Geometry Module 9 Characteristics of Geometric Shapes Lesson 3 Circles

Transcription

1 Geometry Module 9 Characteristics of Geometric Shapes Lesson 3 Circles

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3 Objectives Model and identify circle, radius, diameter, center, circumference, and chord. Draw, label, and determine relationships among the radius, diameter, center, and circumference (e.g. radius is half the diameter) of a circle. Teacher Notes 9.3 Model and develop the concept that pi is the ratio of the circumference to the diameter of any circle. Power Send Talk Prerequisites Identifying and naming line segments Rounding numbers to a given place value Multiplying fractions and decimals Voca Circl e bula Equi ry d Plan istant e Cent (8.1) er (o Radi fac u ircle ) Line s segm Chor e nt (8 d.1) Diam eter Infin it Cong e (3.1) ruen Endp t (8. 2 o Circu int (8.1 ) ) mfer Perim en eter ce Pi (7.1) Ratio ( Term 4.1) i Repe nating d ecim ating Irrat a iona decima l (5.5) l num l (5. Prop 5) o Com rtion (7 ber.2) mut ative Mult P Imp iplicatio roperty rope n (1 of. r fra ction 3) (4.1 ) Get Started Divide the class into pairs or groups of three or four. Give each group a compass and a pair of scissors. Have each person in the group use the compass to draw a circle. Tell the students to make circles of different sizes and have students use the point of the compass to mark the center of the circle. Have each student fold their circle in half and make a crease on the fold. Have them open their circle and then, fold the circle in half again but this time making a different crease line than before. Have students make five or six similar creases. Ask what they notice about these creases. Possible answers: They all pass through the center mark. They all appear to be the same length. 9.3A Module 9 Lesson 3 Teacher Notes

4 Measure the line segments created by the folds. Ask what they notice. Possible answers: All the creases are the same length. The center mark divides each crease in half. Subtopic 1 Circles Expand Their Horizons In this subtopic, students learn how to give the formal definition of a circle, how to name a circle, and how to identify parts of a circle. A circle is defined as a set of points on a plane that are equidistant, or the same distance, from a given point. Relate this definition to the circles the students drew with a compass in the Get Started activity. While drawing a circle, the length of the compass opening did not change. Point out that without the phrase, on a plane, the definition would become the definition of a sphere. Show students that a compass is not the only way to make a circle. Tie a string to a piece of chalk and use it to make a circle on the board by holding the empty end of the string in one place. Show how changing the length of the string changes the size of the circle just as the adjusting the length of a compass opening did. Point out that this distance is the radius. The terms radius and diameter each have two meanings. They can refer to the actual segments or to the lengths of those segments. A circle is named by its center point. Knowing the center point is important because without it, it would be impossible to identify radii and diameters. In the figure on the left, the segment may appear to be a diameter, but the same figure on the right, which includes the center point, shows that it is not. Common Error Alert: Students often confuse the terms radius and diameter. Tell students that the prefix dia- means through or across and that a diameter goes all the way through, or across, a circle. For a given circle, the diameter is twice the radius. This translates to the mathematical equation d =2r, which is the formula given in the lesson. Teachers may also want to introduce the formula r = 1 d which translates to the radius is half the diameter. Some 2 students may prefer the latter when they know the diameter and are finding the radius. 9.3B Module 9 Lesson 3 Teacher Notes

5 A radius must have one endpoint at T and the other endpoint on the circle. TR and TB are radii. RB is a diameter because it connects two points on the circle and passes through T. RB, XB, and XR are chords because they each connect two points on the circle. Point E is the center of the circle. Segments EB, EA, EC, and ED are radii because they each have E as one endpoint and a point on the circle as the other endpoint. BC and AD are diameters because they each begin and end on the circle and pass through point E. Because they are diameters, they are also chords. For a given circle, the radius is half the diameter. Divide 30 by two; the radius is 15 feet. A radius can never be a chord because the endpoints of a chord must always lie on the circle while one endpoint of a radius must be the center of the circle. By definition, a diameter is always a chord because its endpoints are on the circle. However, a chord is not always a diameter. A chord is only a diameter if it happens to pass through the center. Additional Examples 1. A circular swimming pool has a radius of 7.5 feet. What is the diameter of the swimming pool? 2. Mandy ordered one medium and one large pizza. The diameter of the medium pizza is 12 inches. The radius of the large pizza is two inches greater than the radius of the medium pizza. What is the diameter of the large pizza? The diameter is twice the radius. d =2r d =2 7.5 d = ft Because the diameter of the medium pizza is 12 inches, the radius is half as much, or six inches. 6in. The diameter is 15 feet. 12 in. The radius of the large pizza is two inches greater than the radius of the medium pizza, which makes the radius eight inches and the diameter 16 inches. 8in. 16 in. The diameter of the large pizza is 16 in. 9.3C Module 9 Lesson 3 Teacher Notes

6 Subtopic 2 Circumference Expand Their Horizons In this subtopic, students learn that pi is the ratio of the circumference of a circle to its diameter. They also use the formula C = d to find the circumference of a circle. Pi is an irrational number. Irrational means not rational; therefore, pi is a nonrepeating and nonterminating decimal. In other words, it goes on forever without repeating. Pi has fascinated mathematicians for over 3,000 years. Many have spent their entire careers approximating the digits of pi. The task became easier with the advent of computers. In 2002, pi was approximated to digits, which is greater than one trillion digits. In this lesson, pi is approximated as 3.14 or 22. Teachers may wish to show students 7 the button on a calculator. Because of the screen space, only the first 10 to 12 digits can be seen. Any calculation that uses pi is an approximation. Diameter of circle: 5 ft. Exact circumference of circle: 5 ft. Approximate circumference of circle: 5(3.14) = 15.7 ft. As more digits of pi are used, the closer the approximation becomes to the actual answer. Circumference, like perimeter, is measured in linear units. If given the option of which approximation to use, students may wish to use the fraction when the diameter is a multiple of seven. This way, the fractions can be simplified. In the lesson, the formula C = d is given. When students know the radius of a circle, they must double it to find the diameter before using the formula. An option is to rewrite the formula with 2r in place of d: C = (2r) orc =2 r. 5 Use the formula C = d. The diameter of the wheel is 28 inches, so C = 28.Use3.14 to approximate. C = inches. Rounding the answer to the nearest whole number gives a circumference of approximately 88 inches. Because 28 is a multiple of seven, students may choose to use 22 7 for C Because the diameter is expressed as a fraction, 22 is an appropriate approximation for 7 1. Change 2 to the improper fraction 5 22 and multiply it by. The circumference is approximately 7 feet. To use 3.14 for, write 2 as D Module 9 Lesson 3 Teacher Notes

7 Additional Examples 5 1. A DVD has a radius of 2 inches. What 16 is the circumference of the DVD? 2. The circumference of a Ferris wheel is approximately 276 feet. What is the diameter of the Ferris wheel? Double the radius to find the diameter. Then, use the formula C = d. 5 d = = = 37 8 C = d Use the formula C = d. Substitute 276 for C and 3.14 for. Then, solve for d. C = d d d 88 d The diameter is approximately 88 feet. 15 The circumference is about 14 inches. 28 Look Beyond Students will use pi later in this course when they find the area of a circle. The area of a circle is pi times the square of the radius. They will also use pi to find the surface area and volume of cones, cylinders, and spheres. In high school geometry, students will study circles and parts of circles in greater detail. They will explore the angles formed by radii and chords, and they will learn about secants and tangents, which are lines that intersect a circle. In statistics, students will use circle graphs to display data. Each category will be a sector of a circle. Connections Tires for cars and light trucks come in different sizes because the wheels are different sizes. Most tires have a code on the side, where the last number given is the diameter of the wheel rim in inches. This number often follows the letter R, but this R does not stand for radius. It means the tire is a radial tire. A common tire code looks like this: P225-75R14. A tire with this code fits a wheel with a diameter of 14 inches. 9.3E Module 9 Lesson 3 Teacher Notes

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9 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles Lesson Notes 9.3 Lesson Objectives Model and identify circle, radius, diameter, center, circumference, and chord. Draw, label, and determine relationships among the radius, diameter, center, and circumference (e.g. radius is half the diameter) of a circle. Model and develop the concept that pi is the ratio of the circumference to the diameter of any circle. Subtopic 1 Circles A circle is the set of points that are equidistant from a special point in the plane called the center. A radius is a line segment that connects the center of the circle to any point on the circle. A chord is a line segment that connects two points on a circle. A diameter is a line segment that connects two points on the circle and passes through the center of the circle. The length of a diameter is twice the length of a radius. 1 Identify the radii, the diameter, and the chords shown in Circle T. R Radii: TR, TB Diameter: RB Chord: RB, XB, XR X T B 27 Module 9 Lesson 3 Lesson Notes

10 2 Identify the radii, the diameters, and the chords shown in circle E. B Radii: EB, EA, EC, ED B A Diameters: BC, AD Chords: BC, AD D E C 3 The diameter of a circle is 30 feet. Find the radius. d =2r 30 = 2r 30 2 = r 30 2 = 15 The radius of the circle is 15 feet. 4 Tell whether each statement is always true, sometimes true, or never true. A radius is a chord. Never A diameter is a chord. Always A chord is a diameter. Sometimes Subtopic 2 Circumference The circumference of a circle is the distance around the circle. Pi is the ratio of the circumference of any circle to its diameter. Pi ( ) Irrational number Approximately 3.14 or 22 7 Module 9 Lesson 3 28 Lesson Notes

11 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles 5 The diameter of a bike wheel is 28 inches. What is the circumference? Round to the nearest inch. C d C C The circumference of the bike wheel is about 88 inches. 6 1 The diameter of a manhole cover is 2 ft. What is the circumference? 2 C d C 11 C = The circumference of the manhole cover is about feet. Module 9 Lesson 3 29 Lesson Notes

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13 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles Guided Practice 9.3 Set 1 1 Identify the radii, the diameter, and the chords shown in Circle N. Radii: NM, NR, NQ M Q Diameter: QR Chords: QR, VS R V N S 2 Identify the radii, the diameter, and the chords shown in Circle W. Radii: WY Chords: VY andwz and VZ Y Z X W V 3 The diameter of a compact disc is 120 millimeters. Find the length of the radius. d =2r 120 = 2r = r = 60 The radius of the compact disc is 60 mm. Module 9 Lesson 3 30 Guided Practice

14 4 Tell whether each statement is always true, sometimes true, or never true. Chords in the same circle are congruent. Sometimes A diameter passes through the center of a circle. Always Set 2 1 The diameter of a coin is 35 mm. What is the circumference? Round to the nearest millimeter. C d C C The coin s circumference is about 110 mm. 2 The radius of the lens of a magnifying glass is 38 millimeters. What is the circumference? Round to the nearest millimeter. C d C C The circumference is about 239 mm. 3 The radius of a circle is inch inches. What is the circumference? Round to the nearest d 2r d C d C The circumference of the circle is about 39 inches. Module 9 Lesson 3 31 Guided Practice

15 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles Challenge Problems 9.3 Set 1 1 Use a calculator to find the value of 22 to six decimal places. Using the key on a 7 calculator, find the value of rounded to six decimal places. Then, order 22 7,, and 3.14 from least to greatest. 2 Explain how to estimate the diameter of a tree trunk if its circumference is 60 inches. 3 Determine if this statement is true or false and explain: If the diameter of a circle is doubled, then the circumference is doubled. Module 9 Lesson 3 32 Challenge Problems

16 Possible Answers Set = < < Use the relationship, circumference divided by diameter equals pi. Round the value of pi to three. Sixty divided by the diameter is about three. That means that the diameter must be about 20 inches. C d This statement is true d d 20 inches Testing with a circle whose diameter is five: C = (5) 15.7 C = (10) = 31.4 Rewriting the formula: C = d C = 2d =2 d Module 9 Lesson 3 33 Challenge Problems

17 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles Independent Practice 9.3 Identify the radii, diameters, and chords shown in each circle. 1. Circle O B C 2. Circle C T X Y G O F D R C U Radii: OG, OB, OF Diameter: BF Chords: CD, BF Radii: none Diameters: none Chords: RY and TU The length of a radius, r,ordiameter,d, is given. Find the missing measure. 3. d =61m 4. r = 1 4 ft r =? d =? r = 30.5 m d = 1 2 ft In each circle, either a radius or diameter is shown. Find the circumference. Round to the nearest inch in. 200 in. About 94 inches About 628 inches 34 Module 9 Lesson 3 Independent Practice

18 Tell whether each statement is always true, sometimes true, or never true. 7. A chord is a radius. Never true 8. Diameters in the same circle are congruent. Always true 9. Chords pass through the center of a circle. Sometimes true 10. A merry-go-round is 630 inches in diameter. Use 22 7 circumference of the merry-go-round. for to approximate the About 1,980 inches. 11. The diameter of a large pizza is 16 inches. To the nearest inch, what is the circumference of the pizza? About 50 inches 12. The circumference of a bowl is about 66 centimeters. To the nearest centimeter, what is the diameter of the bowl? About 21 centimeters 35 Module 9 Lesson 3 Independent Practice

19 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles Use the circle below for problems Draw and label the center point P. 14. Draw and label diameters JT and AM. J A 15. Draw and label chord HK so that it is not a diameter. 16. Name all the radii shown in circle P. H P K PJ, PA, PT,and PM M T Journal 1. Tell how chords and diameters are alike. Tell how they are different. 2. Describe the relationship between a radius and diameter of the same circle. How can youfindoneifyouaregiventheother? 3. Explain what pi represents in a circle. Give two approximations for pi. Then, explain which approximation would be most appropriate for estimating the circumference of a circle with a diameter of 10 feet and which would be most appropriate for estimating a circle with a diameter of 14 feet. Cumulative Review Use the diagram on the right for Problems 1 6. A T 1. What point is coplanar with points M, A, ande? M E H L Point S S P 2. Describe MA and HT as parallel, perpendicular, or neither. Parallel 36 Module 9 Lesson 3 Independent Practice

20 3. Describe EL and HP as parallel, perpendicular, or neither. Neither A T 4. Describe HP and PL as parallel, perpendicular, or neither. M E H L Perpendicular S P 5. Classify PSL. Acute 6. The opposite sides of parallelogram PSEL are congruent. Tell why PLS ESL. SSS Congruence Tell if each figure is a polygon. If so, classify it by its number of sides and tell if it is concave or convex Concave pentagon Not a polygon Convex octagon Not a polygon 37 Module 9 Lesson 3 Independent Practice

21 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles Possible Journal Answers 1. Chords and diameters are both line segments, and they both have their endpoints on a circle. A chord may or may not pass through the center of the circle. A diameter always passes through the center of the circle. 2. In the same circle, the length of a radius is half the length of a diameter. If given the length of the radius, double it to find the length of the diameter. Because a diameter is twice as long as a radius, if given the length of the diameter, divide it by two to find the length of the radius. 3. In a circle, pi represents the ratio of the circumference to its diameter. The ratio is the same for any circle. Pi is an irrational number but can be approximated by 3.14 or 22. The circumference of a circle can be found by multiplying pi times 7 the diameter of the circle: C = d. For a circle with a circumference of 10 feet, the decimal approximation would be most appropriate because the circumference can be found by simply moving the decimal point one place to the right: C 3.14(10) = 31.4 feet. For a circle with a circumference of 14 feet, the fraction approximation would be most appropriate because the numerator and denominator have a common factor: C 44 ft Module 9 Lesson 3 Independent Practice

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23 NAME Module 9 Lesson 3 Characteristics of Geometric Shapes Circles Additional Practice 9.3 Use the circle at right for questions 1 6. Point Q is the center of the circle. 1. Name the circle. A B Circle Q 2. Name all the chords shown in the circle. Q AD and BC 3. Name all the diameters shown in the circle. F AD E D C 4. Name all the radii shown in the circle. QA, QF, QE, and QD 5. Classify FQE by its sides. Explain why it is classified in this way. Isosceles: QF and QE are radii, and all radii in the same circle are congruent. An isosceles triangle has at least two congruent sides. 6. Find the length of AD if EQ =6.5cm. AD =13cm Find the circumference of each circle. Round to the nearest tenth mm About 50.2 mm 42 in. About inches 39 Module 9 Lesson 3 Additional Practice

24 9. The circumference of a circular rug is about 57 inches. To the nearest inch, what is the radius of the rug? The radius of the rug is about nine inches. 10. The two circles on the right have the same center point. The diameter of the smaller circle is seven meters, and AB is 10 meters long. Find the circumference of the larger circle. Round to the nearest meter. The circumference of the larger circle is about 85 meters. A B 40 Module 9 Lesson 3 Additional Practice