their Central Angles Next you will discover some properties of chords and central angles. You will also see a relationship between chords and arcs. B O D Construct a large circle. Label the center O. Using your compass, construct two congruent chords in your circle. Label the chords AB and CD, then construct radii OA, OB, OC, and OD. With your protractor, measure BOA and COD. How do they compare? Share your results with others in your group. Then copy and complete the conjecture. C A LESSON 6.1 Chord Properties 307 Chord Central Angles Conjecture C-54 If two chords in a circle are congruent, then they determine two central angles that are?. Step 3 How can you fold your circle construction to check the conjecture? As you learned in Chapter 1, the measure of an arc is defined as the measure of its central angle. For example, the central angle, BOA at right, has a measure of 40°. The measure of a 40°, so mAB major arc is 360° minus the measure of the minor arc making up the remainder of the circle. For example, the measure of major arc BCA is 360° 40°, or 320°. A 40° B Intercepted arc 40° O C Your next conjecture follows from the Chord Central Angles Conjecture and the definition of arc measure. Step 4 Two congruent chords in a circle determine two central angles that are congruent. If two central angles are congruent, their intercepted arcs must be congruent. Combine these two statements to complete the conjecture.? Chord Arcs Conjecture If two chords in a circle are congruent, then their? are congruent.? C-55 “Pull the cord?! Don’t I need to construct it first?” 308 CHAPTER 6 Discovering and Proving Circle Properties You will need ● a compass |
● a straightedge Investigation 3 Chords and the Center of the Circle In this investigation, you will discover relationships about a chord and the center of its circle. Step 1 On a sheet of paper, construct a large circle. Mark the center. Construct two nonparallel congruent chords. Then, construct the perpendiculars from the center to each chord. Step 2 How does the perpendicular from the center of a circle to a chord divide the chord? Copy and complete the conjecture. Perpendicular to a Chord Conjecture The perpendicular from the center of a circle to a chord is the? of the chord. C-56 Let’s continue this investigation to discover a relationship between the length of congruent chords and their distances from the center of the circle. Step 3 With your compass, compare the distances (measured along the perpendicular) from the center to the chords. Are the results the same if you change the size of the circle and the length of the chords? State your observations as your next conjecture. Chord Distance to Center Conjecture Two congruent chords in a circle are? from the center of the circle. C-57 Investigation 4 Perpendicular Bisector of a Chord Next you will discover a property of perpendicular bisectors of chords. You will need ● a compass ● a straightedge Step 1 On another sheet of paper, construct a large circle and mark the center. Construct two nonparallel chords that are not diameters. Then, construct the perpendicular bisector of each chord and extend the bisectors until they intersect. LESSON 6.1 Chord Properties 309 Step 2 What do you notice about the point of intersection? Compare your results with the results of others near you. Copy and complete the conjecture. Perpendicular Bisector of a Chord Conjecture The perpendicular bisector of a chord?. C-58 With the perpendicular bisector of a chord, you can find the center of any circle, and therefore the vertex of the central angle to any arc. All you have to do is construct the perpendicular bisectors of nonparallel chords. EXERCISES Solve Exercises 1–7. State which conjecture or definition you used to support your conclusion. You will need Construction tools for Exercises 13–15, 17, and 21 1. x? x 165° 4. y? 72° y 7. AB is a diameter. Find and mB. mAC B 68° O C A 2. z? |
20° 128° z 5. AB CD PO 8 cm OQ? 3. w? 70° w 6. AB 6 cm OP 4 cm CD 8 cm OQ 3 cm BD 6 cm What is the perimeter of OPBDQ. What’s wrong with 9. What’s wrong with this picture? this picture? 37 cm 18 cm O 310 CHAPTER 6 Discovering and Proving Circle Properties 10. Draw a circle and two chords of unequal length. Which is closer to the center of the circle, the longer chord or the shorter chord? Explain. 11. Draw two circles with different radii. In each circle, draw a chord so that the chords have the same length. Draw the central angle determined by each chord. Which central angle is larger? Explain. 12. Polygon MNOP is a rectangle inscribed in a circle centered at the origin. Find the coordinates of points M, N, and O. M (?, 3) P (4, 3) 13. Construction Construct a triangle. Using the sides of the triangle as chords, construct a circle passing through all three vertices. Explain. Why does this seem familiar? N (?,?) O (?,?) 14. Construction Trace a circle onto a blank sheet of paper without using your compass. Locate the center of the circle using a compass and straightedge. Trace another circle onto patty paper and find the center by folding. 15. Construction Adventurer Dakota Davis digs up a piece of a circular ceramic plate. Suppose he believes that some ancient plates with this particular design have a diameter of 15 cm. He wants to calculate the diameter of the original plate to see if the piece he found is part of such a plate. He has only this piece of the circular plate, shown at right, to make his calculations. Trace the outer edge of the plate onto a sheet of paper. Help him find the diameter. 16. Complete the flowchart proof shown, which proves that if two chords of a circle are congruent, then they determine two congruent central angles. Given: Circle O with chords AB CD Show: AOB COD A B C O D Flowchart Proof 1 AB CD? 2 AO CO All radii of a circle are congruent 3 BO DO? 4??? 5 AOB?? km 5 km 17. Construction The satellite photo at right shows only a portion of a lunar crater. How can cartographers use the photo to find its center? Trace the crater and locate its center. Using the scale |
shown, find its radius. To learn more about satellite photos, go to www.keymath.com/DG. LESSON 6.1 Chord Properties 311 18. Circle O has center (0, 0) and passes through points A(3, 4) and B(4, 3). Find an equation to show that the perpendicular bisector of AB passes through the center of the circle. Explain your reasoning. Review 19. Identify each quadrilateral from the given characteristics. a. Diagonals are perpendicular and bisect each other. b. Diagonals are congruent and bisect each other, but it is not a square. c. Only one diagonal is the perpendicular bisector of the other diagonal. d. Diagonals bisect each other. y O A (3, 4) x B (4, –3) 20. A family hikes from their camp on a bearing of 15°. (A bearing is an angle measured clockwise from the north, so a bearing of 15° is 15° east of north.) They hike 6 km and then stop for a swim in a lake. Then they continue their hike on a new bearing of 117°. After another 9 km, they meet their friends. What is the measure of the angle between the path they took to arrive at the lake and the path they took to leave the lake? 21. Construction Use a protractor and a centimeter ruler to make a careful drawing of the route the family in Exercise 20 traveled to meet their friends. Let 1 cm represent 1 km. To the nearest tenth of a kilometer, how far are they from their first camp? 22. Explain why x equals y. 55° x y 40° 23. What is the probability of randomly selecting three points from the 3-by-3 grid below that form the vertices of a right triangle? IMPROVING YOUR ALGEBRA SKILLS Algebraic Sequences II Find the next two terms of each algebraic pattern. 1. x6, 6x5y, 15x4y2, 20x3y3, 15x2y4,?,? 2. x7, 7x6y, 21x5y2, 35x4y3, 35x3y4, 21x2y5,?,? 3. x8, 8x7y, 28x6y2, 56x5y3, 70x4y4, 56x3y5 |
, 28x2y6,?,? 312 CHAPTER 6 Discovering and Proving Circle Properties L E S S O N 6.2 We are, all of us, alone Though not uncommon In our singularity. Touching, We become tangent to Circles of common experience, Co-incident, Defining in collective tangency Circles Reciprocal in their subtle Redefinition of us. In tangency We are never less alone, But no longer Only. GENE MATTINGLY You will need ● a compass ● a straightedge Step 1 Step 2 Step 3 Tangent Properties Each wheel of a train theoretically touches only one point on the rail. Rails act as tangent lines to the wheels of a train. Each point where the rail and the wheel meet is a point of tangency. Why can’t a train wheel touch more than one point at a time on the rail? Can a car wheel touch more than one point at a time on the road? The rail is tangent to the wheels of the train. The penguins’ heads are tangent to each other. Investigation 1 Going Off on a Tangent In this investigation, you will discover the relationship between a tangent line and the radius drawn to the point of tangency. Construct a large circle. Label the center O. Using your straightedge, draw a line that appears to touch the circle at only one point. Label the point T. Construct OT. T O Use your protractor to measure the angles at T. What can you conclude about the radius OT and the tangent line at T? Step 4 Share your results with your group. Then copy and complete the conjecture. Tangent Conjecture A tangent to a circle? the radius drawn to the point of tangency. C-59 LESSON 6.2 Tangent Properties 313 Technology A series of rockets burning chemical fuels provide the thrust to launch satellites into orbit. Once the satellite reaches its proper orientation in space, it provides its own power for the duration of its mission, sometimes staying in space for five to ten years with the help of solar energy and a battery backup. At right is the Mir space station and the space shuttle Atlantis in orbit in 1995. According to the United States Space Command, there are over 8,000 objects larger than a softball circling Earth at speeds of over 18,000 miles per hour! If gravity were suddenly “turned off” somehow, these objects would travel off into space on a straight line tang |
ent to their orbits, and not continue in a curved path. You will need ● a compass ● a straightedge The Tangent Conjecture has important applications related to circular motion. For example, a satellite maintains its velocity in a direction tangent to its circular orbit. This velocity vector is perpendicular to the force of gravity, which keeps the satellite in orbit. v g Investigation 2 Tangent Segments In this investigation, you will discover something about the lengths of segments tangent to a circle from a point outside the circle. Construct a circle. Label the center E. Choose a point outside the circle and label it N. Draw two lines through point N tangent to the circle. Mark the points where these lines appear to touch the circle and label them A and G. Use your compass to compare segments NA and NG. These segments are called tangent segments. Step 1 Step 2 Step 3 Step 4 A E G Step 5 Share your results with your group. Copy and complete the conjecture. Tangent Segments Conjecture Tangent segments to a circle from a point outside the circle are?. N C-60 314 CHAPTER 6 Discovering and Proving Circle Properties Tangent circles are two circles that are tangent to the same line at the same point. They can be internally tangent or externally tangent, as shown. Externally tangent circles Internally tangent circles EXERCISES 1. Rays m and n are tangents to circle P. w? 2. Rays r and s are tangent to circle Q. x? m n P 130° w 70° x Q r s You will need Construction tools for Exercises 8–12 and 15 Geometry software for Exercises 13 and 26 3. Ray k is tangent to circle R. y? 4. Line t is tangent to both circles. z? y R 120° k 75° S z M t 5. Quadrilateral POST is circumscribed about circle Y. OR 13 and ST 12. What is the perimeter of POST. Pam participates in the hammer-throw event. She swings a 16 lb ball at arm’s length, about eye-level. Then she releases the ball at the precise moment when the ball will travel in a straight line toward the target area. Draw an overhead view that shows the ball’s circular path, her arms at the moment she releases it, and the ball’s straight path toward the target area. 7. Explain how you could use only a T-square |
, like the one shown, to find the center of a Frisbee. Pam Dukes competes in the hammer-throw event. LESSON 6.2 Tangent Properties 315 Construction For Exercises 8–12, first make a sketch of what you are trying to construct and label it. Then use the segments below, with lengths r, s, and t. r s t 8. Construct a circle with radius r. Mark a point on the circle. Construct a tangent through this point. 9. Construct a circle with radius t. Choose three points on the circle that divide it into three minor arcs and label points X, Y, and Z. Construct a triangle that is circumscribed about the circle and tangent at points X, Y, and Z. 10. Construct two congruent, externally tangent circles with radius s. Then construct a third circle that is both congruent and externally tangent to the two circles. 11. Construct two internally tangent circles with radii r and t. 12. Construct a third circle with radius s that is externally tangent to both the circles you constructed in Exercise 11. 13. Technology Use geometry software to construct a circle. Label three points on the circle and construct tangents through them. Drag the three points and write your observations about where the tangent lines intersect and the figures they form. 14. Find real-world examples (different from the examples shown below) of two internally tangent circles and of two externally tangent circles. Either sketch the examples or make photocopies from a book or a magazine for your notebook. This astronomical clock in Prague, Czech Republic, has one pair of internally tangent circles. What other circle relationships can you find in the clock photo? The teeth in the gears shown extend from circles that are externally tangent. 15. Construction In Taoist philosophy, all things are governed by one of two natural principles, yin and yang. Yin represents the earth, characterized by darkness, cold, or wetness. Yang represents the heavens, characterized by light, heat, or dryness. The two principles, when balanced, combine to produce the harmony of nature. The symbol for the balance of yin and yang is shown at right. Construct the yin-and-yang symbol. Start with one large circle. Then construct two circles with half the diameter that are internally tangent to the large circle and externally tangent to each other. Finally, construct small circles that are concentric to the two inside |
circles. Shade or color your construction. 316 CHAPTER 6 Discovering and Proving Circle Properties 16. A satellite in geostationary orbit remains above the same point on the earth’s surface even as the earth turns. If such a satellite has a 30° view of the equator of the earth, what percentage of the equator is observable from the satellite?? 30° 17. Circle P is centered at the origin. AT is tangent to circle P at A(8, 15). Find the equation of AT. 18. PA is tangent to circle Q. The line containing chord CB passes through P. Find mP. y A (8, 15) T x P (0, 0) 19. TA and TB are tangent to circle O. What’s wrong with this picture? Review B Q 168° C 78° P A T 36° A 148° O B 20. Circle U passes through points (3, 11), (11, 1), and (14, 4). Find the coordinates of its center. Explain your method. 21. Complete the flowchart proof or write a paragraph proof of the Perpendicular to a Chord Conjecture: The perpendicular from the center of a circle to a chord is the bisector of the chord. Given: Circle O with chord CD, radii OC and OD, and OR CD Show: OR bisects CD Flowchart Proof O D R C 1 OR CD? 2 ORC and ORD are right angles? 3 mORC 90° mORD 90°? 4 mORC mORD (ORC ORD)? 5 OC OD 6 OCD is isosceles 7 C? 8 OCR????? 9 CR?? 10 OR bisects CD? LESSON 6.2 Tangent Properties 317 22. Identify each of these statements as true or false. If the statement is true, explain why. If it is false, give a counterexample. a. If the diagonals of a quadrilateral are congruent, but only one is the perpendicular bisector of the other, then the quadrilateral is a kite. b. If the quadrilateral has exactly one line of reflectional symmetry, then the quadrilateral is a kite. c. If the diagonals of a quadrilateral are congruent and bisect each other, then it is a square. 23. Rachel and Yulia are building an |
art studio above their back bedroom. There will be doors on three sides leading to a small deck that surrounds the studio. They need to place an electrical junction box in the ceiling of the studio so that it is equidistant from the three light switches shown by the doors. Copy the diagram of the room and find the most efficient location for the junction box. 24. What will the units digit be when you evaluate 323? s s s 25. A small light-wing aircraft has made an emergency landing in a remote portion of a wildlife refuge and is sending out radio signals for help. Ranger Station Alpha receives the signal on a bearing of 38° and Station Beta receives the signal on a bearing of 312°. (Recall that a bearing is an angle measured clockwise from the north.) Stations Alpha and Beta are 8.2 miles apart, and Station Beta is on a bearing of 72° from Station Alpha. Which station is closer to the downed aircraft? Explain your reasoning. 26. Technology Use geometry software to pick any three points. Construct an arc through all three points. (Can it be done?) How do you find the center of the circle that passes through all three points? IMPROVING YOUR VISUAL THINKING SKILLS Colored Cubes Sketch the solid shown, but with the red cubes removed and the blue cube moved to cover the starred face of the green cube. 318 CHAPTER 6 Discovering and Proving Circle Properties L E S S O N 6.3 You will do foolish things, but do them with enthusiasm. SIDONIE GABRIELLA COLETTE Arcs and Angles Many arches that you see in structures are semicircular, but Chinese builders long ago discovered that arches don’t have to have this shape. The Zhaozhou bridge, shown below, was completed in 605 C.E. It is the world’s first stone arched bridge in the shape of a minor arc, predating other minor-arc arches by about 800 years. In this lesson you’ll discover properties of arcs and the angles associated with them. You will need ● a compass ● a straightedge ● a protractor Investigation 1 Inscribed Angle Properties In this investigation, you will compare an inscribed angle and a central angle, both inscribed in the same arc. Refer to the diagram of circle O, with central angle COR and inscribed angle CAR. Step 1 Step 2 Measure COR with your protractor to find mCR intercepted arc. Measure |
CAR. How does mCAR compare with mCR, the? Construct a circle of your own with an inscribed angle. Draw the central angle that intercepts the same arc. What is the measure of the inscribed angle? How do the two measures compare? A C O Step 3 Share your results with others near you. Copy and complete the conjecture. Inscribed Angle Conjecture The measure of an angle inscribed in a circle?. R C-61 LESSON 6.3 Arcs and Angles 319 Investigation 2 Inscribed Angles Intercepting the Same Arc Next, let’s consider two inscribed angles that intercept the same arc. In the figure at right, AQB and APB both intercept AB inscribed in APB.. Angles AQB and APB are both A P B You will need ● a compass ● a straightedge ● a protractor Step 1 Step 2 Step 3 Step 4 Construct a large circle. Select two points on the circle. Label them A and B. Select a point P on the major arc and construct inscribed angle APB. With your protractor, measure APB. Select another point Q on APB inscribed angle AQB. Measure AQB. How does mAQB compare with mAPB? and construct Repeat Steps 1 and 2 with points P and Q selected on minor arc AB. Compare results with your group. Then copy and complete the conjecture. Q A Q Inscribed Angles Intercepting Arcs Conjecture Inscribed angles that intercept the same arc?. Investigation 3 Angles Inscribed in a Semicircle Next, you will investigate a property of angles inscribed in semicircles. This will lead you to a third important conjecture about inscribed angles. You will need ● a compass ● a straightedge ● a protractor B P C-62 B Step 1 Construct a large circle. Construct a diameter AB. Inscribe three angles in the same semicircle. Make sure the sides of each angle pass through A and B. A Step 2 Measure each angle with your protractor. What do you notice? Compare your results with the results of others and make a conjecture. Angles Inscribed in a Semicircle Conjecture Angles inscribed in a semicircle?. C-63 Now you will discover a property of the angles of a quadrilateral inscribed in a circle. 320 CHAPTER 6 Discovering and Proving Circle Properties You will need ● a compass ● a straightedge ● a protractor Step 1 Step 2 Investigation 4 Cyclic |
Quadrilaterals A quadrilateral inscribed in a circle is called a cyclic quadrilateral. Each of its angles is inscribed in the circle, and each of its sides is a chord of the circle. Construct a large circle. Construct a cyclic quadrilateral by connecting four points anywhere on the circle. Measure each of the four inscribed angles. Write the measure in each angle. Look carefully at the sums of various angles. Share your observations with students near you. Then copy and complete the conjecture. Cyclic Quadrilateral Conjecture The? angles of a cyclic quadrilateral are?. You will need ● patty paper ● a compass ● a double-edged straightedge Investigation 5 Arcs by Parallel Lines Next, you will investigate arcs formed by parallel lines that intersect a circle. A line that intersects a circle in two points is called a secant. A secant contains a chord of the circle, and passes through the interior of a circle, while a tangent line does not. C-64 Secant Step 1 Step 2 Step 3 On a piece of patty paper, construct a large circle. Lay your straightedge across the circle so that its parallel edges pass through the circle. Draw secants AB and DC along both edges of the straightedge. Fold your patty paper to compare AD can you say about AD and BC and BC. What D? A Repeat Steps 1 and 2, using either lined paper or another object with parallel edges to construct different parallel secants. Share your results with other students. Then copy and complete the conjecture. Parallel Lines Intercepted Arcs Conjecture Parallel lines intercept? arcs on a circle. C B C-65 LESSON 6.3 Arcs and Angles 321 Review these conjectures and ask yourself which quadrilaterals can be inscribed in a circle. Can any parallelogram be a cyclic quadrilateral? If two sides of a cyclic quadrilateral are parallel, then what kind of quadrilateral will it be? You will need Geometry software for Exercises 19 and 21 Construction tools for Exercise 24 5. d? e? 96° e 20° d 8. CALM is a rectangle. x? x M L C 32° A 11. r? s? s r 14. What is the sum of a, b, c, d, and e? EXERCISES Use your new conjectures to solve Exercises 1–17. 1. a? 2. b? a |
130° b 60° 3. c? 4. h? 95° 120° c 6. f? g? g 75° f 110° 9. DOWN is a kite. y? D O y W 136° N 12. m? n? m 40° n 98° 40° h 7. JUST is a rhombus. w? J w U 130° T S 10. k? k 38° 13. AB CD p? q? A p C 120° 98° q B D e d a b c 322 CHAPTER 6 Discovering and Proving Circle Properties 15. y? 16. What’s wrong with this picture? 17. Explain why AC CE. 80° y 70° 37° 35 18. How can you find the center of a circle, using only the corner of a piece of paper? 19. Technology Chris Chisholm, a high school student in Whitmore, California, used the Angles Inscribed in a Semicircle Conjecture to discover a simpler way to find the orthocenter in a triangle. Chris constructs a circle using one of the sides of the triangle as the diameter, then he immediately finds an altitude to each of the triangle’s other two sides. Use geometry software and Chris’s method to find the orthocenter of a triangle. Does this method work on all kinds of triangles? 20. APPLICATION The width of a view that can be captured in a photo depends on the camera’s picture angle. Suppose a photographer takes a photo of your class standing in one straight row with a camera that has a 46° picture angle. Draw a line segment to represent the row. Draw a 46° angle on a piece of patty paper. Locate at least eight different points on your paper where a camera could be positioned to include all the students, filling as much of the picture as possible. What is the locus of all such camera positions? What conjecture does this activity illustrate? 46° 21. Technology Construct a circle and a diameter. Construct a point on one of the semicircles, and construct two chords from it to the endpoints of the diameter to create a right triangle. Locate the midpoint of each of the two chords. Predict, then sketch the locus of the two midpoints as the vertex of the right angle is moved around the circle. Finally, use your computer to animate the point on the circle and trace the locus of the two midpoints. What do |
you get? Review 22. Find the measure of each lettered angle LESSON 6.3 Arcs and Angles 323 23. Use the diagram at right and the flowchart below to write a paragraph proof explaining why two congruent chords in a circle are equidistant from the center of the circle. Given: Circle O with PQ RS and OT PQ and OV RS Show: OT OV P T 2 Q O OT PQ OV RS OTQ and OVS are right angles OTQ OVS R V 1 S OP OR OQ OS OTQ OVS OT OV PQ RS OPQ ORS 2 1 24. Construction Use your construction tools 25. What’s wrong with this picture? to re-create this design of three congruent circles all tangent to each other and internally tangent to a larger circle. A B C 26. Consider the figure at right with line AB. As P moves from left to right along line, which of these lengths or distances always increases? A. The distance PB B. The distance from D to AB C. DC D. Perimeter of ABP E. None of the above P D A C B IMPROVING YOUR REASONING SKILLS Think Dinosaur If the letter in the word dinosaur that is three letters after the word’s second vowel is also found before the sixteenth letter of the alphabet, then print the word dinosaur horizontally. Otherwise, print the word dinosaur vertically and cross out the second letter after the first vowel. 324 CHAPTER 6 Discovering and Proving Circle Properties L E S S O N 6.4 Mistakes are a fact of life. It’s the response to the error that counts. NIKKI GIOVANNI Proving Circle Conjectures In Lesson 6.3, you discovered the Inscribed Angle Conjecture: The measure of an angle inscribed in a circle equals half the measure of its intercepted arc. Many other circle conjectures are logical consequences of the Inscribed Angle Conjecture. Let’s start this lesson by proving it. When you inscribe an angle in a circle, the angle will relate to the circle’s center in one of the three ways described below. These three possible relationships to the center are the three cases for which you prove the Inscribed Angle Conjecture. To prove the conjecture, you must prove all three cases. Case 1 The circle’s center is on the angle. Case 2 The |
center is outside the angle. Case 3 The center is inside the angle. You will first prove that the conjecture is true for Case 1. Case 1 Conjecture: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc when a side of the angle passes through the center of the circle. Given: Circle O with inscribed angle MDR on diameter DR Let z mDMO, x mMDR, and y mMOR (y mMR ) M x D z y O R Show: mMDR 1 mMR 2 Work backward to formulate a plan. Ask yourself what you’re trying to show and what you would need to do that. Plan ● You need to show that mMDR 1 mMR 2 given, this can be restated as x 1 y. 2. Using the variables defined in the ● You want to show that x 1 y, so you need to show that 2x y. 2 ● You know that y x z because of the Exterior Angle Conjecture. ● You also know that x z because DOM is isosceles. ● So, start your flowchart proof by establishing that DOM has two congruent sides. From this plan you create a flowchart proof. LESSON 6.4 Proving Circle Conjectures 325 Flowchart Proof of Case 1 DO OM x z y All radii of a circle are congruent Triangle Exterior Angle Conjecture x x y (substitution property of equality) 2x y (combine like terms) x y (division property of equality) 1_ 2 DOM is isosceles Isosceles triangle definition x z Algebra operations Isosceles Triangle Conjecture mMDR m MR 1_ 2 Substitution You will use Case 1 to write a paragraph proof for Case 2. Case 2 Conjecture: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc when the center of the circle is outside the angle. Given: Circle O with inscribed angle MDK on one side of diameter DR Show: mMDK 1 mMK 2 Paragraph Proof of Case. Then, mMR Let z mMDR, x mMDK, and w mKDR. Then, mKDR mMDR mMDK, so w x z. mMK Let a mMR and b mMK = a b. So, mKR Stated in terms of x and b, you wish to show that x b. From Case 1 |
, you 2 know that w (a b) and that z a. You know that w x z, so x w z 2 2 by the subtraction property of equality. Substitute (a get x (a b) for w and a for z to 2 2 ) a b (a b b) a. Therefore, mMDK 1 mMK. 2 2 2 2 2 mKR. You will prove Case 3 in the exercises. Pattern of light rays from distant objects as focused on the retina of a normal eye. Science Light rays from distant objects are focused on the retina of a normal eye. The rays converge short of the retina on a myopic (near-sighted) eye, causing blurry vision. With the proper lens, light rays from distant objects will focus sharply on the retina, giving the nearsighted person clear vision. When the rays are focused on the retina, what kind of angle do they form inside the eye? 326 CHAPTER 6 Discovering and Proving Circle Properties Pattern of light rays from distant objects as focused in a myopic eye. The rays are focused short of the retina. EXERCISES 1. Prove Case 3 of the Inscribed Angle Conjecture. Case 3 Conjecture: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc when the center of the circle is inside the angle. z D Given: Circle O with inscribed angle MDK whose sides DM and DK lie on either side of diameter DR Show: mMDK 1 mMK 2 In Exercises 2–5 the four conjectures are consequences of the Inscribed Angle Conjecture. Prove each conjecture by writing a paragraph proof or a flowchart proof. 2. Inscribed angles that intercept the same arc are congruent. Given: Circle O with ACD and ABD inscribed in ACD Show: ACD ABD 3. Angles inscribed in a semicircle are right angles. Given: Circle O with diameter AB, and ACB inscribed in semicircle ACB Show: ACB is a right angle. The opposite angles of a cyclic quadrilateral are supplementary. L Given: Circle O with inscribed quadrilateral LICY Show: L and C are supplementary 5. Parallel lines intercept congruent arcs on a circle. Given: Circle O with chord BD and AB CD Show: BC DA For Exercises 6 and 7, determine whether each conjecture is true or false. If the conjecture is false, draw a counterex |
ample. If the conjecture is true, prove it by writing either a paragraph or flowchart proof. 6. If a parallelogram is inscribed within a circle, then the parallelogram is a rectangle. Given: Circle Y with inscribed parallelogram GOLD Show: GOLD is a rectangle LESSON 6.4 Proving Circle Conjectures 327 7. If a trapezoid is inscribed within a circle, then the trapezoid is isosceles. G E Given: Circle R with inscribed trapezoid GATE Show: GATE is an isosceles trapezoid R A T Review 8. For each of the statements below, choose the letter for the word that best fits (A stands for always, S for sometimes, and N for never). If the answer is S, give two examples, one showing how the statement is true and one showing how the statement can be false. a. An equilateral polygon is (A/S/N) equiangular. b. If a triangle is a right triangle, then the acute angles are (A/S/N) complementary. c. The diagonals of a kite are (A/S/N) perpendicular bisectors of each other. d. A regular polygon (A/S/N) has both reflectional symmetry and rotational symmetry. e. If a polygon has rotational symmetry, then it (A/S/N) has more than one line of reflectional symmetry. 9. Explain why m is parallel to n. 39° 141° m n 10. What is the probability of randomly selecting three collinear points from the points in the 3-by-3 grid below? 11. How many different 3-edge routes are possible from R to G along the wire frame shown? A R L T N E G C IMPROVING YOUR VISUAL THINKING SKILLS Rolling Quarters One of two quarters remains motionless while the other rotates around it, never slipping and always tangent to it. When the rotating quarter has completed a turn around the stationary quarter, how many turns has it made around its own center point? Try it! 328 CHAPTER 6 Discovering and Proving Circle Properties ALGEBRA SKILLS 6 ● USING YOUR ALGEBRA SKILLS 6 ● USING YOUR ALGEBRA SKILLS 6 ● USING YO USING YOUR ALGEBRA SKILLS 6 Finding the Circumcenter Suppose |
you know the coordinates of the vertices of a triangle. How can you find the coordinates of the circumcenter? You can graph the triangle, construct the perpendicular bisectors of the sides, and then estimate the coordinates of the point of concurrency. However, to find the exact coordinates, you need to use algebra. Let’s look at an example. EXAMPLE Find the coordinates of the circumcenter of ZAP with Z(0, 4), A(4, 4), and P(8, 8). Solution To review midpoint, slopes of perpendicular lines, or solving systems of equations, see the Table of Contents for those Using Your Algebra Skills topics. A To find the coordinates of the circumcenter, you can write equations for the perpendicular bisectors of two of the sides of the triangle and then find the point where the bisectors intersect. To find the equation for the perpendicular bisector of ZA, first find the midpoint of ZA, then find its slope. 4 (2, 0) Midpoint of ZA 0 (4), 4 2 2 4 ) ( Slope of ZA The slope of the perpendicular bisector of ZA is the negative reciprocal of 2, or 1, and it passes through point (2, 0). So the equation of the perpendicular 2 y 0. Solving for y gives the equation y 1 1 x 1. bisector is x ( 2 2 2) You can use the same technique to find the equation of the perpendicular bisector of ZP. The midpoint of ZP is (4, 2), and the slope is 3. So the slope of 2 the perpendicular bisector of ZP is 2 and it passes through the point (4, 2). 3 y 4. x 1, or y 2 2 The equation of the perpendicular bisector is 3 3 3 x 2 4 Since all the perpendicular bisectors intersect at the same point, you can solve these two equations to find that point. To find the point where the perpendicular bisectors intersect, solve this system by substitution Perpendicular bisector of ZA. Perpendicular bisector of ZP Original first equation. 4 (from the second equation) for y. x 1 Substitute 2 3 3 USING YOUR ALGEBRA SKILLS 6 Finding the Circumcenter 329 ALGEBRA SKILLS 6 ● USING YOUR ALGEBRA SKILLS 6 ● USING YOUR ALGEBRA SKILLS 6 ● USING |
YO 4x 28 3x 6 Multiply both sides by 6. 22 7x Add 4x to both sides. Subtract 6 from both sides. Divide both sides by 7. Substitute 2 for x in the first equation. Simplify. The circumcenter is 2 8. You can check this result by writing the equation for 2, 1 7 7 the perpendicular bisector of AP and verifying that 2 8 is a point on 2, 1 7 7 this line. EXERCISES 1. Triangle RES has vertices R(0, 0), E(4, 6), and S(8, 4). Find the equation of the perpendicular bisector of RE. In Exercises 2–5, find the coordinates of the circumcenter of each triangle. 2. Triangle TRM with vertices T(2, 1), R(4, 3), and M(4, 1) 3. Triangle FGH with vertices F(0, 6), G(3, 6), and H(12, 0) 4. Right triangle MNO with vertices M(4, 0), N(0, 5), and O(10, 3) 5. Isosceles triangle CDE with vertices C(0, 6), D(0, 6), and E(12, 0) 6. If a triangle is a right triangle, there is a shorter method to finding the circumcenter. What is it? Explain. 7. If a triangle is an isosceles triangle, then there is a different, perhaps shorter method to finding the circumcenter. Explain. 8. Circle P with center at (6, 6) and circle Q with center at (11, 0) have a common internal tangent AB. Find the coordinates of B if A has coordinates (3, 2). y –12 (–3, –2) A P (–6, –6) –12 Q (11, 0) 14 x B (?,?) 330 CHAPTER 6 Discovering and Proving Circle Properties L E S S O N 6.5 No human investigations can ever be called true science without going through mathematical tests. LEONARDO DA VINCI The Circumference/ Diameter Ratio The distance around a polygon is called the perimeter. The distance around a circle is called the circumference. Here is a nice visual puzzle. Which is greater, the height of a tennis-ball can or the circumference of the can? The height is approximately three tennis-ball |
diameters tall. The diameter of the can is approximately one tennis-ball diameter. If you have a tennis-ball can handy, try it. Wrap a string around the can to measure its circumference, then compare this measurement with the height of the can. Surprised? If you actually compared the measurements, you discovered that the circumference of the can is greater than three diameters of the can. In this lesson you are going to discover (or perhaps rediscover) the relationship between the diameter and the circumference of every circle. Once you know this relationship, you can measure a circle’s diameter and calculate its circumference. If you measure the circumference and diameter of a circle and divide the circumference by the diameter, you get a number slightly larger than 3. The more accurate your measurements, the closer your ratio will come to a special number called (pi), pronounced “pie,” like the dessert. History In 1897, the Indiana state assembly tried to legislate the value of. The vague language of the state’s House Bill No. 246, which became known as the “Indiana Pi Bill,” implies several different incorrect values for —3.2, 3.232, 3.236, 3.24, and 4. With a unanimous vote of 67-0, the House passed the bill to the state senate, where it was postponed indefinitely. LESSON 6.5 The Circumference/Diameter Ratio 331 You will need ● several round objects (cans, mugs, bike wheel, plates) ● a meterstick or metric measuring tape ● sewing thread or thin string Investigation A Taste of Pi In this investigation you will find an approximate value of by measuring circular objects and calculating the ratio of the circumference to the diameter. Let’s see how close you come to the actual value of. Step 1 Measure the circumference of each round object by wrapping the measuring tape, or string, around its perimeter. Then measure the diameter of each object with the meterstick or tape. Record each measurement to the nearest millimeter (tenth of a centimeter). Step 2 Make a table like the one below and record the circumference (C) and diameter (d) measurements for each round object. Object Circumference (C) Diameter (d) Ratio C d Can Mug Wheel Step 3 Step 4 Calculate the ratio C for each object. Record the answers in your table. d Calculate the average of your ratios of C. d Compare your average with the averages |
of other groups. Are the C ratios close? d You should now be convinced that the ratio C is very close to 3 for every circle. d We define as the ratio C. If you solve this formula for C, you get a formula for d the circumference of a circle in terms of the diameter, d. The diameter is twice the radius (d 2r), so you can also get a formula for the circumference in terms of the radius, r. Step 5 Copy and complete the conjecture. Circumference Conjecture C-66 If C is the circumference and d is the diameter of a circle, then there is a number such that C?. If d 2r where r is the radius, then C?. 332 CHAPTER 6 Discovering and Proving Circle Properties Mathematics The number is an irrational number—its decimal form never ends. It is also a transcendental number—the pattern of digits does not repeat. The symbol is a letter of the Greek alphabet. Perhaps no other number has more fascinated 4 mathematicians throughout history. Mathematicians in ancient Egypt used 4 3 as their approximation of circumference to diameter. Early Chinese and Hindu mathematicians used 10. By 408 C.E., Chinese mathematicians were using 3 5 5. 3 1 1 Today, computers have calculated approximations of to billions of decimal places, and there are websites devoted to! See www.keymath.com/DG. Accurate approximations of have been of more interest intellectually than practically. Still, what would a carpenter say if you asked her to cut a board 3 feet long? Most calculators have a button that gives to eight or ten decimal places. You can use this value for most calculations, then round your answer to a specified decimal place. If your calculator doesn’t have a button, or if you don’t have access to a calculator, use the value 3.14 for. If you’re asked for an exact answer instead of an approximation, state your answer in terms of. How do you use the Circumference Conjecture? Let’s look at two examples. EXAMPLE A If a circle has diameter 3.0 meters, what is the circumference? Use a calculator and state your answer to the nearest 0.1 meter. Solution C d C (3.0) Original formula. Substitute the value of d. In terms of, the answer is 3. The circumference is about 9.4 meters. EXAMPLE B If a circle |
has circumference 12 meters, what is the radius? Solution C 2r 12 2r r 6 Original formula. Substitute the value of C. Solve. The radius is 6 meters. 3 m r EXERCISES Use the Circumference Conjecture to solve Exercises 1–12. In Exercises 1–6, leave your answer in terms of. You will need A calculator for Exercises 7–10 1. If C 5 cm, find d. 2. If r 5 cm, find C. 3. If C 24 m, find r. 4. If d 5.5 m, find C. 5. If a circle has a diameter of 12 cm, what is its circumference? 6. If a circle has a circumference of 46 m, what is its diameter? LESSON 6.5 The Circumference/Diameter Ratio 333 In Exercises 7–10, use a calculator. Round your answer to the nearest 0.1 unit. Use the symbol to show that your answer is an approximation. 7. If d 5 cm, find C. 8. If r 4 cm, find C. 9. If C 44 m, find r. 10. What’s the circumference of a bicycle wheel with a 27-inch diameter? 11. If the distance from the center of a Ferris wheel to one of the seats is approximately 90 feet, what is the distance traveled by a seated person, to the nearest foot, in one revolution? 12. If a circle is inscribed in a square with a perimeter of 24 cm, what is the circumference of the circle? 13. If a circle with a circumference of 16 inches is circumscribed about a square, what is the length of a diagonal of the square? 14. Each year a growing tree adds a new ring to its cross section. Some years the ring is thicker than others. Why do you suppose this happens? Suppose the average thickness of growth rings in the Flintstones National Forest is 0.5 cm. About how old is “Old Fred,” a famous tree in the forest, if its circumference measures 766 cm? Science Trees can live hundreds to thousands of years, and we can determine the age of one tree by counting its growth rings. A pair of rings—a light ring formed in the spring and summer and a dark one formed in the fall and early winter—represent the growth for one year. We can learn a lot about the climate of a region over a period of years |
will not land on any lines? The length of the needle and the distance between the lines will affect these probabilities. Start with a toothpick of any length L as your “needle” and construct parallel lines a distance L apart. Write your predictions, then experiment. Using N as the number of times you dropped the needle, and C as the number of times the needle crossed the line, enter your results into the expression 2N/C. As you drop the needle more and more times, the value of this expression seems to be getting close to what number? Your project should include Your predictions and data. The calculated probabilities and your prediction of the theoretical probability. Any other interesting observations or conclusions. Using Fathom, you can simulate many experiments. See the demonstration Buffon’s Needle that comes with the software package. 336 CHAPTER 6 Discovering and Proving Circle Properties L E S S O N 6.6 Love is like —natural, irrational, and very important. LISA HOFFMAN EXAMPLE Solution Around the World Many application problems are related to. Satellite orbits, the wheels of a vehicle, tree trunks, and round pizzas are just a few of the real-world examples that involve the circumference of circles. Here is a famous example from literature. Literature In the novel Around the World in Eighty Days (1873), Jules Verne (1828–1905) recounts the adventures of brave Phileas Fogg and his servant Passerpartout. They begin their journey when Phileas bets his friends that he can make a trip around the world in 80 days. Phileas’s precise behavior, such as monitoring the temperature of his shaving water or calculating the exact time and location of his points of travel, reflects Verne’s interest in the technology boom of the late nineteenth century. His studies in geology, engineering, and astronomy aid the imaginative themes in this and his other novels, including A Journey to the Center of the Earth (1864) and Twenty Thousand Leagues Under the Sea (1870). If the diameter of the earth is 8000 miles, find the average speed in miles per hour Phileas Fogg needs to circumnavigate the earth about the equator in 80 days. To find the speed, you need to know the distance and the time. The distance around the equator is equal to the circumference C of a circle with a diameter of 8,000 miles. C d The equation for |
circumference. (8,000) 25,133 Substitute 8,000 for d. Round to nearest mile. So, Phileas must travel 25,133 miles in 80 days. To find the speed v in mi/hr, you need to divide distance by time and convert days into hours. ce an t s v di m e i t d ay i 1 3 13 25, r h 24 d ay 0 8 m s 13 mi/hr The formula for speed, or velocity. Substitute values and convert units of time. Evaluate and round to the nearest mile per hour. If the earth’s diameter were exactly 8,000 miles, you could evaluate an exact answer of 25 in terms of. 6 8,000 80 24 and get LESSON 6.6 Around the World 337 EXERCISES In Exercises 1–6, round answers to the nearest unit. You may use 3.14 as an approximate value of. If you have a button on your calculator, use that value and then round your final answer. You will need A calculator for Exercises 1–8 1. A satellite in a nearly circular orbit is 2000 km above Earth’s surface. The radius of Earth is approximately 6400 km. If the satellite completes its orbit in 12 hours, calculate the speed of the satellite in kilometers per hour. 2. Wilbur Wrong is flying his remote-control plane in a circle with a radius of 28 meters. His brother, Orville Wrong, clocks the plane at 16 seconds per revolution. What is the speed of the plane? Express your answer in meters per second. The brothers may be wrong, but you could be right! 3. Here is a tiring problem. The diameter of a car tire is approximately 60 cm (0.6 m). The warranty is good for 70,000 km. About how many revolutions will the tire make before the warranty is up? More than a million? A billion? (1 km 1000 m) 4. If the front tire of this motorcycle has a diameter of 50 cm (0.5 m), how many revolutions will it make if it is pushed 1 km to the nearest gas station? In other words, how many circumferences of the circle are there in 1000 meters? 5. Goldi’s Pizza Palace is known throughout the city. The small Baby Bear pizza has a 6-inch radius and sells for $9.75. The savory medium Mama Bear pizza sells for $12.00 and has |
an 8-inch radius. The large Papa Bear pizza is a hefty 20 inches in diameter and sells for $16.50. The edge is stuffed with cheese, and it’s the best part of a Goldi’s pizza. What size has the most pizza edge per dollar? What is the circumference of this pizza? 6. Felicia is a park ranger, and she gives school tours through the redwoods in a national park. Someone in every tour asks, “What is the diameter of the giant redwood tree near the park entrance?” Felicia knows that the arm span of each student is roughly the same as his or her height. So in response, Felicia asks a few students to arrange themselves around the circular base of the tree so that by hugging the tree with arms outstretched, they can just touch fingertips to fingertips. She then asks the group to calculate the diameter of the tree. In one group, four students with heights of 138 cm, 136 cm, 128 cm, and 126 cm were able to ring the tree. What is the approximate diameter of the redwood? 338 CHAPTER 6 Discovering and Proving Circle Properties 7. APPLICATION Zach wants a circular table so that 12 chairs, each 16 inches wide, can be placed around it with at least 8 inches between chairs. What should be the diameter of the table? Will the table fit in a 12-by-14-foot dining room? Explain. A 45 rpm record has a 7-inch diameter and spins at 45 revolutions per minute. A 33 rpm record has a 12-inch diameter and spins at 33 revolutions per minute. Find the difference in speeds of a point on the edge of a 33 rpm record to that of a point on the edge of a 45 rpm record, in ft/sec. Recreation Using tinfoil records, Thomas Edison (1847–1931) invented the phonograph, the first machine to play back recorded sound, in 1877. Commonly called record players, they weren’t widely reproduced until high-fidelity amplification (hi-fi) and advanced speaker systems came along in the 1930s. In 1948, records could be played at slower speeds to allow more material on the disc, creating longer-playing records (LPs). When compact discs became popular in the early 1990s, most record companies stopped making LPs. Some disc jockeys still use records instead of CDs. S T N T Review 9. Mini-Investigation In these diagrams of circle O |
, find the relationship between ECA formed by the secants and the difference of the intercepted arc measures. Then copy and complete the conjecture. mAE 35° mECA 20° E A O mNTS 75° T mNTS 200° mAE 40° mECA 80° C E A S O N mAE 25° E A mECA 61° C S N O mNTS 147° Conjecture: The measure of an angle formed by two secants through a circle is?. LESSON 6.6 Around the World 339 10. Copy the diagram below. Construct SA. Prove the conjecture you made in Exercise 9. 11. Explain why a equals b. S N E A O C a 12. As P moves from A to B along the semicircle ATB, which of these measures constantly increases? A. The perimeter of ABP B. The distance from P to AB C. mABP D. mAPB P A B O b T 13. a? 14. b? 96° 15. d? 44° a 32° b 168° 16. A helicopter has three blades each measuring about 26 feet. What is the speed in feet per second at the tips of the blades when they are moving at 400 rpm? 17. Two sides of a triangle are 24 cm and 36 cm. Write an inequality that represents the range of values for the third side. Explain your reasoning. d 24 cm 18 cm IMPROVING YOUR VISUAL THINKING SKILLS Picture Patterns I Draw the next picture in each pattern. Then write the rule for the total number of squares in the nth picture of the pattern. 1. 2. 340 CHAPTER 6 Discovering and Proving Circle Properties L E S S O N 6.7 Some people transform the sun into a yellow spot, others transform a yellow spot into the sun. PABLO PICASSO Arc Length You have learned that the measure of an arc is equal to the measure of its central angle. On a clock, the measure of the arc from 12:00 to 1:00 is equal to the measure of the angle formed by the hour and minute hands. A circular clock is divided into 0°, or 30°. 6 12 equal arcs, so the measure of this arc is 3 1 2 Notice that because the minute hand is longer, the tip of the minute hand must travel farther than the tip of the hour hand even though they both move 30° from 12 to 1. So the arc |
length is different even though the arc measure (the degree measure) is the same! Let’s take another look at the arc measure. 10 9 8 12 1 11 7 6 5 2 3 4 EXAMPLE A What fraction of its circle is each arc? a. AB is what fraction b. CED is what fraction c. EF is what fraction of circle T? of circle O? of circle P 120° F Solution In part a, you probably “just knew” that the arc is one-fourth of the circle because you have seen one-fourth of a circle so many times. Why is it one-fourth? The arc measure is 90°, a 1 ° 0 9. full circle measures 360°, and 4 6 ° 0 3 The arc in part b is half of the circle because 1 1 8 0 °. In part c, you may 2 ° 0 6 3 or may not have recognized right away that the arc is one-third of the circle. The arc is one-third of the circle 1 2 0 ° because 1. 3 ° 0 6 3 Cultural This modern mosaic shows the plan of an ancient Mayan observatory. On certain days of the year light would shine through openings, indicating the seasons. This sculpture includes blocks of marble carved into arcs of concentric circles. LESSON 6.7 Arc Length 341 What do these fractions have to do with arc length? If you traveled halfway around a circle, you’d cover 1 of its perimeter, or circumference. If you went a quarter of 2 the way around, you’d travel 1 of its circumference. The length of an arc, or arc 4 length, is some fraction of the circumference of its circle. The measure of an arc is calculated in units of degrees, but arc length is calculated in units of distance. Investigation Finding the Arcs In this investigation you will find a method for calculating the arc length. Step 1 For AB, CED, and GH B, find what fraction of the circle each arc is. A 12 m T E 4 in. O C 4 in. D G 36 ft F P 140° H Step 2 Step 3 Step 4 Find the circumference of each circle. Combine the results of Steps 1 and 2 to find the length of each arc. Share your ideas for finding the length of an arc. Generalize this method for finding the length of any arc, and state it as a conjecture. Arc Length Conjecture The length of an arc equals the?. C- |
67 How do you use this new conjecture? Let’s look at a few examples. EXAMPLE B If the radius of the circle is 24 cm and mBTA 60°, what is the length of AB? B Solution 120° by the Inscribed Angle mBTA 60°, so mAB, so the arc length is 1 1 2 0 Conjecture. Then 1 of the 3 3 0 6 3 circumference, by the Arc Length Conjecture. 60° T A Arc length 1 C 3 1 (48) 3 16 Substitute 2r for C, where r 24. Simplify. The arc length is 16 cm, or approximately 50.3 cm. 342 CHAPTER 6 Discovering and Proving Circle Properties EXAMPLE C If the length of ROT the circle? is 116 meters, what is the radius of O Solution mROT 240°, so ROT 116 2 C 3 116 2 (2r) 3 348 4r 87 r The radius is 87 m., or 2 4 0 is 2 of the circumference. 3 0 6 3 R 120° T Apply the Arc Length Conjecture. Substitute 2r for C. Multiply both sides by 3. Divide both sides by 4. EXERCISES For Exercises 1–8, state your answers in terms of. 1. Length of CD is?. 2. Length of EF is?. D 80° C 3 F 120° 12 m E You will need A calculator for Exercises 9–14 Construction tools for Exercise 16 Geometry software for Exercise 16 3. Length of BIG is?. 4. Length of AB is 6 m. The radius is?. 5. The radius is 18 ft. is?. Length of RT G 150° 12 cm I B 6. The radius is 9 m. Length of SO is?. S O 100° T C A r 120° B R T 30° A 7. Length of TV is 12 in. The diameter is?. T 8. Length of AR RE CA is 40 cm.. The radius is?. A 146° R 70° C E V 80° A LESSON 6.7 Arc Length 343 9. A go-cart racetrack has 100-meter straightaways and semicircular ends with diameters of 40 meters. Calculate the average speed in meters per minute of a go-cart if it completes 4 laps in 6 minutes. Round your answer to the nearest m/min. 10. |
Astronaut Polly Hedra circles Earth every 90 minutes in a path above the equator. If the diameter of Earth is approximately 8000 miles, what distance along the equator will she pass directly over while eating a quick 15-minute lunch? 11. APPLICATION The Library of Congress reading room has desks along arcs of concentric circles. If an arc on the outermost circle with eight desks is about 12 meters long and makes up 1 of the circle, how far 9 are these desks from the center of the circle? How many desks would fit along an arc with the same central angle but that is half as far from the center? Explain. 12. A Greek mathematician who lived in the The Library of Congress, Washington, D.C. third century B.C.E., Eratosthenes, devised a clever method to calculate the circumference of Earth. He knew that the distance between Aswan (then called Syene) and Alexandria was 5000 Greek stadia (a stadium was a unit of distance at that time), or about 500 miles. At noon of the summer solstice, the Sun cast no shadow on a vertical pole in Syene, but at the same time in Alexandria a vertical pole did cast a shadow. Eratosthenes found that the angle between the vertical pole and the ray from the tip of the pole to the end of the shadow was 7.2°. From this he was able to calculate the ratio of the distance between the two cities to the circumference of Earth. Use this diagram to explain Eratosthenes’ method, then use it to calculate the circumference of Earth in miles. Alexandria 7.2° x° Syene 5000 stadia Sun’s rays Review 13. Angular velocity is a measure of the rate at which an object revolves around an axis, and can be expressed in degrees per second. Suppose a carousel horse completes a revolution in 20 seconds. What is its angular velocity? Would another horse on the carousel have a different angular velocity? Why or why not? 344 CHAPTER 6 Discovering and Proving Circle Properties 14. Tangential velocity is a measure of the distance an object travels along a circular path in a given amount of time. Like speed, it can be expressed in meters per second. Suppose two carousel horses complete a revolution in 20 seconds. The horses are 8 m and 6 m from the center of the carousel, respectively. What are the tangential velocities of the two horses? Round your answers to the nearest 0 |
.1 m/sec. Explain why the horses have equal angular velocities but different tangential velocities. 15. Calculate the measure of each lettered angle. 110 2a a Art The traceries surrounding rose windows in Gothic cathedrals were constructed with only arcs and straight lines. The photo at right shows a rose window from Reims cathedral, which was built in the thirteenth century, in Reims, a city in northeastern France. The overlaid diagram shows its constructions. 16. Construction Read the Art Connection above. Reproduce the constructions shown with your compass and straightedge or with geometry software. 17. Find the measure of the angle formed by a clock’s hands at 10:20. RACETRACK GEOMETRY If you had to start and finish at the same line of a racetrack, which lane would you choose? The inside lane has an obvious advantage. For a race to be fair, runners in the outside lanes must be given head starts, as shown in the photo. Design a four-lane oval track with straightaways and semicircular ends. Show start and finish lines so that an 800-meter race can be run fairly in all four lanes. The semicircular ends must have inner diameters of 50 meters. The distance of one lap in the inner lane must be 800 meters. Your project should contain A detailed drawing with labeled lengths. An explanation of the part that radius, lane width, and straightaway length plays in the design. LESSON 6.7 Arc Length 345 Cycloids Imagine a bug gripping your bicycle tire as you ride down the street. What would the bug’s path look like? What path does the Moon make as it rotates around Earth while Earth rotates around the Sun? Using animation in Sketchpad, you can model a rotating wheel. Activity Turning Wheels In this activity, you’ll investigate the path of a point on a wheel as it rolls along the ground or around another wheel. You’ll start by constructing a stationary circle with a rotating spoke. Step 1 Step 2 Step 3 Step 4 Construct two points A and B. Select them in that order, and choose Circle By Center+Point from the Construct menu. Construct radius AB. Construct a second radius, AC. Select point C and choose Animation from the Action Buttons submenu of the Edit menu. Make point C move counterclockwise around the circle at fast speed. Press the Animation button. Now you have a |
circle with one spoke that rotates in a counterclockwise direction. (Press it again to stop.) How can you make a wheel that will roll? You can’t roll your circle with the spinning spoke because if the circle moves, the spoke would have to move with it. But you can make a different circle and, as you move it, use circle A to rotate it. Here’s how. Step 5 Construct a long horizontal segment DE going from right to left and a point F on the segment. Step 6 Select points A and F, in order, and choose Mark Vector from the Transform menu. 346 CHAPTER 6 Discovering and Proving Circle Properties Animate Point A B C These circles in the sand were created by the wind blowing blades of grass as if they were spokes on a wheel. Step 7 Step 8 Select circle A, point C, and AC, and use the Transform menu to translate by the marked vector. Construct a line FC overlapping FC and a point G anywhere on the line. Hide the line and construct FG. Animate Point E F D A B C C´ G Step 9 Press the animation button. The rotating spoke AC should cause the spoke FG to spin at the same time. Now you’re ready to make that circle roll. Step 10 Step 11 Step 12 Select DE and point F, and choose Perpendicular Line from the Construct menu. To construct the road, select point H, the lower point where the line intersects the circle, and DE and choose Parallel Line from the Construct menu. Hide the perpendicular line and point H. Select points F and C and create an action button that animates both point F backward along the segment at fast speed and point C counterclockwise around the circle at fast speed. Animate Point Animate Points E A B C D F H C' G Step 13 Press this new animation button. Circle F should move to the left, rotating at the same time so that it appears to be rolling. Drag point G so that it is on the circle and choose Trace Point from the Display menu. Investigate these questions. Step 14 What does the path of a point on a rolling circle look like? On your paper, sketch the curve point G makes as the circle rolls. This curve is called a cycloid. EXPLORATION Cycloids 347 Step 15 Step 16 Step 17 Step 18 Step 19 Experiment with different cycloids made when point G is inside the circle and outside the circle. Sketch these curves onto your paper. A cycl |
oid is an example of a periodic curve. What do you think that means? Adjust the radius AB or the length DE so that point G traces one period, or cycle, of the curve. Adjust these lengths so that point G traces two cycles or three cycles. How are the lengths DE and AB related to the number of cycles of the curve? If you apply a car’s brakes on a slippery road, you’ll skid, because your wheels won’t turn fast enough to keep up with the car’s movement down the road. If you try to accelerate on a slippery road, your wheels will spin—they’ll turn faster than the car is able to go. Experiment with different speeds for points C and F by selecting the animation button and choosing Properties from the Edit menu. What combinations cause the wheel to spin? To skid? Explain why the wheel has good traction when the animation speeds are the same. Add to your sketch so that it shows a traveling bicycle or car. To make a second wheel, you can simply translate the wheel you have by a fixed distance. When you construct the vehicle on these wheels, you’ll need to make sure all the parts move when the wheels do! The figure below shows an epicycloid, the path of a point on a circle that is rolling around another circle. See if you can make a construction that traces an epicycloid. (Hint: The dashed circle is important in the construction. The circle inside it is just drawn for show.) Animate Points 348 CHAPTER 6 Discovering and Proving Circle Properties VIEW ● CHAPTER 11 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHA CHAPTER 6 R E V I E W In this chapter you learned some new circle vocabulary and solved real-world application problems involving circles. You discovered the relationship between a radius and a tangent line. You discovered special relationships between angles and their intercepted arcs. And you learned about the special ratio and how to use it to calculate the circumference of a circle and the length of an arc. You should be able to sketch these terms from memory: chord, tangent, central angle, inscribed angle, and intercepted arc. And you should be able to explain the difference between arc measure and arc length. EXERCISES 1. What do you think is the most important or useful circle property you learned in this chapter? Why? 2. How can you find the center |
of a circle with a compass and a straightedge? With patty paper? With the right-angled corner of a carpenter’s square? You will need Construction tools for Exercises 21–24, 67, and 70 A calculator for Exercises 11, 12, and 27–33 3. What does the path of a satellite have to do with the Tangent Conjecture? 4. Explain the difference between the degree measure of an arc and its arc length. Solve Exercises 5–19. If the exercise uses the “” sign, answer in terms of. If the exercise uses the “” sign, give your answer accurate to one decimal place. 5. b? 35° b 8. e? 60° e 64° a 6. a? 110° 155° 9. d? 89° d 7. c? 104° c f 10. f? 88° 118° CHAPTER 6 REVIEW 349 EW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTE 11. Circumference? 12. Circumference is 132 cm. d? 13. r 27 cm. The arc length is?. B of AB 20 cm d A r 100° T 14. r 36 ft. The arc length is?. of CD 15. What’s wrong with this picture? D r C 60° 50° O L 35° 57° 16. What’s wrong with this picture? 84° 56° 158° 17. Explain why KE YL. 18. Explain why JIM is isosceles. 19. Explain why KIM is isosceles. 108° K L 36° E Y I 152° J 56° M I K 70° M 70° E 20. On her latest archaeological dig, Ertha Diggs has unearthed a portion of a cylindrical column. All she has with her is a pad of paper. How can she use it to locate the diameter of the column? 21. Construction Construct a scalene obtuse triangle. Construct the circumscribed circle. 22. Construction Construct a scalene acute triangle. Construct the inscribed circle. 23. Construction Construct a rectangle. Is it possible to construct the circumscribed circle, the inscribed circle, neither, or both? 350 CHAPTER 6 Discovering and Proving Circle Properties ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 |
REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 24. Construction Construct a rhombus. Is it possible to construct the circumscribed circle, the inscribed circle, neither, or both? 25. Find the equation of the line tangent to circle S centered at (1, 1) if the point of tangency is (5, 4). 26. Find the center of the circle passing through the points (7, 5), (0, 6), and (1, 1). 27. Rashid is an apprentice on a road crew for a civil engineer. He needs to find a trundle wheel similar to but larger than the one shown at right. If each rotation is to be 1 m, what should be the diameter of the trundle wheel? 28. Melanie rides the merry-go-round on her favorite horse on the outer edge, 8 meters from the center of the merry-go-round. Her sister, Melody, sits in the inner ring of horses, 3 meters in from Melanie. In 10 minutes, they go around 30 times. What is the average speed of each sister? 29. Read the Geography Connection below. Given that the polar radius of Earth is 6357 kilometers and that the equatorial radius of Earth is 6378 kilometers, use the original definition to calculate one nautical mile near a pole and one nautical mile near the equator. Show that the international nautical mile is between both values. Geography One nautical mile was originally defined to be the length of one minute of arc of a great circle of Earth. (A great circle is the intersection of the sphere and a plane that cuts through its center. There are 60 minutes of arc in each degree.) But Earth is not a perfect sphere. It is wider at the great circle of the equator than it is at the great circle through the poles. So defined as one minute of arc, one nautical mile could take on a range of values. To remedy this, an international nautical mile was defined as 1.852 kilometers (about 1.15 miles). 30. While talking to his friend Tara on the phone, Dmitri sees a lightning flash, and 5 seconds later he hears thunder. Two seconds after that, Tara, who lives 1 mile away, hears it. Sound travels at 1100 feet per second. Draw and label a diagram showing the possible locations of the lightning strike. 31. King Arthur wishes to seat all his knights at a round table. He instructs Merlin to design and create an oak |
table large enough to seat 100 people. Each knight is to have 2 ft along the edge of the table. Help Merlin calculate the diameter of the table. Short nautical mile Polar radius Equatorial radius Long nautical mile CHAPTER 6 REVIEW 351 EW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTE 32. If the circular moat should have been a circle of radius 10 meters instead of radius 6 meters, how much greater should the larger moat’s circumference have been? 33. The part of a circle enclosed by a central angle and the arc it intercepts is called a sector. The sector of a circle shown below can be curled into a cone by bringing the two straight 45-cm edges together. What will be the diameter of the base of the cone? 48° 45 cm MIXED REVIEW In Exercises 34–56, identify the statement as true or false. For each false statement, explain why it is false or sketch a counterexample. 34. If a triangle has two angles of equal measure, then the third angle is acute. 35. If two sides of a triangle measure 45 cm and 36 cm, then the third side must be greater than 9 cm and less than 81 cm. 36. The diagonals of a parallelogram are congruent. 37. The measure of each angle of a regular dodecagon is 150°. 38. The perpendicular bisector of a chord of a circle passes through the center of the circle. 39. If CD is the midsegment of trapezoid PLYR with PL one of the bases, then CD 1 (PL YR). 2 40. In BOY, BO 36 cm, mB 42°, and mO 28°. In GRL, GR 36 cm, mR 28°, and mL 110°. Therefore, BOY GRL. 41. If the sum of the measures of the interior angles of a polygon is less than 1000°, then the polygon has fewer than seven sides. 42. The sum of the measures of the three angles of an obtuse triangle is greater than the sum of the measures of the three angles of an acute triangle. 43. The sum of the measures of one set of exterior angles of a polygon is always less than the sum of the measures of interior angles. 352 CHAPTER 6 Discovering and Proving Circle Properties ● CHAPTER 6 REVIEW |
● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 44. Both pairs of base angles of an isosceles trapezoid are supplementary. 45. If the base angles of an isosceles triangle each measure 48°, then the vertex angle has a measure of 132°. 46. Inscribed angles that intercept the same arc are supplementary. 47. The measure of an inscribed angle in a circle is equal to the measure of the arc it intercepts. 48. The diagonals of a rhombus bisect the angles of the rhombus. 49. The diagonals of a rectangle are perpendicular bisectors of each other. 50. If a triangle has two angles of equal measure, then the triangle is equilateral. 51. If a quadrilateral has three congruent angles, then it is a rectangle. 52. In two different circles, arcs with the same measure are congruent. 53. The ratio of the diameter to the circumference of a circle is. 54. If the sum of the lengths of two consecutive sides of a kite is 48 cm, then the perimeter of the kite is 96 cm. 55. If the vertex angles of a kite measure 48° and 36°, then the nonvertex angles each measure 138°. 56. All but seven statements in Exercises 34–56 are false. The concentric circles in the sky are actually a time exposure photograph of the movement of the stars in a night. 57. Find the measure of each lettered angle in the diagram below. a 58 116° f g h k iq 1 2 3 u v n m 75° l p r CHAPTER 6 REVIEW 353 EW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTE In Exercises 58–60, from the information given, determine which triangles, if any, are congruent. State the congruence conjecture that supports your congruence statement. 58. STARY is a regular pentagon. 59. FLYT is a kite. 60. PART is an isosceles trapezoid. T R L P A 61. Adventurer Dakota Davis has uncovered a piece of triangular tile from a mosaic. A corner is broken off. Wishing to repair the mosaic, he lays the broken tile piece down on paper and traces the straight edges. With a ruler |
he then extends the unbroken sides until they meet. What triangle congruence shortcut guarantees that the tracing reveals the original shape? 62. Circle O has a radius of 24 inches. Find the measure and the length of AC. 63. EC and ED are tangent to the circle, and AB CD. Find the measure of each lettered angle. C E 54° A O 42° B C A w x D y B 64. Use your protractor to draw and label a pair of supplementary angles that is not a linear pair. 65. Find the function rule f(n) of this sequence and find the 20th term. n f(n 11 15...... n...... 20 354 CHAPTER 6 Discovering and Proving Circle Properties ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 66. The design at right shows three hares joined by three ears, although each hare appears to have two ears of its own. a. Does the design have rotational symmetry? b. Does the design have reflectional symmetry? 67. Construction Construct a rectangle whose length is twice its width. 68. If AB 15 cm, C is the midpoint of AB, D is the midpoint of AC, and E is the midpoint of DC, what is the length of EB? 69. Draw the next shape in this pattern. Private Collection, Berkeley, California Ceramist, Diana Hall 70. Construction Construct any triangle. Then construct its centroid. TAKE ANOTHER LOOK 1. Show how the Tangent Segments Conjecture follows logically from the Tangent Conjecture and the converse of the Angle Bisector Conjecture. 2. Investigate the quadrilateral formed by two tangent segments to a circle and the two radii to the points of tangency. State a conjecture. Explain why your conjecture is true, based on the properties of radii and tangents. 3. State the Cyclic Quadrilateral Conjecture in “if-then” form. Then state the converse of the conjecture in “if-then” form. Is the converse also true? 4. A quadrilateral that can be inscribed in a circle is also called a cyclic quadrilateral. Which of these quadrilaterals are always cyclic: parallelograms kites, isosceles trapezoids, rhomb |
uses, rectangles, or squares? Which ones are never cyclic? Explain why each is or is not always cyclic. N A E G 5. Use graph paper or a graphing calculator to graph the data collected from the investigation in Lesson 6.5. Graph the diameter on the x-axis and the circumference on the y-axis. What is the slope of the best-fit line through the data points? Does this confirm the Circumference Conjecture? Explain. CHAPTER 6 REVIEW 355 EW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTER 6 REVIEW ● CHAPTE Assessing What You’ve Learned With the different assessment methods you’ve used so far, you should be getting the idea that assessment means more than a teacher giving you a grade. All the methods presented so far could be described as self-assessment techniques. Many are also good study habits. Being aware of your own learning and progress is the best way to stay on top of what you’re doing and to achieve the best results. WRITE IN YOUR JOURNAL You may be at or near the end of your school year’s first semester. Look back over the first semester and write about your strengths and needs. What grade would you have given yourself for the semester? How would you justify that grade? Set new goals for the new semester or for the remainder of the year. Write them in your journal and compare them to goals you set at the beginning of the year. How have your goals changed? Why? ORGANIZE YOUR NOTEBOOK Review your notebook and conjectures list to be sure they are complete and well organized. Write a one-page chapter summary. UPDATE YOUR PORTFOLIO Choose a piece of work from this chapter to add to your portfolio. Document the work according to your teacher’s instructions. PERFORMANCE ASSESSMENT While a classmate, a friend, a family member, or a teacher observes, carry out one of the investigations or Take Another Look activities from this chapter. Explain what you’re doing at each step, including how you arrived at the conjecture. WRITE TEST ITEMS Divide the lessons from this chapter among group members and write at least one test item per lesson. Try out the test questions written by your classmates and discuss them. GIVE A PRESENTATION Give a presentation on an investigation, exploration, Take Another Look project, or puzzle. Work with your group |
, or try giving a presentation on your own. 356 CHAPTER 6 Discovering and Proving Circle Properties CHAPTER 7 Transformations and Tessellations I believe that producing pictures, as I do, is almost solely a question of wanting so very much to do it well. M. C. ESCHER Magic Mirror, M. C. Escher, 1946 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● discover some basic properties of transformations and symmetry ● learn more about symmetry in art and nature ● create tessellations L E S S O N 7.1 Symmetry is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. HERMANN WEYL Transformations and Symmetry By moving all the points of a geometric figure according to certain rules, you can create an image of the original figure. This process is called transformation. Each point on the original figure corresponds to a point on its image. The image of point A after a transformation of any type is called point A (read “A prime”), as shown in the transformation of ABC to ABC on the facing page. If the image is congruent to the original figure, the process is called rigid transformation, or isometry. A transformation that does not preserve the size and shape is called nonrigid transformation. For example, if an image is reduced or enlarged, or if the shape changes, its transformation is nonrigid. Frieze of bowmen from the Palace of Artaxerxes II in Susa, Iran Three types of rigid transformation are translation, rotation, and reflection. You have been doing translations, rotations, and reflections in your patty-paper investigations and in exercises on the coordinate plane, using (x, y) rules. Each light bulb is an image of every other light bulb. Image Original Image Original Original Image Translation Rotation Reflection Translation is the simplest type of isometry. You can model a translation by tracing a figure onto patty paper, then sliding it along a straight path without turning it. Notice that when you slide the figure, all points move the same distance along parallel paths to form its image. That is, each point in the image is equidistant from the point that corresponds to it in the original figure. This distance, because it is the same for all points, is called the distance of the translation. A translation also has a particular direction. So you |
Conjecture The line of reflection is the? of every segment joining a point in the original figure with its image. C-68 If a figure can be reflected over a line in such a way that the resulting image coincides with the original, then the figure has reflectional symmetry. The reflection line is called the line of symmetry. The Navajo rug shown below has two lines of symmetry. The letter T has reflectional symmetry. You can test a figure for reflectional symmetry by using a mirror or by folding it. Navajo rug (two lines of symmetry) The letter Z has 2-fold rotational symmetry. When it is rotated 180° and 360° about a center of rotation, the image coincides with the original figure. 180° If a figure can be rotated about a point in such a way that its rotated image coincides with the original figure before turning a full 360°, then the figure has rotational symmetry. Of course, every image is identical to the original figure after a rotation of any multiple of 360°. However, we don’t call a figure symmetric if this is the only kind of symmetry it has. You can trace a figure to test it for rotational symmetry. Place the copy exactly over the original, put your pen or pencil point on the center to hold it down, and rotate the copy. Count the number of times the copy and the original coincide until the copy is back in its original position. Two-fold rotational symmetry is also called point symmetry. LESSON 7.1 Transformations and Symmetry 361 Some polygons have no symmetry, or only one kind of symmetry. Regular polygons, however, are symmetric in many ways. A square, for example, has 4-fold reflectional symmetry and 4-fold rotational symmetry. This tile pattern has both 5-fold rotational symmetry and 5-fold reflectional symmetry. 90° 180° 270° 360° Art Reflecting and rotating a letter can produce an unexpected and beautiful design. With the aid of graphics software, a designer can do this quickly and inexpensively. To see how, go to www.keymath.com/DG by geometry student Michelle Cotter.. This design was created EXERCISES In Exercises 1–3, say whether the transformations are rigid or nonrigid. Explain how you know. You will need Construction tools for Exercises 9 and 10 1. B A 3. C 2. A B C In Exercises 4–6, copy the |
figure onto graph or square dot paper and perform each transformation. 4. Reflect the figure over the line of reflection, line. 5. Rotate the figure 180° about 6. Translate the figure by the the center of rotation, point P. translation vector. P 362 CHAPTER 7 Transformations and Tessellations 7. An ice skater gliding in one direction creates several translation transformations. Give another real-world example of translation. 8. An ice skater twirling about a point creates several rotation transformations. Give another real-world example of rotation. In Exercises 9–11, perform each transformation. Attach your patty paper to your homework. 9. Construction Use the semicircular figure and its reflected image. a. Copy the figure and its reflected image onto a piece of patty paper. Locate the line of reflection. Explain your method. b. Copy the figure and its reflected image onto a sheet of paper. Locate the line of reflection using a compass and straightedge. Explain your method. 10. Construction Use the rectangular figure and the reflection line next to it. a. Copy the figure and the line of reflection onto a sheet of paper. Use a compass and straightedge to construct the reflected image. Explain your method. b. Copy the figure and the line of reflection onto a piece of patty paper. Fold the paper and trace the figure to construct the reflected image. 11. Trace the circular figure and the center of rotation, P. Rotate the design 90° clockwise about point P. Draw the image of the figure, as well as the dotted line. In Exercises 12–14, identify the type (or types) of symmetry in each design. 12. 13. 14. P Butterfly Hmong textile, Laos The Temple Beth Israel, San Diego’s first synagogue, built in 1889 LESSON 7.1 Transformations and Symmetry 363 15. All of the woven baskets from Botswana shown below have rotational symmetry and most have reflectional symmetry. Find one that has 7-fold symmetry. Find one with 9-fold symmetry. Which basket has rotational symmetry but not reflectional symmetry? What type of rotational symmetry does it have? A B C D E I Cultural F G H J K L For centuries, women in Botswana, a country in southern Africa, have been weaving baskets like the ones you see above, to carry and store food. Each generation passes on the tradition of weaving choice shoots from the m |
words will all go the right way again.” ALICE IN THROUGH THE LOOKING-GLASS BY LEWIS CARROLL Properties of Isometries In many earlier exercises, you used ordered pair rules to transform polygons on a coordinate plane by relocating their vertices. For any point on a figure, the ordered pair rule (x, y) → (x h, y k) results in a horizontal move of h units and a vertical move of k units for any numbers h and k. That is, if (x, y) is a point on the original figure, (x h, y k) is its corresponding point on the image. Let’s look at an example. EXAMPLE A Transform the polygon at right using the rule (x, y) → (x 2, y 3). Describe the type and direction of the transformation. Solution Apply the rule to each ordered pair. Every point of the polygon moves right 2 units and down 3 units. This is a translation of (2, 3). –10 –10 y 5 –5 x 3 x 2 means to move right 2 units. (–5, 5) y 5 Original y 3 means to move down 3 units. (–3, 2) x 3 Image –5 So the ordered pair rule (x, y) → (x h, y k) results in a translation of (h, k). 366 CHAPTER 7 Transformations and Tessellations Investigation 1 Transformations on a Coordinate Plane In this investigation you will discover (or rediscover) how four ordered pair rules transform a polygon. Each person in your group can choose a different polygon for this investigation. You will need ● graph paper ● patty paper Step 1 Step 2 Step 3 On graph paper, create and label four sets of coordinate axes. Draw the same polygon in the same position in a quadrant of each of the four graphs. Write one of these four ordered pair rules below each graph. a. (x, y) → (x, y) b. (x, y) → (x, y) c. (x, y) → (x, y) d. (x, y) → (y, x) Use the ordered pair rule you assigned to each graph to relocate the vertices of your polygon and create its image. 7 6 x y Use patty paper to see if your transformation is a reflection, translation, or rotation. Compare your results with those |
of your group members. Complete the conjecture. –6 Coordinate Transformations Conjecture The ordered pair rule (x, y) → (x, y) is a? over?. The ordered pair rule (x, y) → (x, y) is a? over?. The ordered pair rule (x, y) → (x, y) is a? about?. The ordered pair rule (x, y) → (y, x) is a? over?. C-69 Let’s revisit “poolroom geometry.” When a ball rolls without spin into a cushion, the outgoing angle is congruent to the incoming angle. This is true because the outgoing and incoming angles are reflections of each other. You will need ● patty paper ● a protractor Investigation 2 Finding a Minimal Path In Chapter 1, you used a protractor to find the path of the ball. In this investigation, you’ll discover some other properties of reflections that have many applications in science and engineering. They may even help your pool game! LESSON 7.2 Properties of Isometries 367 Step 1 Step 2 Step 3 Draw a segment, representing a pool table cushion, near the center of a piece of patty paper. Draw two points, A and B, on one side of the segment. Imagine you want to hit a ball at point A so that it bounces off the cushion and hits another ball at point B. Use your protractor to find the point C on the cushion that you should aim for. Draw AC and CB to represent the ball’s path. Step 4 Fold your patty paper to draw the reflection of point B. Label the image point B Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Unfold the paper and draw a segment from point A to point B. What do you notice? Does point C lie on segment AB? How does the path from A to B compare to the twopart path from A to C to B? Can you draw any other path from point A to the cushion to point B that is shorter than AC CB? Why or why not? The shortest path from point A to the cushion to point B is called the minimal path. Copy and complete the conjecture. B A B C Minimal Path Conjecture If points A and B are on one side of line, then the minimal path from point A to line to point B is found by?. C-70 How can this discovery |
help your pool game? Suppose you need to hit a ball at point A into the cushion so that it will bounce off the cushion and pass through point B. To what point on the cushion should you aim? Visualize point B reflected across the cushion. Then aim directly at the reflected image. A B B' 368 CHAPTER 7 Transformations and Tessellations Let’s look at a miniature-golf example. EXAMPLE B How can you hit the ball at T around the corner and into the hole at H in one shot? H T Solution First, try to get a hole-in-one with a direct shot or with just one bounce off a wall. For one bounce, decide which wall the ball should hit. Visualize the image of the hole across that wall and aim for the reflected hole. There are two possibilities. H H H Oops, it hits the corner! Case 1 T It still hits the corner. T Case 2 H In both cases the path is blocked. It looks like you need to try two bounces. Visualize a path from the tee hitting two walls and into the hole. H T Visualize the path. Now you can work backward. Which wall will the ball hit last? Reflect the hole across that wall creating image H. Reflect H over this wall. H H T LESSON 7.2 Properties of Isometries 369 H Reflect Hover this wall. H H T Which wall will the ball hit before it approaches the second wall? Reflect the image of the hole H across that wall creating image H (read “H double prime”). Draw the path from the tee to H, H, and H. Can you visualize other possible paths with two bounces? Three bounces? What do you suppose is the minimal path from T to H? H H H Draw the path to H, H, and H. T EXERCISES In Exercises 1–5, copy the figure and draw the image according to the rule. Identify the type of transformation. 1. (x, y) → (x 5, y) y 2. (x, y) → (x, y) y You will need Construction tools for Exercises 19 and 20 3. (x, y) → (y, x) 5 5 5 x –5 –4 x y 5 –5 x 7 4. (x, y) → (8 x, y) y 5 5. (x, y) → (x, y) 6. |
Look at the rules in y 5 x 5 Exercises 1–5 that produced reflections. What do these rules have in common? How about the ones that produce translations? Rotations? 5 x 370 CHAPTER 7 Transformations and Tessellations In Exercises 7 and 8, complete the ordered pair rule that transforms the black triangle to its image, the red triangle. 7. (x, y) → (?,? ) 8. (x, y) → (?,? ) V –10 V y 10 Y R R Y –10 x 10 –10 H In Exercises 9–11, copy the position of each ball and hole onto patty paper and draw the path of the ball. 9. What point on the W cushion can a player aim for so that the cue ball bounces and strikes the 8-ball? What point can a player aim for on the S cushion? W y 10 R D D H x 10 R –10 N S Cue ball E 10. Starting from the tee (point T), what point on a wall should a player aim for so that the golf ball bounces off the wall and goes into the hole at H? N 11. Starting from the tee (point T), plan a shot so that the golf ball goes into the hole at H. Show all your work Proposed freeway 12. A new freeway is being built near the two towns of Perry and Mason. The two towns want to build roads to one junction point on the freeway. (One of the roads will be named Della Street.) Locate the junction point and draw the minimal path from Perry to the freeway to Mason. How do you know this is the shortest path? Mason Perry LESSON 7.2 Properties of Isometries 371 Review In Exercises 13 and 14, sketch the next two figures.,,,,,,?? 13. 14.???????? 15. The word DECODE remains unchanged when it is reflected over its horizontal line of symmetry. Find another such word with at least five letters. 16. How many reflectional symmetries does an isosceles triangle have? 17. How many reflectional symmetries does a rhombus have? 18. Write what is actually on the T-shirt shown at right. 19. Construction Construct a kite circumscribed about a circle. 20. Construction Construct a rhombus circumscribed about a circle. In Exercises 21 and 22, identify each statement |
as true or false. If true, explain why. If false, give a counterexample. 21. If two angles of a quadrilateral are right angles, then it is a rectangle. 22. If the diagonals of a quadrilateral are congruent, then it By Holland. ©1976, Punch Cartoon Library. is a rectangle. IMPROVING YOUR REASONING SKILLS Chew on This for a While If the third letter before the second consonant after the third vowel in the alphabet is in the twenty-sixth word of this puzzle, then print the fortieth word of this puzzle and then print the twenty-second letter of the alphabet after this word. Otherwise, list three uses for chewing gum. 372 CHAPTER 7 Transformations and Tessellations L E S S O N 7.3 There are things which nobody would see unless I photographed them. DIANE ARBUS Compositions of Transformations In Lesson 7.2, you reflected a point, then reflected it again to find the path of a ball. When you apply one transformation to a figure and then apply another transformation to its image, the resulting transformation is called a composition of transformations. Let’s look at an example of a composition of two translations. EXAMPLE Triangle ABC with vertices A(1, 0), B(4, 0), and C(2, 6) is first translated by the rule (x, y) → (x 6, y 5), and then its image, ABC, is translated by the rule (x, y) → (x 14, y 3). a. What single translation is equivalent to the composition of these two translations? b. What single translation brings the second image, ABC, back to the position of the original triangle, ABC? Solution Draw ABC on a set of axes and relocate its vertices using the first rule to get ABC. Then relocate the vertices of ABC using the second rule to get ABC. The translation (x, y) → (x 6, y 5) moves C (2, 6) to C (4, 1). y 7 C C The single translation (x, y) → (x 8, y 2) moves C (2, 6) directly to C (10, 4). C –8 A B 8 A The translation (x, y) → (x 14, y 3) moves C (4, 1) to C(10, 4). A B –7 x B a. |
Each vertex is moved left 6 then right 14, and down 5 then up 3. So the equivalent single translation would be (x, y) → (x 6 14, y 5 3) or (x, y) → (x 8, y 2). You can also write this as (8, 2). LESSON 7.3 Compositions of Transformations 373 b. Reversing the steps, the translation (8, 2) brings the second image, ABC, back to ABC. In the investigations you will see what happens when you compose reflections. Investigation 1 Reflections over Two Parallel Lines First consider the case of parallel lines of reflection. You will need ● patty paper Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 1 Step 2 Step 3 On a piece of patty paper, draw a figure and a line of reflection that does not intersect it. Fold to reflect your figure over the line of reflection and trace the image. On your patty paper, draw a second reflection line parallel to the first so that the image is between the two parallel reflection lines. Fold to reflect the image over the second line of reflection. Turn the patty paper over and trace the second image. How does the second image compare to the original figure? Name the single transformation that transforms the original to the second image. Use a compass or patty paper to measure the distance between a point in the original figure and its second image point. Compare this distance with the distance between the parallel lines. How do they compare? Step 7 Compare your findings with others in your group and state your conjecture. Reflections over Parallel Lines Conjecture C-71 A composition of two reflections over two parallel lines is equivalent to a single?. In addition, the distance from any point to its second image under the two reflections is? the distance between the parallel lines. Is a composition of reflections always equivalent to a single reflection? If you reverse the reflections in a different order, do you still get the original figure back? Can you express a rotation as a set of reflections? 374 CHAPTER 7 Transformations and Tessellations Investigation 2 Reflections over Two Intersecting Lines Next, you will explore the case of intersecting lines of reflection. You will need ● patty paper ● a protractor Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 On a piece of patty paper, draw a figure and a reflection line that does not intersect it. Fold to reflect your figure over the line and trace the image. On |
your patty paper, draw a second reflection line intersecting the first so that the image is in an acute angle between the two intersecting reflection lines. Step 4 Fold to reflect the first image over the second line and trace the second image.?? Step 5 Step 6 Step 7 Step 5 Step 6 Step 7 Draw two rays that start at the point of intersection of the two intersecting lines and that pass through corresponding points on the original figure and its second image. How does the second image compare to the original figure? Name the single transformation from the original to the second image. With a protractor or patty paper, compare the angle created in Step 5 with the acute angle formed by the intersecting reflection lines. How do the angles compare? Step 8 Compare findings in your group and state your next conjecture. Reflections over Intersecting Lines Conjecture C-72 A composition of two reflections over a pair of intersecting lines is equivalent to a single?. The angle of? is? the acute angle between the pair of intersecting reflection lines. LESSON 7.3 Compositions of Transformations 375 There are many other ways to combine transformations. Combining a translation with a reflection gives a special two-step transformation called a glide reflection. A sequence of footsteps is a common example of a glide reflection. You will explore a few other examples of glide reflection in the exercises and later in this chapter. Glide-reflectional symmetry EXERCISES 1. Name the single translation that can replace the composition of these three translations: (2, 3), then (–5, 7), then (13, 0). 2. Name the single rotation that can replace the composition of these three rotations about the same center of rotation: 45°, then 50°, then 85°. What if the centers of rotation differ? Draw a figure and try it. 3. Lines m and n are parallel and 10 cm apart. a. Point A is 6 cm from line m and 16 cm from line n. Point A is reflected over line m, then its image, A, is reflected over line n to create a second image, point A. How far is point A from point A? b. What if A is reflected over n, and then its image is reflected over m? Find the new image and distance from A. 4. Two lines m and n intersect at point P, forming a 40° angle. a. You reflect point B over line m, then reflect the image of B over line |
n. What angle of rotation about point P rotates the second image of point B back to its original position? b. What if you reflect B first over n, and then reflect the image of B over m? Find the angle of rotation that rotates the second image back to the original position. 5. Copy the figure and PAL onto patty paper. Reflect the figure over AP. Reflect the image over AL. What is the equivalent rotation? A 6 cm 10 cm m n P 40° n P m B A L 376 CHAPTER 7 Transformations and Tessellations 6. Copy the figure and the pair of parallel lines onto patty paper. Reflect the figure over PA. Reflect the image over RL. What is the length of the equivalent translation vector? 7. Copy the hexagonal figure and its translated image onto patty paper. Find a pair of parallel reflection lines that transform the original onto the image. R P L A 8. Copy the original figure and its translated image onto patty paper. Find a pair of intersecting reflection lines that transform the original onto the image. N O O A H N A H Center of rotation 9. Copy the two figures below onto graph paper. Each figure is the glide-reflected image of the other. Continue the pattern with two more glide-reflected figures. Review In Exercises 10 and 11, sketch the next two figures. 10. 11.,,,,,,?????????? LESSON 7.3 Compositions of Transformations 377 12. If you draw a figure on an uninflated balloon and then blow up the balloon the figure will undergo a nonrigid transformation. Give another example of a nonrigid transformation. 13. List two objects in your home that have rotational symmetry but not reflectional symmetry. List two objects in your classroom that have reflectional symmetry but not rotational symmetry. 14. Have you noticed that some letters have both horizontal and vertical symmetries? Have you also noticed that all the letters that have both horizontal and vertical symmetries also have point symmetry? Is this a coincidence? Use what you have learned about transformations to explain why. 15. Is it possible for a triangle to have exactly one line of symmetry? Exactly two? Exactly three? Support your answers with sketches. 16. Draw two points onto a piece of paper and connect them with a curve that is point symmetric. KALEIDOSCOPES You have probably looked through kaleidoscopes |
and enjoyed their beautiful designs, but do you know how they work? For a simple kaleidoscope, hinge two mirrors with tape, place a small object or photo between the mirrors and adjust them until you see four objects (the original and three images). What is the angle between the mirrors? At what angle should you hold the mirrors to see six objects? Eight objects? The British physicist Sir David Brewster invented the tube kaleidoscope in 1816. Some tube kaleidoscopes have colored glass or plastic pieces that tumble around in their end chambers. Some have colored liquid. Others have only a lens in the chamber—the design you see depends on where you aim it. Design and build your own kaleidoscope using a plastic or cardboard cylinder and glass or plastic as reflecting surfaces. Try various items in the end chamber. Your project should include Your kaleidoscope (pass it around!). A report with diagrams that show the geometry properties you used, a list of the materials and tools you used, and a description of problems you had and how you solved them. 378 CHAPTER 7 Transformations and Tessellations Glass Colored glass Cardboard spacers or rubber washers Glass Tube with three reflecting surfaces Glass Cardboard eye piece L E S S O N 7.4 I see a certain order in the universe and math is one way of making it visible. MAY SARTON Tessellations with Regular Polygons Honeycombs are remarkably geometric structures. The hexagonal cells that bees make are ideal because they fit together perfectly without any gaps. The regular hexagon is one of many shapes that can completely cover a plane without gaps or overlaps. Mathematicians call such an arrangement of shapes a tessellation or a tiling. A tessellation that uses only one shape is called a monohedral tiling. You can find tessellations in every home. Decorative floor tiles have tessellating patterns of squares. Brick walls, fireplaces, and wooden decks often display creative tessellations of rectangles. Where do you see tessellations every day? The hexagon pattern in the honeycomb of the bee is a tessellation of regular hexagons. Regular hexagons and equilateral triangles combine in this tiling from the 17th-century Topkapi Palace in Istanbul, Turkey. You already know that squares and regular hexagons create monohedral tessellations. Because each regular hexagon can be divided into six equilateral triangles, we can logically |
conclude that equilateral triangles also create monohedral tessellations. Will other regular polygons tessellate? Let’s look at this question logically. For shapes to fill the plane without gaps or overlaps, their angles, when arranged around a point, must have measures that add up to exactly 360°. If the sum is less than 360°, there will be a gap. If the sum is greater, the shapes will overlap. Six 60° angles from six equilateral triangles add up to 360°, as do four 90° angles from four squares or three 120° angles from three regular hexagons. What about regular pentagons? Each angle in a regular pentagon measures 108°, and 360 is not divisible by 108. So regular pentagons cannot be arranged around a point without overlapping or leaving a gap. What about regular heptagons? Triangles Squares Pentagons Hexagons Heptagons? LESSON 7.4 Tessellations with Regular Polygons 379 In any regular polygon with more than six sides, each angle has a measure greater than 120°, so no more than two angles can fit about a point without overlapping. So the only regular polygons that create monohedral tessellations are equilateral triangles, squares, and regular hexagons. A monohedral tessellation of congruent regular polygons is called a regular tessellation. Tessellations can have more than one type of shape. You may have seen the octagon-square combination at right. In this tessellation, two regular octagons and a square meet at each vertex. Notice that you can put your pencil on any vertex and that the point is surrounded by one square and two octagons. So you can call this a 4.8.8 or a 4.82 tiling. The numbers give the vertex arrangement, or numerical name for the tiling. When the same combination of regular polygons (of two or more kinds) meet in the same order at each vertex of a tessellation, it is called a semiregular tessellation. Below are two more examples of semiregular tessellations. The same polygons appear in the same order at each vertex: square, hexagon, dodecagon. The same polygons appear in the same order at each vertex: triangle, dodecagon, dodecagon. There are eight different semiregular tessellations. Three of them are shown |
above. In this investigation, you will look for the other five. To make this easier, the remaining five use only combinations of triangles, squares, or hexagons. Investigation The Semiregular Tessellations Find or create a set of regular triangles, squares, and hexagons for this investigation. Then work with your group to find the remaining five of the eight semiregular tessellations. Remember, the same combination of regular polygons must meet in the same order at each vertex for the tiling to be semiregular. Also remember to check that the sum of the measures at each vertex is 360°. You will need ● triangles, squares and hexagons from a set of pattern blocks or, ● geometry software such as GSP or, ● a set of triangles, squares and hexagons created from the set shown 380 CHAPTER 7 Transformations and Tessellations Step 1 Step 2 Step 3 Investigate which combinations of two kinds of regular polygons you can use to create a semiregular tessellation. Next, investigate which combinations of three kinds of regular polygons you can use to create a semiregular tessellation. Summarize your findings by sketching all eight semiregular tessellations and writing their vertex arrangements (numerical names). The three regular tessellations and the eight semiregular tessellations you just found are called the Archimedean tilings. They are also called 1-uniform tilings because all the vertices in a tiling are identical. Mathematics Greek mathematician and inventor Archimedes (ca. 287–212 B.C.E.) studied the relationship between mathematics and art with tilings. He described 11 plane tilings made up of regular polygons, with each vertex being the same type. Plutarch (ca. 46–127 C.E.) wrote of Archimedes’ love of geometry, “... he neglected to eat and drink and took no care of his person; that he was often carried by force to the baths, and when there he would trace geometrical figures in the ashes of the fire.” Often, different vertices in a tiling do not have the same vertex arrangement. If there are two different types of vertices, the tiling is called 2-uniform. If there are three different types of vertices, the tiling is called 3-uniform. Two examples are shown below. 3 |
.4.3.12 3.12.12 3.3.3.4.4 or 33.42 3.3.4.3.4 or 32.4.3.4 4.4.4.4 or 44 A 2-uniform tessellation: 3.4.3.12/ 3.122 A 3-uniform tessellation: 33.42/32.4.3.4/ 44 There are 20 different 2-uniform tessellations of regular polygons, and 61 different 3-uniform tilings. The number of 4-uniform tessellations of regular polygons is still an unsolved problem. LESSON 7.4 Tessellations with Regular Polygons 381 EXERCISES 1. Sketch two objects or designs you see every day that are monohedral tessellations. You will need Geometry software for Exercises 11–14 2. List two objects or designs outside your classroom that are semiregular tessellations. In Exercises 3–5, write the vertex arrangement for each semiregular tessellation in numbers. 3. 4. 5. In Exercises 6–8, write the vertex arrangement for each 2-uniform tessellation in numbers. 6. 7. 8. 9. When you connect the center of each triangle across the common sides of the tessellating equilateral triangles at right, you get another tessellation. This new tessellation is called the dual of the original tessellation. Notice the dual of the equilateral triangle tessellation is the regular hexagon tessellation. Every regular tessellation of regular polygons has a dual. a. Draw a regular square tessellation and make its dual. What is the dual? b. Draw a hexagon tessellation and make the dual of it. What is the dual? c. What do you notice about the duals? 10. You can make dual tessellations of semiregular tessellations, but they may not be tessellations of regular polygons. Try it. Sketch the dual of the 4.8.8 tessellation, shown at right. Describe the dual. Technology In Exercises 11–14, use geometry software, templates of regular polygons, or pattern blocks. 11. Sketch and color the 3. |
6.3.6 tessellation. Continue it to fill an entire sheet of paper. 12. Sketch the 4.6.12 tessellation. Color it so it has reflectional symmetry but not rotational symmetry. 382 CHAPTER 7 Transformations and Tessellations 13. Show that two regular pentagons and a regular decagon fit about a point but that 5.5.10 does not create a semiregular tessellation. 14. Create the tessellation 3.12.12/3.4.3.12. Draw your design onto a full sheet of paper. Color your design to highlight its symmetries. Review 15. Design a logo with rotational symmetry for Happy Time Ice Cream Company. Or design a logo for your group or for a made-up company. y 4 16. Reflect y 1 x 4 over the x-axis and find the equation of the 2 image line. –3 5 x 17. Words like MOM, WOW, TOOT, and OTTO all have a vertical line of symmetry when you write them in capital letters. Find another word that has a vertical line of symmetry. –6 The design at left comes from Inversions, a book by Scott Kim. Not only does the design spell the word mirror, it does so with mirror symmetry! 18. Frisco Fats needs to sink the 8-ball into the NW corner pocket, but he seems trapped. Can he hit the cue ball to a point on the N cushion so that it bounces out, strikes the S cushion, and taps the 8-ball into the corner pocket? Copy the table and construct the path of the ball. W N S Cue ball E IMPROVING YOUR REASONING SKILLS Scrambled Arithmetic In the equation 65 28 43, all the digits are correct but they are in the wrong places! Written correctly, the equation is 23 45 68. In each of the three equations below, the operations and the digits are correct, but some of the digits are in the wrong places. Find the correct equations. 1. 11 66 457 2. 39 11 75 7 8 31 3. 3 2 5 LESSON 7.4 Tessellations with Regular Polygons 383 L E S S O N 7.5 The most uniquely personal of all that he knows is that which he has discovered for himself. JEROME BRUNER Tessellations with Nonregular Polygons In Lesson 7.4, you t |
essellated with regular polygons. You drew both regular and semi-regular tessellations with them. What about tessellations of nonregular polygons? For example, will a scalene triangle tessellate? Let’s investigate.? Step 1 Step 2 Step 3 Investigation 1 Do All Triangles Tessellate? Make 12 congruent scalene triangles and use them to try to create a tessellation Look at the angles about each vertex point. What do you notice? What is the sum of the measures of the three angles of a triangle? What is the sum of the measures of the angles that fit around each point? Compare your results with the results of others and state your next conjecture. Making Congruent Triangles 1. Stack three pieces of paper and fold them in half. 2. Draw a scalene triangle on the top half-sheet and cut it out, cutting through all six layers to get six congruent scalene triangles. 3. Use one triangle as a template and repeat. You now have 12 congruent triangles. 4. Label the corresponding angles of each triangle a, b, and c, as shown. Tessellating Triangles Conjecture? triangle will create a monohedral tessellation. C-73 You have seen that squares and rectangles tile the plane. Can you visualize tiling with parallelograms? Will any quadrilateral tessellate? Let’s investigate. 384 CHAPTER 7 Transformations and Tessellations Investigation 2 Do All Quadrilaterals Tessellate? You want to find out if any quadrilateral can tessellate, so you should not choose a special quadrilateral for this investigation. Step 1 Step 2 Step 3 Cut out 12 congruent quadrilaterals. Label the corresponding angles in each quadrilateral a, b, c, and d. Using your 12 congruent quadrilaterals, try to create a tessellation. Notice the angles about each vertex point. How many times does each angle of your quadrilateral fit at each point? What is the sum of the measures of the angles of a quadrilateral? Compare your results with others. State a conjecture. Tessellating Quadrilaterals Conjecture? quadrilateral will create a monohedral tessellation. C-74 A regular pentagon does not tessellate, but are there any pentagons that tessellate? How many |
? Mathematics In 1975, when Martin Gardner wrote about pentagonal tessellations in Scientific American, experts thought that only eight kinds of pentagons would tessellate. Soon another type was found by Richard James III. After reading about this new discovery, Marjorie Rice began her own investigations. With no formal training in mathematics beyond high school, Marjorie Rice investigated the tessellating problem and discovered four more types of pentagons that tessellate. Mathematics professor Doris Schattschneider of Moravian College verified Rice’s research and brought it to the attention of the mathematics community. Rice had indeed discovered what professional mathematicians had been unable to uncover! In 1985, Rolf Stein, a German graduate student, discovered a fourteenth type of tessellating pentagon. Are all the types of convex pentagons that tessellate now known? The problem remains unsolved. Shown at right are Marjorie Rice (left) and Dr. Doris Schattschneider Type 13, discovered in December 1977 B E 90°, 2A D 360° 2C D 360° a e, a e d One of the pentagonal tessellations discovered by Marjorie Rice. Capital letters represent angle measures in the shaded pentagon. Lowercase letters represent lengths of sides. You will experiment with some pentagon tessellations in the exercises. LESSON 7.5 Tessellations with Nonregular Polygons 385 EXERCISES 1. Construction The beautiful Cairo street tiling shown below uses equilateral You will need Construction tools for Exercise 1 pentagons. One pentagon is shown below left. Use a ruler and a protractor to draw the equilateral pentagon on poster board or heavy cardboard. (For an added challenge, you can try to construct the pentagon, as Egyptian artisans likely would have done.) Cut out the pentagon and tessellate with it. Color your design Rice’s first discovery, February 1976 2E B 2D C 360° a b c d 45° 45° M Point M is the midpoint of the base. 2. At right is Marjorie Rice’s first pentagonal tiling discovery. Another way to produce a pentagonal tessellation is to make the dual of the tessellation shown in Lesson 7.4, Exercise 5. Try it. 3. A tessellation of regular hexagons can be used to create a |
pentagonal tessellation by dividing each hexagon as shown. Create this tessellation and color it. Cultural Mats called tatami are used as a floor covering in traditional Japanese homes. Tatami is made from rush, a flowering plant with soft fibers, and has health benefits, such as removing carbon dioxide and regulating humidity and temperature. When arranging tatami, you want the seams to form T-shapes. You avoid arranging four at one vertex forming a cross because it is difficult to get a good fit within a room this way. You also want to avoid fault lines—straight seams passing all the way through a rectangular arrangement—because they make it easier for the tatami to slip. Room sizes are often given in tatami numbers (for example, a 6-mat room or an 8-mat room). 4.5-mat room 6-mat room 8-mat room 386 CHAPTER 7 Transformations and Tessellations 4. Can a concave quadrilateral like the one at right tile the plane? Try it. Create your own concave quadrilateral and try to create a tessellation with it. Decorate your drawing. 5. Write a paragraph proof explaining why you can use any triangle to create a monohedral tiling. Review Refer to the Cultural Connection on page 386 for Exercises 6 and 7. 6. Use graph paper to design an arrangement of tatami for a 10-mat room. In how many different ways can you arrange the mats so that there are no places where four mats meet at a point (no cross patterns)? Assume that the mats measure 3-by-6 feet and that each room must be at least 9 feet wide. Show all your solutions. 7. There are at least two ways to arrange a 15-mat rectangle with no fault lines. One is shown. Can you find the other? Fault line 8. Reflect y 2x 3 across the y-axis and find the equation of the image line. y 8 –2 x 5 IMPROVING YOUR VISUAL THINKING SKILLS Picture Patterns II Draw what comes next in each picture pattern. 1. 2. LESSON 7.5 Tessellations with Nonregular Polygons 387 PENROSE TILINGS When British scientist Sir Roger Penrose of the University of Oxford is not at work on quantum mechanics or relativity theory, he’s inventing mathematical games. Penrose came up with a special tiling that uses two shapes |
, a kite and a dart. (The dart is a concave kite.) The tiles must be placed so that each vertex with a dot always touches only other vertices with dots. By adding this extra requirement, Penrose’s tiles make a nonperiodic tiling. That is, as you tessellate, the pattern does not repeat by translations. Penrose tilings decorate the Storey Hall building in Melbourne, Australia. Try it. Copy the two tiles shown below—the kite and the dart with their dots—onto patty paper. Use the patty-paper tracing to make two cardboard tiles. Create your own unique Penrose tiling and color it. Or, use geometry software to create and color your design. 388 CHAPTER 7 Transformations and Tessellations Penrose tiling at the Center for Mathematics and Computing, Carleton College, Northfield, Minnesota L E S S O N 7.6 There are three kinds of people in this world: those who make things happen, those who watch things happen, and those who wonder what happened. ANONYMOUS Tessellations Using Only Translations In 1936, M. C. Escher traveled to Spain and became fascinated with the tile patterns of the Alhambra. He spent days sketching the tessellations that Islamic masters had used to decorate the walls and ceilings. Some of his sketches are shown at right. Escher wrote that the tessellations were “the richest source of inspiration” he had ever tapped. Brickwork, Alhambra, M. C. Escher ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. Escher spent many years learning how to use translations, rotations, and glide reflections on grids of equilateral triangles and parallelograms. But he did not limit himself to pure geometric tessellations. The four steps below show how Escher may have created his Pegasus tessellation, shown at left. Notice how a partial outline of the Pegasus is translated from one side of a square to another to complete a single tile that fits with other tiles like itself. You can use steps like this to create your own unique tessellation. Start with a tessellation of squares, rectangles, or parallelograms, and try translating curves on opposite sides of the tile. It may take a few tries to get a shape that looks like a person, animal, or plant. Use your |
imagination! Symmetry Drawing E105, M. C. Escher, 1960 ©2002 Cordon Art B. V.–Baarn– Holland. All rights reserved. Step 1 Step 2 Step 3 Step 4 LESSON 7.6 Tessellations Using Only Translations 389 You can also use the translation technique with regular hexagons. The only difference is that there are three sets of opposite sides on a hexagon. So you’ll need to draw three sets of curves and translate them to opposite sides. The six steps below show how student Mark Purcell created his tessellation, Monster Mix. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 The Escher designs and the student tessellations in this lesson took a great deal of time and practice. When you create your own tessellating designs of recognizable shapes, you’ll appreciate the need for this practice! For more about tessellations, and resources to help you learn how to make them, go to www.keymath.com/DG. Monster Mix, Mark Purcell EXERCISES In Exercises 1–3, copy each tessellating shape and fill it in so that it becomes a recognizable figure. 1. 2. 3. 390 CHAPTER 7 Transformations and Tessellations In Exercises 4–6, identify the basic tessellation grid (squares, parallelograms, or regular hexagons) that each geometry student used to create each translation tessellation. 4. 5. 6. Cat Pack, Renee Chan Dog Prints, Gary Murakami Snorty the Pig, Jonathan Benton In Exercises 7 and 8, copy the figure and the grid onto patty paper. Create a tessellation on the grid with the figure. 7. 8. Now it’s your turn. In Exercises 9 and 10, create a tessellation of recognizable shapes using the translation method you learned in this lesson. At first, you will probably end up with shapes that look like amoebas or spilled milk, but with practice and imagination, you will get recognizable images. Decorate and title your designs. 9. Use squares as the basic structure. 10. Use regular hexagons as the basic structure. Review 11. The route of a rancher takes him from the house at point A to the south fence, then over to the east fence, then to the corral at point B. |
Copy the figure at right onto patty paper and locate the points on the south and east fences that minimize the rancher’s route. A S B E LESSON 7.6 Tessellations Using Only Translations 391 12. Reflect y 2 x 3 over the y-axis. Write an equation for the image. How does it 3 compare with the original equation? 13. Give the vertex arrangement for the tessellation at right. 14. A helicopter has four blades. Each blade measures about 26 feet from the center of rotation to the tip. What is the speed in feet per second at the tips of the blades when they are moving at 400 rpm? 15. Identify each of the following statements as true or false. If true, explain why. If false, give a counterexample explaining why it is false. a. If the two diagonals of a quadrilateral are congruent but only one is the perpendicular bisector of the other, then the quadrilateral is a kite. b. If the quadrilateral has exactly one line of reflectional symmetry, then the quadrilateral is a kite. c. If the diagonals of a quadrilateral are congruent and bisect each other, then it is a square. d. If a trapezoid is cyclic, then it is isosceles. IMPROVING YOUR VISUAL THINKING SKILLS 3-by-3 Inductive Reasoning Puzzle I Sketch the figure missing in the lower right corner of this 3-by-3 pattern. 392 CHAPTER 7 Transformations and Tessellations L E S S O N 7.7 Einstein was once asked, “What is your phone number?” He answered, “I don’t know, but I know where to find it if I need it.” ALBERT EINSTEIN Tessellations That Use Rotations In Lesson 7.6, you created recognizable shapes by translating curves from opposite sides of a regular hexagon or square. In tessellations using only translations, all the figures face in the same direction. In this lesson you will use rotations of curves on a grid of parallelograms, equilateral triangles, or regular hexagons. The resulting tiles will fit together when you rotate them, and the designs will have rotational symmetry about points in the tiling. For example, in this Escher print, each reptile is made by |
rotating three different curves about three alternating vertices of a regular hexagon Step 1 X N Step 3 Step 1 Step 2 Step 3 Step 4 Step 5 Step Step 2 X N Step 4 Symmetry Drawing E25, M. C. Escher, 1939 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved Step 5 N Step 6 Connect points S and I with a curve. Rotate curve SI about point I so that point S rotates to coincide with point X. Connect points G and X with a curve. Rotate curve GX about point G so that point X rotates to coincide with point O. Create curve NO. Rotate curve NO about point N so that point O rotates to coincide with point S. LESSON 7.7 Tessellations That Use Rotations 393 Escher worked long and hard to adjust each curve until he got what he recognized as a reptile. When you are working on your own design, keep in mind that you may have to redraw your curves a few times until something you recognize appears. Escher used his reptile drawing in this famous lithograph. Look closely at the reptiles in the drawing. Escher loved to play with our perceptions of reality! Reptiles, M. C. Escher, 1943 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. Another method used by Escher utilizes rotations on an equilateral triangle grid. Two sides of each equilateral triangle have the same curve, rotated about their common point. The third side is a curve with point symmetry. The following steps demonstrate how you might create a tessellating flying fish like that created by Escher Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 Connect points F and I with a curve. Then rotate the curve 60° clockwise about point I so that it becomes curve IH. Find the midpoint S of FH and draw curve SH. Rotate curve SH 180° about S to produce curve FS. Together curve FS and curve SH become the point-symmetric curve FH. 394 CHAPTER 7 Transformations and Tessellations With a little added detail, the design becomes a flying fish. Or, with just a slight variation in the curves, the resulting shape will appear more like a bird than a flying fish. Symmetry Drawing E99, M. C. Escher, 1954 ©2002 Cordon Art B. V.–Baarn– Holland |
. All rights reserved. EXERCISES In Exercises 1 and 2, identify the basic grid (equilateral triangles or regular hexagons) that each geometry student used to create the tessellation. 1. 2. You will need Geometry software for Exercise 13 Merlin, Aimee Plourdes Snakes, Jack Chow LESSON 7.7 Tessellations That Use Rotations 395 In Exercises 3 and 4, copy the figure and the grid onto patty paper. Show how you can use other pieces of patty paper to tessellate the figure on the grid. 3. 4. In Exercises 5 and 6, create tessellation designs by using rotations. You will need patty paper, tracing paper, or clear plastic, and grid paper or isometric dot paper. 5. Create a tessellating design of recognizable shapes by using a grid of regular hexagons. Decorate and color your art. 6. Create a tessellating design of recognizable shapes by using a grid of equilateral or isosceles triangles. Decorate and color your art. Review 7. Study these knot designs by Rinus Roelofs. Now try creating one of your own. Select a tessellation. Make a copy and thicken the lines. Make two copies of this thick-lined tessellation. Lay one of them on top of the other and shift it slightly. Trace the one underneath onto the top copy. Erase where they overlap. Then create a knot design using what you learned in Lesson 0.5. Dutch artist Rinus Roelofs (b 1954) experiments with the lines between the shapes rather than looking at the plane-filling figures. In these paintings, he has made the lines thicker and created intricate knot designs. (Above) Impossible Structures–III, structure 24 (At left) Interwoven Patterns–V, structure 17 Rinus Roelofs/Courtesy of the artist & ©2002 Artist Rights Society (ARS), New York/Beeldrecht, Amsterdam. 396 CHAPTER 7 Transformations and Tessellations For Exercises 8–11, identify the statement as true or false. For each false statement, explain why it is false or sketch a counterexample. 8. If the diagonals of a quadrilateral are congruent, the quadrilateral is a parallelogram. 9. If the diagonals of a quadrilateral |
are congruent and bisect each other, the quadrilateral is a rectangle. 10. If the diagonals of a quadrilateral are perpendicular and bisect each other, the quadrilateral is a rhombus. 11. If the diagonals of a quadrilateral are congruent and perpendicular, the quadrilateral is a square. 12. Earth’s radius is about 4000 miles. Imagine that you travel from the equator to the South Pole by a direct route along the surface. Draw a sketch of your path. How far will you travel? How long will the trip take if you travel at an average speed of 50 miles per hour? 13. Technology Use geometry software to construct a line and two points A and B not on the line. Reflect A and B over the line and connect the four points to form a trapezoid. a. Is it isosceles? Why? b. Choose a random point C inside the trapezoid and connect it to the four vertices with segments. Calculate the sum of the distances from C to the four vertices. Drag point C around. Where is the sum of the distances the greatest? The least? IMPROVING YOUR REASONING SKILLS Logical Liars Five students have just completed a logic contest. To confuse the school’s reporter, Lois Lang, each student agreed to make one true and one false statement to her when she interviewed them. Lois was clever enough to figure out the winner. Are you? Here are the students’ statements. Frances: Kai was second. I was fourth. Leyton: I was third. Charles was last. Denise: Kai won. I was second. Kai: Charles: I came in second. Kai was third. Leyton had the best score. I came in last. LESSON 7.7 Tessellations That Use Rotations 397 Tessellations That Use Glide Reflections In this lesson you will use glide reflections to create tessellations. In Lesson 7.6, you saw Escher’s translation tessellation of the winged horse Pegasus. All the horses are facing in the same direction. In the drawings below and below left, Escher used glide reflections on a grid of glide-reflected kites to get his horsemen facing in opposite directions. L E S S O N 7.8 The young do not know enough to be prudent, and therefore they attempt the impossible, and achieve |
it, generation after generation. PEARL S. BUCK Horseman, M. C. Escher, 1946 ©2002 Cordon Art B. V.–Baarn– Holland. All rights reserved. The steps below show how you can make a tessellating design similar to Escher’s Horseman. (The symbol indicates a glide reflection.) Step 1 Step 2 Horseman Sketch, M. C. Escher ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. Step 3 Step 4 398 CHAPTER 7 Transformations and Tessellations In the tessellation of birds below left, you can see that Escher used a grid of squares. You can use the same procedure on a grid of any type of glide-reflected parallelograms. The steps below show how you might create a tessellation of birds or fishes on a parallelogram grid. Step 1 Step 2 Step 3 Step 4 Symmetry Drawing E108, M. C. Escher, 1967 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. EXERCISES In Exercises 1 and 2, identify the basic tessellation grid (kites or parallelograms) that the geometry student used to create the tessellation. 1. 2. You will need Construction tools for Exercise 8 A Boy with a Red Scarf, Elina Uzin Glide Reflection, Alice Chan LESSON 7.8 Tessellations That Use Glide Reflections 399 In Exercises 3 and 4, copy the figure and the grid onto patty paper. Show how you can use other patty paper to tessellate the figure on the grid. 3. 4. 5. Create a glide-reflection tiling design of recognizable shapes by using a grid of kites. Decorate and color your art. 6. Create a glide-reflection tiling design of recognizable shapes by using a grid of parallelograms. Decorate and color your art. Review 7. Find the coordinates of the circumcenter and orthocenter of FAN with F(6, 0), A(7, 7), and N(3, 9). 8. Construction Construct a circle and a chord of the circle. With compass and straightedge construct a second chord parallel and congruent to the first chord. 9. Remy’s friends are pulling him on a sled. One of his friends is |
stronger and exerts more force. The vectors in this diagram represent the forces his two friends exert on him. Copy the vectors, complete the vector parallelogram, and draw the resultant vector force on his sled. 10. The green prism below right was built from the two solids below left. Copy the figure on the right onto isometric dot paper and shade in one of the two pieces to show how the complete figure was created. F1 47° F2 IMPROVING YOUR ALGEBRA SKILLS Fantasy Functions If a ° b ab then 3 ° 2 32 9, and if a b a2 b2 then 5 2 52 22 29. If 8 x 17 ° 2, find x. 400 CHAPTER 7 Transformations and Tessellations ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 7 ● USING YO USING YOUR ALGEBRA SKILLS 7 EXAMPLE A Solution Finding the Orthocenter and Centroid Suppose you know the coordinates of the vertices of a triangle. You have seen that you can find the coordinates of the circumcenter by writing equations for the perpendicular bisectors of two of the sides and solving the system. Similarly, you can find the coordinates of the orthocenter by finding equations for two lines containing altitudes of the triangle and solving the system. Find the coordinates of the orthocenter of PDQ with P(0, 4), D(4, 4), and Q(8, 4). y 5 D Q x 8 –5 P –6 The altitude of a triangle passes through one vertex and is perpendicular to the opposite side. To find the equation for the altitude from Q to PD, first you calculate the slope of PD, getting 2. So the slope of the altitude to PD is 1, the negative 2 reciprocal of 2. The altitude from Q to PD passes through point Q(8, 4), so its equation is y 4. Solving for y gives y 1 1 x. 2 2 8 x Use the same technique to find the equation of the altitude from D to PQ. The slope of PQ is 1. So, the slope of the altitude to PQ is 1. The altitude y 4 passes through point D(4, 4), so its equation is 1, or y x. ( x 4) To find the point where the altitudes intersect, use elimination to solve this system Equation of the line |
containing the altitude from Q to PD. Equation of the line containing the altitude from D to PQ. Subtract the second equation from the first equation to eliminate y. Multiply both sides by 2. 3 The x-coordinate of the point of the intersection is 0. Substitute 0 for x in either original equation and you will get y 0. So, the orthocenter is (0, 0). You can verify your result by writing the equation of the line containing the third altitude and making sure (0, 0) satisfies it. You can also find the coordinates of the centroid of a triangle by solving a system of two lines containing medians. However, as you will see in the next example, there is a more efficient method. USING YOUR ALGEBRA SKILLS 7 Finding the Orthocenter and Centroid 401 ALGEBRA SKILLS 7 ● USING YOUR ALGEBRA SKILLS 7 ● USING YOUR ALGEBRA SKILLS 7 ● USING YO EXAMPLE B Consider ABC with A(5, 3), B(3, 5), and C(1, 2). a. Find the coordinates of the centroid of ABC by writing equations for two lines containing medians and finding their point of intersection. b. Find the mean of the x-coordinates and the mean of the y-coordinates of the triangle’s vertices. What do you notice? y 3 C –6 A x 5 B –6 Solution The median of a triangle joins a vertex with the midpoint of the opposite side. a. First, find the equation of the line containing the median from A to BC. The midpoint of BC is 1, 3. The slope from A(5, 3) to this midpoint is 2 3 3 y ( ) 3 2 1, or 1. Solving for y gives. The equation of the line is as the equation for the median. 4 4 Next, find the equation of the line containing the median from B(3, 5) to AC. The midpoint of AC is 3, 1. The slope is 3. So you get the 4 2 ( equation y 5) 3 1 as the equation x 1. Solving for y gives y 3 x 4 4 4 3 for the median. Finally, use elimination to solve this system Equation of the line containing the median from A to BC. Equation of the line containing the median from B to AC. Sub |
tract the second equation from the first. Subtract 1 from both sides. The x-coordinate of the point of intersection is 1. Use substitution to find the y-coordinate. (1) 7 y 1 4 4 y 2 Substitute 1 for x in the first equation. Simplify. The centroid is (1, 2). You can verify your result by writing the equation for the third median and making sure (1, 2) satisfies it. (1) 5 3 3 1. b. The mean of the x-coordinates is 3 3 5) 2 3 ( 6 2. The mean of the y-coordinates is 3 3 Notice that these means give you the coordinates of the centroid: (1, 2). 402 CHAPTER 7 Transformations and Tessellations ALGEBRA SKILLS 7 ● USING YOUR ALGEBRA SKILLS 7 ● USING YOUR ALGEBRA SKILLS 7 ● USING YO You can generalize the findings from Example B to all triangles. The easiest way to find the coordinates of the centroid is to find the mean of the vertex coordinates. EXERCISES In Exercises 1 and 2, use RES with vertices R(0, 0), E(4, 6), and S(8, 4). 1. Find the equation of the line containing the median from R to ES. 2. Find the equation of the line containing the altitude from E to RS. y 4 S (8, 4) R (0, 0) x 8 In Exercises 3 and 4, use algebra to find the coordinates of the centroid and the orthocenter for each triangle. –6 E (4, –6) 3. Right triangle MNO 4. Isosceles triangle CDE y 5 N (0, 5) M (–4, 0) –5 –5 5 x 10 O (10, –3) C (0, 6) y 4 4 8 E (12, 0) x –4 D (0, –6) 5. Find the coordinates of the centroid of the triangle formed by the x-axis, the y-axis, and the line 12x 9y 36. 6. The three lines 8x 3y 12, 6y 7x 24, and x 9y 33 0 intersect to form a triangle. Find the coordinates of its centroid. IMPROVING YOUR VISUAL THINKING SKILLS Painted Faces I |
Suppose some unit cubes are assembled into a large cube, then some of the faces of this large cube are painted. After the paint dries, the large cube is disassembled into the unit cubes and you discover that 32 of these have no paint on any of their faces. How many faces of the large cube were painted? USING YOUR ALGEBRA SKILLS 7 Finding the Orthocenter and Centroid 403 ● CHAPTER 11 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER CHAPTER 7 R E V I E W How is your memory? In this chapter you learned about rigid transformations in the plane—called isometries—and you revisited the principles of symmetry that you first learned in Chapter 0. You applied these concepts to create tessellations. Can you name the three rigid transformations? Can you describe how to compose transformations to make other transformations? How can you use reflections to improve your miniature-golf game? What types of symmetry do regular polygons have? What types of polygons will tile the plane? Review this chapter to be sure you can answer these questions. EXERCISES For Exercises 1–12, identify each statement as true or false. For each false statement, sketch a counterexample or explain why it is false. 1. The two transformations in which the orientation (the order of points as you move clockwise) does not change are translation and rotation. 2. The two transformations in which the image has the opposite orientation from the original are reflection and glide reflection. 3. A translation of (5, 12) followed by a translation of (8, 6) is equivalent to a single translation of (3, 6). 4. A rotation of 140° followed by a rotation of 260° about the same point is equivalent to a single rotation of 40° about that point. 5. A reflection across a line followed by a second reflection across a parallel line that is 12 cm from the first is equivalent to a translation of 24 cm. 6. A regular n-gon has n reflectional symmetries and n rotational symmetries. 7. The only three regular polygons that create monohedral tessellations are equilateral triangles, squares, and regular pentagons. 8. Any triangle can create a monohedral tessellation. 9. Any quadrilateral can create a monohedral tessellation. 10. No pentagon can create a monohedral tes |
sellation. 11. No hexagon can create a monohedral tessellation. 12. There are at least three times as many true statements as false statements in Exercises 1–12. King by Minnie Evans (1892–1987) 404 CHAPTER 7 Transformations and Tessellations ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 In Exercises 13–15, identify the type or types of symmetry, including the number of symmetries, in each design. For Exercise 15, describe how you can move candles on the menorah to make the colors symmetrical, too. 13. 14. 15. Mandala, Gary Chen, geometry student 16. The façade of Chartres Cathedral in France does not have reflectional symmetry. Why not? Sketch the portion of the façade that does have bilateral symmetry. 17. Find or create a logo that has reflectional symmetry. Sketch the logo and its line or lines of reflectional symmetry. 18. Find or create a logo that has rotational symmetry but not reflectional symmetry. Sketch it. In Exercises 19 and 20, classify the tessellation and give the vertex arrangement. 19. 20. CHAPTER 7 REVIEW 405 EW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTE 21. Experiment with a mirror to find the smallest vertical portion ( y) in which you can still see your full height (x). How does y compare to x? Can you explain, with the help of a diagram and what you know about reflections, why a “full-length” mirror need not be as tall as you? y x 22. Miniature-golf pro Sandy Trapp wishes to impress her fans with a hole in one on the very first try. How should she hit the ball at T to achieve this feat? Explain. T H In Exercises 23–25, identify the shape of the tessellation grid and a possible method that the student used to create each tessellation. 23. 24. 25. Perian Warriors, Robert Bell Doves, Serene Tam Sightings, Peter Chua and Monica Grant In Exercises 26 and 27, copy the figure and grid onto patty paper. Determine whether or not you can use the figure to create a tessellation on the |
grid. Explain your reasoning. 26. 27. 406 CHAPTER 7 Transformations and Tessellations ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 28. In his woodcut Day and Night, Escher gradually changes the shape of the patches of farmland into black and white birds. The birds are flying in opposite directions, so they appear to be glide reflections of each other. But notice that the tails of the white birds curve down, while the tails of the black birds curve up. So, on closer inspection, it’s clear that this is not a glide-reflection tiling at all! When two birds are taken together as one tile (a 2-motif tile), they create a translation tessellation. Use patty paper to find the 2-motif tile. Day and Night, M. C. Escher, 1938 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved. Career Commercial tile contractors use tessellating polygons to create attractive designs for their customers. Some designs are 1-uniform or 2-uniform, and others are even more complex. CHAPTER 7 REVIEW 407 EW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTER 7 REVIEW ● CHAPTE Assessing What You’ve Learned Try one or more of these assessment suggestions. UPDATE YOUR PORTFOLIO Choose one of the tessellations you did in this chapter and add it to your portfolio. Describe why you chose it and explain the transformations you used and the types of symmetry it has. ORGANIZE YOUR NOTEBOOK Review your notebook to be sure it’s complete and well organized. Are all the types of transformation and symmetry included in your definition list or conjecture list? Write a one-page chapter summary. WRITE IN YOUR JOURNAL This chapter emphasizes applying geometry to create art. Write about connections you see between geometry and art. Does creating geometric art give you a greater appreciation for either art or geometry? Explain. PERFORMANCE ASSESSMENT While a classmate, a friend, a family member, or a teacher observes, carry out one of the investigations from this chapter. Explain what you’re doing at each step, including how you arrive at the conjecture. GIVE A PRESENTATION Give a presentation about one of the investigations or projects you did or about |
one of the tessellations you created. 408 CHAPTER 7 Transformations and Tessellations CHAPTER 8 Area I could fill an entire second life with working on my prints. M. C. ESCHER Square Limit, M. C. Escher, 1964 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● discover area formulas for rectangles, parallelograms, triangles, trapezoids, kites, regular polygons, circles, and other shapes ● use area formulas to solve problems ● learn how to find the surface areas of prisms, pyramids, cylinders, and cones L E S S O N 8.1 A little learning is a dangerous thing—almost as dangerous as a lot of ignorance. ANONYMOUS Areas of Rectangles and Parallelograms People work with areas in many occupations. Carpenters calculate the areas of walls, floors, and roofs before they purchase materials for construction. Painters calculate surface areas so that they know how much paint to buy for a job. Decorators calculate the areas of floors and windows to know how much carpeting and drapery they will need. In this chapter you will discover formulas for finding the areas of the regions within triangles, parallelograms, trapezoids, kites, regular polygons, and circles. Tile layers need to find floor area to determine how many tiles to buy. The area of a plane figure is the measure of the region enclosed by the figure. You measure the area of a figure by counting the number of square units that you can arrange to fill the figure completely. 1 1 1 Length: 1 unit Area: 1 square unit You probably already know many area formulas. Think of the investigations in this chapter as physical demonstrations of the formulas that will help you understand and remember them. It’s easy to find the area of a rectangle. To find the area of the first rectangle, you can simply count squares. To find the areas of the other rectangles, you could draw in the lines and count the squares, but there’s an easier method. 410 CHAPTER 8 Area Any side of a rectangle can be called a base. A rectangle’s height is the length of the side that is perpendicular to the base. For each pair of parallel bases, there is a corresponding height. Height Height Base Base If we call the bottom side of each rectangle in the figure the base, then the length of the base is the number |
of squares in each row and the height is the number of rows. So you can use these terms to state a formula for the area. Add this conjecture to your list. Rectangle Area Conjecture The area of a rectangle is given by the formula?, where A is the area, b is the length of the base, and h is the height of the rectangle. C-75 The area formula for rectangles can help you find the areas of many other shapes. EXAMPLE A Find the area of this shape. Solution The middle section is a rectangle with an area of 4 8, or 32 square units. You can divide the remaining pieces into right triangles, so each piece is actually half a rectangle. The area of the figure is 32 2(2) 2(6) 48 square units. There are other ways to find the area of this figure. One way is to find the area of an 8-by-8 square and subtract the areas of four right triangles. 1 2 (4) 2 1 2 (4) 2 32 in.2 1 2 (12) 6 1 2 (12) 6 LESSON 8.1 Areas of Rectangles and Parallelograms 411 You can also use the area formula for a rectangle to find the area formula for a parallelogram. Just as with a rectangle, any side of a parallelogram can be called a base. But the height of a parallelogram is not necessarily the length of a side. An altitude is any segment from one side of a parallelogram perpendicular to a line through the opposite side. The length of the altitude is the height. Altitude to the base or Altitude to the base Base Base The altitude can be inside or outside the parallelogram. No matter where you draw the altitude to a base, its height should be the same, because the opposite sides are parallel. h h h b You will need ● heavy paper or cardboard ● a straightedge ● a compass Investigation Area Formula for Parallelograms Using heavy paper, investigate the area of a parallelogram. Can the area be rearranged into a more familiar shape? Different members of your group should investigate different parallelograms so you can be sure your formula works for all parallelograms. b Construct a parallelogram on a piece of heavy paper or cardboard. From the vertex of the obtuse angle adjacent to the base, draw an altitude to the side opposite the base. Label the parallelogram as shown. h s s Cut out |
the parallelogram and then cut along the altitude. You will have two pieces—a triangle and a trapezoid. Try arranging the two pieces into other shapes without overlapping them. Is the area of each of these new shapes the same as the area of the original parallelogram? Why? Step 1 Step 2 Step 3 Is one of your new shapes a rectangle? Calculate the area of this rectangle. What is the area of the original parallelogram? State your next conjecture. Parallelogram Area Conjecture The area of a parallelogram is given by the formula?, where A is the area, b is the length of the base, and h is the height of the parallelogram. C-76 412 CHAPTER 8 Area If the dimensions of a figure are measured in inches, feet, or yards, the area is measured in in.2 (square inches), ft2 (square feet), or yd2 (square yards). If the dimensions are measured in centimeters or meters, the area is measured in cm2 (square centimeters) or m2 (square meters). Let’s look at an example. EXAMPLE B Find the height of a parallelogram that has area 7.13 m2 and base length 2.3 m. h 2.3 m Solution A bh 7.13 (2.3).1 The height measures 3.1 m. Write the formula. Substitute the known values. Solve for the height. Divide. EXERCISES In Exercises 1–6, each quadrilateral is a rectangle. A represents area and P represents perimeter. Use the appropriate unit in each answer. 1. A? 12 m 19 m 4. A 273 cm2 h? h 13 cm 2. A? 4.5 cm 5. P 40 ft A? 7 ft 9.3 cm 3. A 96 yd2 b? 12 yd b 6. Shaded area? 21 m 5 m 12 m 11 m In Exercises 7–9, each quadrilateral is a parallelogram. 7. A? 9 in. 8 in. 12 in. 8. A 2508 cm2 P? 9. Find the area of the shaded region. 44 cm 48 cm 9 ft 7 ft 12 ft LESSON 8.1 Areas of Rectangles and Parallelograms 413 10. Sketch and label two different rectangles, each with area 48 cm2. In Exercises 11 and 12, find |
the area of the figure and explain your method. 11. 12. 13. Sketch and label two different parallelograms, each with area 64 cm2. 14. Draw and label a figure with area 64 cm2 and perimeter 64 cm. 15. The photo shows a Japanese police koban. An arch forms part of the roof and one wall. The arch is made from rectangular panels that each measure 1 m by 0.7 m. The arch is 11 panels high and 3 panels wide. What’s the total area of the arch? Cultural Koban is Japanese for “mini-station,” a small police station. These stations are located in several parts of a city, and officers who work in them know the surrounding neighborhoods and people well. The presence of kobans in Japan helps reduce crime and provides communities with a sense of security. 16. What is the total area of the four walls of a rectangular room 4 meters long by 5.5 meters wide by 3 meters high? Ignore all doors and windows. 17. APPLICATION Ernesto plans to build a pen for his pet iguana. What is the area of the largest rectangular pen that he can make with 100 meters of fencing? 18. The big event at George Washington High School’s May Festival each year is the Cow Drop Contest. A farmer brings his well-fed bovine to wander the football field until—well, you get the picture. Before the contest, the football field, which measures 53 yards wide by 100 yards long, is divided into square yards. School clubs and classes may purchase square yards. If one of their squares is where the first dropping lands, they win a pizza party. If the math club purchases 10 squares, what is the probability that the club wins? 19. APPLICATION Sarah is tiling a wall in her bathroom. It is rectangular and measures 4 feet by 7 feet. The tiles are square and measure 6 inches on each side. How many tiles does Sarah need? 414 CHAPTER 8 Area The figure at right demonstrates that (a b)2 a2 2ab b2. In Exercises 20 and 21, sketch and label a rectangle that demonstrates each algebraic expression. 20. (x 3)(x 5) x2 8x 15 21. (3x 2)(2x 5) 6x2 19x 10 a b a b 22. A right triangle with sides measuring 6 cm, 8 cm, and 10 cm has a square constructed on each of its three |
sides, as shown. Compare the area of the square on the longest side to the sum of the areas of the two squares on the two shorter legs. a b a a2 ab a b ab b 2 b 6 10 8 23. What is the area of the parallelogram? 24. What is the area of the trapezoid? y (8, 16) 5 cm 12 cm 21 cm 20 cm (0, 0) (6, 0) x Art The design at right is a quilt design called the Ohio Star. Traditional quilt block designs range from a simple nine-patch, based on 9 squares, to more complicated designs like Jacob’s Ladder or Underground Railroad, which are based on 16, 25, or even 36 squares. Quiltmakers need to calculate the total area of each different type of material before they make a complete quilt. 25. APPLICATION The Ohio Star is a 16-square quilt design. Each block measures 12 inches by 12 inches. One block is shown above. Assume you will need an additional 20% of each fabric to allow for seams and errors. a. Calculate the sum of the areas of all the red patches, the sum of the areas of all the blue patches, and the area of the yellow patch in a single block. b. How many Ohio Star blocks will you need to cover an area that measures 72 inches by 84 inches, the top surface area of a king-size mattress? c. How much fabric of each color will you need? How much fabric will you need for a 15-inch border to extend beyond the edges of the top surface of the mattress? LESSON 8.1 Areas of Rectangles and Parallelograms 415 Review 26. Copy the figure at right. Find the lettered angle measures and arc measures. AB and AC are tangents. CD is a diameter. 27. Given AM as the length of the altitude of an equilateral triangle, construct the triangle. A M A a D d h 94° g E B 52° e c m F b k f C 28. Sketch what the figure at right looks like when viewed from a. Above the figure, looking straight down b. In front of the figure, that is, looking straight at the red-shaded side c. The side, looking at the blue-shaded side RANDOM RECTANGLES What does a typical rectangle look like? A randomly generated rectangle could be long and narrow, or square-like. |
It could have a large perimeter, but a small area. Or it could have a large area, but a small perimeter. In this project you will randomly generate rectangles and study their characteristics using scatter plots and histograms. Your project should include A description of how you created your random rectangles, including any constraints you used. A scatter plot of base versus height, a perimeter histogram, an area histogram, and a scatter plot of perimeter versus area. Any other studies or graphs you think might be interesting. Your predictions about the data before you made each graph. An explanation of why each graph looks the way it does. You can use Fathom to generate random base and height values from 0 to 10. Then you can sort them by various characteristics and make a wide range of interesting graphs. 416 CHAPTER 8 Area L E S S O N 8.2 When you add to the truth, you subtract from it. THE TALMUD You will need ● heavy paper or cardboard Areas of Triangles, Trapezoids, and Kites In Lesson 8.1, you learned the area formula for rectangles, and you used it to discover an area formula for parallelograms. In this lesson you will use those formulas to discover or demonstrate the formulas for the areas of triangles, trapezoids, and kites. Investigation 1 Area Formula for Triangles s h d b Step 1 Step 2 Step 3 Cut out a pair of congruent triangles. Label their corresponding parts as shown. Arrange the triangles to form a figure for which you already have an area formula. Calculate the area of the figure. What is the area of one of the triangles? Make a conjecture. Write a brief description in your notebook of how you arrived at the formula. Include an illustration. Triangle Area Conjecture The area of a triangle is given by the formula?, where A is the area, b is the length of the base, and h is the height of the triangle. C-77 Investigation 2 Area Formula for Trapezoids You will need ● heavy paper or cardboard b2 b1 h s Step 1 Construct any trapezoid and an altitude perpendicular to its bases. Label the trapezoid as shown. Step 2 Cut out the trapezoid. Make and label a copy. LESSON 8.2 Areas of Triangles, Trapezoids, and Kites 417 Step 3 Arrange the two trapezoids to form a figure for which you already have an area formula. What type of polygon |
is this? What is its area? What is the area of one trapezoid? State a conjecture. Trapezoid Area Conjecture The area of a trapezoid is given by the formula?, where A is the area, b1 and b2 are the lengths of the two bases, and h is the height of the trapezoid. C-78 Investigation 3 Area Formula for Kites Can you rearrange a kite into shapes for which you already have the area formula? Do you recall some of the properties of a kite? Create and carry out your own investigation to discover a formula for the area of a kite. Discuss your results with your group. State a conjecture. Kite Area Conjecture The area of a kite is given by the formula?. C-79 EXERCISES In Exercises 1–12, use your new area conjectures to solve for the unknown measures. 1. A? 2. A? 5 cm 6 cm 9 m 8 cm 11 m 5. A 39 cm2 h? 6 cm h 13 cm 4. A? 8 cm 6 cm 14 cm 418 CHAPTER 8 Area 3. A? 15 15 20 9 12 12 16 20 6. A 31.5 ft2 b? b 9 ft 10 ft 7. A 420 ft2 LE? E 17 ft U 8 ft 20 ft 25 ft 17 ft L 25 ft B 10. A 924 cm2 P? 8. A 50 cm2 h? 7 cm 6 cm h 13 cm 9. A 180 m2 b? 9 m 24 m b 11. A 204 cm2 P 62 cm h? 12. x? y? 51 cm 24 cm 40 cm 15 cm h 13 cm 10 cm 13. Sketch and label two different triangles, each with area 54 cm2. 14. Sketch and label two different trapezoids, each with area 56 cm2. 15. Sketch and label two different kites, each with area 1092 cm2. B x 6 ft 5 ft A y 9 ft 15 ft C 16. Sketch and label a triangle and a trapezoid with equal areas and equal heights. How does the base of the triangle compare with the two bases of the trapezoid? 17. P is a random point on side AY of rectangle ARTY. The shaded area is what fraction of the area of the rectangle? Why? Y P A T R 19. APPLICATION Eduardo has designed this kite for a contest. He plans to |
cut the kite from a sheet of Mylar plastic and use balsa wood for the diagonals. He will connect all the vertices with string, and fold and glue flaps over the string. a. How much balsa wood and Mylar will he need? b. Mylar is sold in rolls 36 inches wide. What length of Mylar does Eduardo need for this kite? 18. One playing card is placed over another, as shown. Is the top card covering half, less than half, or more than half of the bottom card? Explain. 20 in. 25 in. 21 in. 1 in. 36 in. 15 in. 39 in. 35 in. LESSON 8.2 Areas of Triangles, Trapezoids, and Kites 419 20. APPLICATION The roof on Crystal’s house is formed by two congruent trapezoids and two congruent isosceles triangles, as shown. She wants to put new wood shingles on her roof. Each shingle will cover 0.25 square foot of area. (The shingles are 1 foot by 1 foot, but they overlap by 0.75 square foot.) How many shingles should Crystal buy? 15 ft 10 ft 20 ft 15 ft 30 ft 21. A trapezoid has been created by combining two congruent right triangles and an isosceles triangle, as shown. Is the isosceles triangle a right triangle? How do you know? Find the area of the trapezoid two ways: first by using the trapezoid area formula, and then by finding the sum of the areas of the three triangles. b c a a c b 22. Divide a trapezoid into two triangles. Use algebra to derive the formula for the area of the trapezoid by expressing the area of each triangle algebraically and finding their algebraic sum. Review 23. A? 24. A? 25. A 264 m2 P? 26. P 52 cm A? 24 m 10 cm 9 cm 56° E 112° T1 24° T2 27. Trace the figure at right. Find the lettered angle measures and arc measures. AB and AC are tangents. CD is a diameter. 28. Two tugboats are pulling a container ship into the harbor. They are pulling at an angle of 24° between the tow lines. The vectors shown in the diagram represent the forces the two tugs are exerting on the container ship. Copy the |
vectors and complete the vector parallelogram to determine the resultant vector force on the container ship. 420 CHAPTER 8 Area 29. Two paths from C to T (traveling on the surface) are shown on the 8 cm-by-8 cm-by-4 cm prism below. M is the midpoint of edge UA. Which is the shorter path from C to T: C-M-T or C-A-T? Explain. 30. Give the vertex arrangement for this 2-uniform tessellation. P 4 cm C 8 cm H E A M U T B 8 cm MAXIMIZING AREA A farmer wants to fence in a rectangular pen using the wall of a barn for one side of the pen and the 10 meters of fencing for the remaining three sides. What dimensions will give her the maximum area for the pen? You can use the trace feature on your calculator to find the value of x that gives the maximum area. Use your graphing calculator to investigate this problem and find the best arrangement. X = 3.5 Y = 10.5 Your project should include An expression for the third side length, in terms of the variable x in the diagram. An equation and graph for the area of the pen. The dimensions of the best rectangular shape for the farmer’s pen. Barn wall x Pen x LESSON 8.2 Areas of Triangles, Trapezoids, and Kites 421 L E S S O N 8.3 Optimists look for solutions, pessimists look for excuses. SUE SWENSON Area Problems By now, you know formulas for finding the areas of rectangles, parallelograms, triangles, trapezoids, and kites. Now let’s see if you can use these area formulas to approximate the areas of irregularly shaped figures. A C B D E F G H You will need ● figures A–H ● centimeter rulers or meterstick Investigation Solving Problems with Area Formulas Find the area of each geometric figure your teacher provides. Before you begin to measure, discuss with your group the best strategy for each step. Discuss what units you should use. Different group members might get different results. However, your results should be close. You may average your results to arrive at one group answer. For each figure, write a sentence or two explaining how you measured the area and how accurate you think it is. Now that you have practiced measuring and calculating area, you’re ready to try some application problems. Many |
everyday projects require you to find the areas of flat surfaces on three-dimensional objects. You’ll learn more about surface area in Lesson 8.7. Career Professional housepainters have a unique combination of skills: For large-scale jobs, they begin by measuring the surfaces that they will paint and use measurements to estimate the quantity of materials they will need. They remove old coating, clean the surface, apply sealer, mix color, apply paint, and add finishes. Painters become experienced and specialize their craft through the on-the-job training they receive during their apprenticeships. 422 CHAPTER 8 Area In the exercises you will learn how to use area in buying rolls of wallpaper, gallons of paint, bundles of shingles, square yards of carpet, and square feet of tile. Keep in mind that you can’t buy 121 1 gallons of paint! You must buy 13 gallons. If your 1 6 calculations tell you that you need 5.25 bundles of shingles, you have to buy 6 bundles. In this type of rounding, you must always round upward. EXERCISES 1. APPLICATION Tammy is estimating how much she should charge for painting 148 rooms in a new motel with one coat of base paint and one coat of finishing paint. The four walls and the ceiling of each room must be painted. Each room measures 14 ft by 16 ft by 10 ft high. a. Calculate the total area of all the surfaces to be painted with each coat. Ignore doors and windows. b. One gallon of base paint covers 500 square feet. One gallon of finishing paint covers 250 square feet. How many gallons of each will Tammy need for the job? 2. APPLICATION Rashad wants to wallpaper the four walls of his bedroom. The room is rectangular and measures 11 feet by 13 feet. The ceiling is 10 feet high. A roll of wallpaper at the store is 2.5 feet wide and 50 feet long. How many rolls should he buy? (Wallpaper is hung from ceiling to floor. Ignore the doors and windows.) 3. APPLICATION It takes 65,000 solar cells, each 1.25 in. by 2.75 in., to power the Helios Prototype, shown below. How much surface area, in square feet, must be covered with the cells? The cells on Helios are 18% efficient. Suppose they were only 12% efficient, like solar cells used in homes. How much more surface area would need to be covered to deliver the same amount of power? |
Technology In August 2001, the Helios Prototype, a remotely controlled, nonpolluting solar-powered aircraft, reached 96,500 feet—a record for nonrocket aircraft. Soon, the Helios will likely sustain flight long enough to enable weather monitoring and other satellite functions. For news and updates, go to www.keymath.com/DG. LESSON 8.3 Area Problems 423 For Exercises 4 and 5, refer to the floor plan at right. 4. APPLICATION Dareen’s family is ready to have wall-to-wall carpeting installed. The carpeting they chose costs $14 per square yard, the padding $3 per square yard, and the installation $3 per square yard. What will it cost them to carpet the three bedrooms and the hallway shown? 5. APPLICATION Dareen’s family now wants to install 1-foot-square terra cotta tiles in the entryway and kitchen, and 4-inch-square blue tiles on each bathroom floor. The terra cotta tiles cost $5 each, and the bathroom tiles cost 45¢ each. How many of each kind will they need? What will the tiles cost? Bedroom 10 ft 9 ft Bedroom 10 ft 8 ft Entryway Kitchen 10 ft 18 ft Bath 10 ft 6 ft Hallway Bath 7 ft 9 ft Master bedroom 13 ft 8 ft Living room 13 ft 13 ft Dining room 13 ft 9 ft 40 ft 70 ft 20 ft 12 ft 7 ft 6. APPLICATION Harold works at a state park. He needs to seal the redwood deck at the information center to protect the wood. He measures the deck and finds that it is a kite with diagonals 40 feet and 70 feet. Each gallon of sealant covers 400 square feet, and the sealant needs to be applied every six months. How many gallon containers should he buy to protect the deck for the next three years? 7. APPLICATION A landscape architect is designing three trapezoidal flowerbeds to wrap around three sides of a hexagonal flagstone patio, as shown. What is the area of the entire flowerbed? The landscape architect’s fee is $100 plus $5 per square foot. What will the flowerbed cost? Career Landscape architects have a keen eye for natural beauty. They study the grade and direction of land slopes, stability of the soil, drainage patterns, and existing structures and vegetation. They look at the various social, economic, and artistic concerns of the client. They |
also use science and engineering to plan environments that harmonize land features with structures, reducing the impact of urban development upon nature. 424 CHAPTER 8 Area 8. APPLICATION Tom and Betty are planning to paint the exterior walls of their cabin (all vertical surfaces). The paint they have selected costs $24 per gallon and, according to the label, covers 150 to 300 square feet per gallon. Because the wood is very dry, they assume the paint will cover 150 square feet per gallon. How much will the project cost? (All measurements shown are in feet.) 16 10 10 18 24 16 28 Review 9. A first-century Greek mathematician named Hero is credited with the following formula for the area of a triangle: A s(s a)(s b)(s c), where A is the area of the triangle, a, b, and c are the lengths of the three sides of the triangle, and s is the semiperimeter (half of the perimeter). Use Hero’s formula to find the area of this triangle. Use the formula A 1 bh to check 2 your answer. 10. Explain why x must be 69° in the diagram at right. 11. As P moves from left to right along, which of the following values changes? A. The area of ABP B. The area of PDC C. The area of trapezoid ABCD D. mA mPCD mCPD E. None of these 15 cm 8 cm 17 cm A x P D A B 20° O 82° C D C B IMPROVING YOUR VISUAL THINKING SKILLS Four-Way Split How would you divide a triangle into four regions with equal areas? There are at least six different ways it can be done! Make six copies of a triangle and try it. LESSON 8.3 Area Problems 425 L E S S O N 8.4 If I had to live my life again, I’d make the same mistakes, only sooner. TALLULAH BANKHEAD Areas of Regular Polygons You can divide a regular polygon into congruent isosceles triangles by drawing segments from the center of the polygon to each vertex. The center of the polygon is actually the center of a circumscribed circle. In this investigation you will divide regular polygons into triangles. Then you will write a formula for the area of any regular polygon. Investigation Area Formula for Regular Polygons Consider a regular pentagon with side length s, divided into con |
gruent isosceles triangles. Each triangle has a base s and a height a. Step 1 Step 2 Step 3 What is the area of one isosceles triangle in terms of a and s? What is the area of this pentagon in terms of a and s? Repeat Steps 1 and 2 with other regular polygons and complete the table below. a s Regular pentagon a s a s Regular hexagon Regular heptagon Number of sides 5 6 7 8 9 10 Area of regular polygon 12...... n...... The distance a always appears in the area formula for a regular polygon, and it has a special name—apothem. An apothem of a regular polygon is a perpendicular segment from the center of the polygon’s circumscribed circle to a side of the polygon. You may also refer to the length of the segment as the apothem. Apothem 426 CHAPTER 8 Area You can restate your last entry in the table as your next conjecture. Regular Polygon Area Conjecture The area of a regular polygon is given by the formula?, where A is the area, a is the apothem, s is the length of each side, and n is the number of sides. The length of each side times the number of sides is the perimeter, P, so sn P. Thus you can also write the formula for area as A? P. C-80 EXERCISES In Exercises 1–8, use the Regular Polygon Area Conjecture to find the unknown length accurate to the nearest unit, or the unknown area accurate to the nearest square unit. Recall that you use the symbol when your answer is an approximation. You will need Construction tools for Exercise 9 Geometry software for Exercises 11 and 17 1. A? s 24 cm a 24.9 cm a s 2. a? s 107.5 cm A 19,887.5 cm2 3. P? a 38.6 cm A 4940.8 cm2 s a a s 4. Regular pentagon: a 3 cm and s 4.4 cm, A? 5. Regular nonagon: a 9.6 cm and A 302.4 cm2, P? 6. Regular n-gon: a 12 cm and P 81.6 cm, A? 7. Find the perimeter of a regular polygon if a 9 m and A 259.2 m2. 8. |
Find the length of each side of a regular n-gon if a 80 feet, n 20, and A 20,000 square feet. 9. Construction Use a compass and straightedge to construct a regular hexagon with sides that measure 4 cm. Use the Regular Polygon Area Conjecture and a centimeter ruler to approximate the hexagon’s area. 10. Draw a regular pentagon with apothem 4 cm. Use the Regular Polygon Area Conjecture and a centimeter ruler to approximate the pentagon’s area. 11. Technology Use geometry software to construct a circle. Inscribe a pentagon that looks regular and measure its area. Now drag the vertices. How can you drag the vertices to increase the area of the inscribed pentagon? To decrease its area? LESSON 8.4 Areas of Regular Polygons 427 12. Find the shaded area of the regular octagon ROADSIGN. The apothem measures about 20 cm. Segment GI measures about 16.6 cm. G I 13. Find the shaded area of the regular hexagonal donut. The apothem and sides of the smaller hexagon are half as long as the apothem and sides of the large hexagon. a 6.9 cm and r 8 cm Career Interior designers, unlike interior decorators, are concerned with the larger planning and technical considerations of interiors, as well as with style and color selection. They have an intuitive sense of spatial relationships. They prepare sketches, schedules, and budgets for client approval and inspect the site until the job is complete. 14. APPLICATION An interior designer created the kitchen plan shown. The countertop will be constructed of colored concrete. What is its total surface area? If concrete countertops 1.5 inches thick cost $85 per square foot, what will be the total cost of this countertop? 60 in. 48 in. 60 in. 36 in. 36 in. 144 in. 120 in. 24 in. 60 in. 48 in. 72 in. 138 in. 186 in. 428 CHAPTER 8 Area Review In Exercises 15 and 16, graph the two lines, then find the area bounded by the x-axis, the y-axis, and both lines. 15. y 1 x 5, y 2x 10 2 x 6, y 4 16. y 1 x 12 3 3 17. Technology Construct a triangle and its three medians. Compare the areas of the six small triangles that the three med |
ians formed. Make a conjecture, and support it with a convincing argument. 18. If the pattern continues, write an expression for the perimeter of the nth figure in the picture pattern 19. Identify the point of concurrency from the construction marks. a. b. c. IMPROVING YOUR VISUAL THINKING SKILLS The Squared Square Puzzle The square shown is called a “squared square.” A square 112 units on a side is divided into 21 squares. The area of square X is 502, or 2500, and the area of square Y is 42, or 16. Find the area of each of the other squares LESSON 8.4 Areas of Regular Polygons 429 Pick’s Formula for Area You know how to find the area of polygon regions, but how would you find the area of the dinosaur footprint at right? You know how to place a polygon on a grid and count squares to find the area. About a hundred years ago, Austrian mathematician Georg Alexander Pick (1859–1943) discovered a relationship, now known as Pick’s formula, for finding the area of figures on a square dot grid. Let’s start by looking at polygons on a square dot grid. The dots are called lattice points. Let’s count the lattice points in the interior of the polygon and those on its boundary and compare our findings to the areas of the polygon that you get by counting squares, as you did in Lesson 8.1. Polygon A 10 boundary points 12 interior points Polygon B 1.5 9 4 Area 2(1.5) 9 4 16 1 6 8 6 6 boundary points 17 interior points Area of rectangle 40 Area of polygon 40 1 8 2(6) 19 430 CHAPTER 8 Area How can the boundary points and interior points help us find the area? There are a lot of things to look at. It seems too difficult to find a pattern with our results. An important technique for finding patterns is to hold one variable constant and see what happens with the other variables. That’s what you’ll do in the activity below. Activity Dinosaur Footprints and Other Shapes You will need ● square dot paper or graph paper ● geoboards (optional Step 1 Step 2 Step 3 Step 4 Step 5 Confirm that each polygon A through J above has area A 12. Let b be the number of boundary points and i be the number of interior points. Create |
and complete a table like this one for polygons A through J. Polygon (A 12) A B C Number of boundary points (b) Number of interior points (i) Study the table for patterns. Do you see a relationship between b and i when A 12? Graph the pairs (b, i) from your table and label each point with its name A through J. What do you notice? Write an equation that fits the points. Consider several polygons with an area of 8. Graph points (b, i) and write an equation. Generalize the formula you found in Steps 3 and 4. When you feel you have enough data, copy and complete the conjecture. Pick’s Formula If A is the area of a polygon whose vertices are lattice points, b is the number of lattice points on the boundary of the polygon, and i is the number of interior lattice points, then A? b? i?. EXPLORATION Pick’s Formula for Area 431 Pick’s formula is especially useful when you apply it to the areas of irregularly shaped regions. Since it relies only on lattice points, you do not need to divide the shape into rectangles or triangles. Step 6 Use Pick’s formula to find the approximate areas of these irregular shapes. 6 in. Dinosaur foot 4 in. Maple leaf Dallas 273 miles San Antonio Texas 1 mi Oil spill 432 CHAPTER 8 Area L E S S O N 8.5 The moon is a dream of the sun. PAUL KLEE You will need ● a compass ● scissors Areas of Circles So far, you have discovered the formulas for the areas of various polygons. In this lesson, you’ll discover the formula for the area of a circle. Most of the shapes you have investigated in this chapter could be divided into rectangles or triangles. Can a circle be divided into rectangles or triangles? Not exactly, but in this investigation you will see an interesting way to think about the area of a circle. Investigation Area Formula for Circles Circles do not have straight sides like polygons do. However, the area of a circle can be rearranged. Let’s investigate. Step 1 Step 2 Step 3 Use your compass to make a large circle. Cut out the circular region. Fold the circular region in half. Fold it in half a second time, then a third time and a fourth time. Unfold your circle and cut it along the folds into 16 wed |
ges. Arrange the wedges in a row, alternating the tips up and down to form a shape that resembles a parallelogram. If you cut the circle into more wedges, you could rearrange these thinner wedges to look even more like a rectangle, with fewer bumps. You would not lose or gain any area in this change, so the area of this new “rectangle,” skimming off the bumps as you measure its length, would be closer to the area of the original circle. If you could cut infinitely many wedges, you’d actually have a rectangle with smooth sides. What would its base length be? What would its height be in terms of C, the circumference of the circle? Step 4 The radius of the original circle is r and the circumference is 2r. Give the base and the height of a rectangle made of a circle cut into infinitely many wedges. Find its area in terms of r. State your next conjecture. LESSON 8.5 Areas of Circles 433 Circle Area Conjecture The area of a circle is given by the formula?, where A is the area and r is the radius of the circle. C-81 How do you use this new conjecture? Let’s look at a few examples. EXAMPLE A The small apple pie has a diameter of 8 inches, and the large cherry pie has a radius of 5 inches. How much larger is the large pie? Solution First, find each area. Small pie A r2 (4)2 (16) 50.2 Large pie A r2 (5)2 (25) 78.5 The large pie is 78.5 in.2, and the small pie is 50.2 in.2. The difference in area is about 28.3 square inches. So the large pie is more than 50% larger than the small pie, assuming they have the same thickness. Notice that we used 3.14 as an approximate value for. EXAMPLE B If the area of the circle at right is 256 m2, what is the circumference of the circle? Solution Use the area to find the radius, then use the radius to find the circumference. 256 m2 C 2r A r 2 256 r 2 256 r 2 r 16 2(16) 32 100.5 m The circumference is 32 meters, or approximately 100.5 meters. 434 CHAPTER 8 Area EXERCISES Use the Circle Area Conjecture to solve for the unknown measures in Ex |
ercises 1–8. Leave your answers in terms of, unless the problem asks for an approximation. For approximations, use the key on your calculator. You will need Geometry software for Exercise 19 1. If r 3 in., A?. 3. If r 0.5 m, A?. 5. If A 3 in.2, then r?. 7. If C 12 in., then A?. 2. If r 7 cm, A?. 4. If A 9 cm2, then r?. 6. If A 0.785 m2, then r?. 8. If C 314 m, then A?. 9. What is the area of the shaded region between the circle and the rectangle? 10. What is the area of the shaded region between the circle and the triangle? y (3, 4) y (–8, 6) (8, 6) x (5, 0) x (10, 0) 11. Sketch and label a circle with an area of 324 cm2. Be sure to label the length of the radius. 12. APPLICATION The rotating sprinkler arms in the photo at right are all 16 meters long. What is the area of each circular farm? Express your answer to the nearest square meter. 13. APPLICATION A small college TV station can broadcast its programming to households within a radius of 60 kilometers. How many square kilometers of viewing area does the station reach? Express your answer to the nearest square kilometer. 14. Sampson’s dog, Cecil, is tied to a post by a chain 7 meters long. How much play area does Cecil have? Express your answer to the nearest square meter. 15. APPLICATION A muscle’s strength is proportional to its cross-sectional area. If the cross section of one muscle is a circular region with a radius of 3 cm, and the cross section of a second, identical type of muscle is a circular region with a radius of 6 cm, how many times stronger is the second muscle? Champion weight lifter Soraya Jimenez extends barbells weighing almost double her own body weight. LESSON 8.5 Areas of Circles 435 16. What would be a good approximation for the area of a regular 100-gon inscribed in a circle with radius r? Explain your reasoning. Review 17. A? 19 cm 24 cm 18. A? 8 ft 10 ft 15 ft 9 ft 19. Technology Construct a parallelogram and a point in its |
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