Tangents and Chords Off On a Tangent


 Arron Green
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1 SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Create Representations, Quickwrite circle is the set of all points in a plane at a given distance from a given point in the plane. Lines and segments that intersect the circle have special names. The following illustrate tangent lines to a circle. 1. On the circle below, draw three unique examples of lines or segments that are not tangent to the circle. nswers may vary. Sample answer: CDEMIC VOCBULRY circle CDEMIC VOCBULRY CTIVITY 4.1 tangent is a line in the plane of a circle that intersects the circle at just one point, called the point of tangency. Investigative ctivity Focus Relationships among tangent lines and diameters in a circle Relationships involving chords of a circle Materials Rulers Scissors BLM 12: Problem 10: Statements for a Proof BLM 13: Problem 10: Reasons for Proof Statements 2. Write a description of tangent lines. line is tangent to a circle if and only if it intersects the circle in exactly one point. 3. Using the circle below, a. Draw a tangent line and a radius to the point of tangency. nswers may vary. Sample answer: b. Describe the relationship between the tangent line and the radius of the circle drawn to the point of tangency. tangent line is perpendicular to a radius that is drawn to the point of tangency. MTH TERMS secant is a line that intersects the circle in two points. MTH TERMS The radius of a circle is a segment, or length of a segment, from the center to any point on the circle Chunking the ctivity #1 2 #9 #14 #3 #10 #15 #4 6 #11 #7 8 # Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Debriefing Have students brainstorm individually before comparing their responses with their group and presenting to the class. Once the two items have been debriefed, the terms tangent line, secant, and point of tangency should be added to the interactive word wall and students should add them to their vocabulary organizer. Unit 4 Circles and Constructions Create Representations, Quickwrite This item is meant as a quick recognition of the theorem: If a radius is drawn to a point of tangency, the radius and tangent line are perpendicular. Unit 4 Circles and Constructions 277
2 Continued 46 Create Representations Before beginning this section, students may need to be reminded about the definitions of chord, arc, and perpendicular bisector. Directing students to use their vocabulary organizer will teach them that when they don t know a definition they should look it up. In Item 4, it is important that students use the outside edge of the wheel. In Item 5, the placement of the two chords is irrelevant. In Item 6, students should be careful when measuring to find the midpoint of each chord before drawing the perpendicular bisectors. Students should record the theorem stated after Item 6 in their notebooks, as this theorem will be important in solving problems throughout Unit Use Manipulatives, Note Taking, Quickwrite, Debriefing To determine the diameter of the scale drawing of the wheel, students must measure the radius of the arc (the distance from the point of intersection of the perpendicular bisectors to the arc). To determine the diameter of the actual wheel, students can write and solve a proportion that uses the measurements made in Item 7 and the scale factor. MTH TERMS The diameter is a segment, or the length of a segment, that contains the center of a circle and two end points on the circle. CDEMIC VOCBULRY chord is a segment whose endpoints are points on a circle. CDEMIC VOCBULRY n arc is part of a circle consisting of two points on the circle and the unbroken part of the circle between the two points. 278 SpringBoard Mathematics with Meaning TM Geometry SUGGESTED LERNING STRTEGIES: Create Representations, Use Manipulatives, Notetaking, Quickwrite Chuck Goodnight dug up part of a wooden wagon wheel. n authentic western wagon has two different sized wheels. The front wheels are 42 inches in diameter while the rear wheels are 52 inches in diameter. Chuck wants to use the part of the wheel that he found to calculate the diameter of the entire wheel, so that he can determine if he has found part of a front or rear wheel. scale drawing of Chuck s wagon wheel part is shown below. 4. Trace the outer edge of the portion of the wheel shown onto a piece of paper. Students should follow the steps as given. 5. Draw two chords on your arc. Students should follow the steps as given. 6. Using a ruler, draw a perpendicular bisector to each of the two chords and extend the bisectors until they intersect. Students should follow the steps as given. The perpendicular bisectors of two chords in a circle intersect at the center of the circle. 7. Determine the diameter of the circle that will be formed. Explain how you arrived at your answer. The diameter of the circle containing the given arc is 3.5 inches. Students will arrive at this answer by measuring the distance from the point of intersection of the two perpendicular bisectors created in Item 6 to a point on the circle, the radius, then doubling it. 8. The scale factor for the drawing is 1:12. Determine which type of wheel can contain the part Chuck found. Justify your answer. front wheel can contain the part that Chuck found. Solving the proportion 1 12 = 3.5 x gives the diameter of the wheel as 42 inches, which is the diameter of the front wheel. 278 SpringBoard Mathematics with Meaning Geometry
3 SUGGESTED LERNING STRTEGIES: Look for a Pattern, Use Manipulatives, Quickwrite 9. In the circle below: Draw a diameter. Draw a chord that is perpendicular to the diameter. nswers may vary. Sample answer: a. Use a ruler to take measurements in this figure. What do you notice? Using their measurements, students should recognize that the diameter bisects the chord. b. Compare your answer with your neighbor s answer. What conjecture can you make based on your investigations of a diameter perpendicular to a chord? When a diameter is perpendicular to a chord, the diameter bisects the chord. Continued 9 Look for a Pattern, Use Manipulatives, Quickwrite Students will use a ruler to create an illustration of and investigate the following theorem: If a diameter is perpendicular to a chord, then it bisects the chord. Students should be encouraged to compose their own conjectures before comparing with a neighbor and with the group. Suggested ssignment CHECK YOUR UNDERSTNDING p. 284, #1 3, 7 UNIT 4 PRCTICE p. 343, #1 2 Unit 4 Circles and Constructions 279 Unit 4 Circles and Constructions 279
4 Continued 0 Debriefing, Create Representations, Note Taking, Self/Peer Revision fter conducting a short class discussion about the term equidistant, distribute BLM 12 and have students cut out the statements of the proof. Next, have students work in pairs to correctly arrange these. Then have each pair compare their work with that of other students by rotating to a new pair s work and examining that work without changing anything. fter a few rotations, students should return to their own statements and make any changes that they think are necessary. Students should be aware that there is more than one correct arrangement of the statements in this proof. It is very important to do the debriefing before having students move on to Item 11, in which students will arrange the reasons for the proof. Statements 1. Draw radii RB and RD. 2. RB RD 3. B CD ; RX CD ; RY B 4. B = CD 5. 1 B = CD 6. BY = 1 B; DX = CD 7. BY = DX 8. BY DX 9. DXR and BYR are right angles. 10. DXR and BYR are right triangles 11. DXR BYR 12. RY RX MTH TERMS If two segments are the same distance from a point, they are equidistant from it. 280 SpringBoard Mathematics with Meaning TM Geometry SUGGESTED LERNING STRTEGIES: Create Representations, Notetaking, Self/Peer Revision 10. For the theorem below, the statements for the proof have been scrambled. Your teacher will give you a sheet that lists these statements. Cut out each of the statements and rearrange them in logical order. R Theorem: In a circle, two congruent chords are equidistant from the center of the circle. Given: B CD ; RX CD RY B Prove: RY RX Y D Draw radii RB and RD. RY RX B CD ; RX CD ; RY B B = CD DXR and BYR are right triangles. RB RD 1 1 B = 2 2 CD BY DX DXR and BYR are right angles. BY = DX X B BY = 1 1 B; DX = 2 2 CD DXR BYR nswers may vary. See sample proof below. C 280 SpringBoard Mathematics with Meaning Geometry
5 Continued SUGGESTED LERNING STRTEGIES: Create Representations, Self/Peer Revision 11. The reasons for the proof in Item 10 are scrambled below. Your teacher will give you a sheet that lists these reasons. Cut out each of the reasons and rearrange them so they match the appropriate statement in your proof. Through any two points there is exactly one line. Definition of right triangle Definition of congruent segments Definition of congruent segments Multiplication Property C.P.C.T.C. HL Theorem Given Definition of perpendicular lines ll radii of a circle are congruent. diameter perpendicular to a chord bisects the chord. Substitution Property nswers may vary. See sample proof below. Unit 4 Circles and Constructions 281 a Debriefing, Create Representations, Self/Peer Revision Distribute BLM 13. Using the same strategy as with the statements in Item 10, have students cut out the reasons for the proof, work in pairs to arrange those reasons to correspond to their statements, and then rotate to compare their work with that of other pairs of students. fter students have revised their proofs and you have debriefed the class or checked the work of each pair, have students copy their proof into their notebooks, or attach the statements and corresponding reasons to a page in their notebooks. Reasons 1. Through any two points there is exactly one line. 2. ll radii of a circle are congruent. 3. Given 4. Definition of congruent segments 5. Multiplication Property 6. diameter perpendicular to a chord bisects the chord. 7. Substitution Property 8. Definition of congruent segments 9. Definition of perpendicular lines 10. Definition of right triangle 11. HL Theorem 12. C.P.C.T.C. Unit 4 Circles and Constructions 281
6 Continued bc Quickwrite, Debriefing, Think/Pair/Share The purpose of these questions is to extend the theorem in Item 10 to include congruent circles. Students should be given the opportunity to think about Items 12 and 13 before discussing a solution with a partner. llowing students to compare answers with classmates will honor multiple correct solutions to these statements. SUGGESTED LERNING STRTEGIES: Quickwrite, Think/Pair/Share E 12. Given EF B, explain how you know that EF and B are not equidistant from the center, R. nswers may vary. Sample answer: EF is a chord of the inner circle with center R, while B is outside the inner circle. So, the distance from B to R is greater than the distance from EF to R. R F B 13. Michael said that if two chords are the same length but are in different circles that are not necessarily concentric circles, then they will not be the same distance from the center of the circle. Is he correct? If he is, give a justification. If not, give a counterexample. Michael is wrong. If the two circles have the same radius, then the two congruent chords will be the same distance from the centers of the circles. 282 SpringBoard Mathematics with Meaning TM Geometry 282 SpringBoard Mathematics with Meaning Geometry
7 Continued SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Create Representations, Self/Peer Revision Theorem: The tangent segments to a circle from a point outside the circle are congruent. 14. Use the theorem above to write the prove statement for the diagram below. Then, prove the theorem. nswers may vary. Students may choose which type of proof to write. sample twocolumn proof is shown. Given: BD and DC are tangent to circle. Prove: DB DC Statements 1. BD and DC are tangent to circle 2. B BD, C CD 3. DB and DC are right angles. 4. DB and DC are right triangles. 5. B C 6. D D 15. In the diagram, RT = 12 cm, RH = 5 cm, and MT = 21 cm. Determine the length of RM. Explain how you arrived at your answer. T 1. Given B C Reasons 2. Tangent lines are perpendicular to the radius drawn to the point of tangency. 3. Definition of perpendicular 4. Definition of right triangle 5. ll radii of a circle are congruent. 6. Reflexive Property 7. DB DC 7. HL Theorem 8. DB DC 8. C.P.C.T.C. D MTH TERMS tangent segment to a circle is part of a tangent line with one endpoint outside the circle and the second endpoint at a point of tangency to the circle. de Think/Pair/Share, Create Representations, Self/Peer Revision The proof of this theorem involves congruent triangles. Encourage students to write a plan before beginning the proof. lthough the answer key shows a twocolumn proof, the proof can be written in paragraph, flow chart or twocolumn format. It is important that students understand the implications of the theorem in Item 14 and how it can be applied in problem situations. Item 15 requires students to apply this theorem. 15. RM = 19 cm. Since two tangent segments to a circle from a point outside the circle are congruent, RD = RH = 5 cm. Similarly, T = HT, which means that T = (RT  RH) = 12 cm  5 cm = 7 cm. M = MT  T = 21 cm  7 = 14 cm. Since M and MD are both tangent to the circle from point M, M = MD = 14 cm. RM = MD + RD = 14 cm + 5 cm = 19 cm. H R O M See answer and explanation given at the right. D Unit 4 Circles and Constructions 283 Unit 4 Circles and Constructions 283
8 Continued Suggested ssignment CHECK YOUR UNDERSTNDING p. 284, #4 6 UNIT 4 PRCTICE p. 343, # nswers may vary. correct answer will consist of a circle and a line that does not intersect the circle cm cm inches 7. nswers may vary. Sample answer: Draw the radii of the circle to the endpoints of the chord. Because all radii in a circle are congruent, the triangle formed is isosceles. In an isosceles triangle an altitude to the base is also a median; therefore, the diameter perpendicular to the chord must also bisect the chord. CHECK YOUR UNDERSTNDING Write your answers on on notebook paper. paper. Show Show your work. 5. In the diagram below, H, D, and DH are your work. each tangent to circle Q. T = 9, H = 13, and D = 15. HD =?. 1. Draw a circle. Draw a line that is neither a secant line nor a tangent line to the circle. H T 2. In the diagram, TP is tangent to circle D. Determine m DTP. T D In the diagram, C is tangent to circle P, the radius of circle P is 8 cm and BC = 9 cm. C =?. P 4. In the diagram, NM and QN are tangent to circle P, the radius of circle P is 5 cm, and MN = 12 cm. QN =?. P M Q B P C N D Q M 6. Suppose a chord of circle is 5 inches from the center and is 24 inches long. Find the length of the radius of the circle. 7. MTHEMTICL Explain how to prove the REFLECTION following conjecture: If a diameter is perpendicular to a chord, then the diameter bisects the chord. 284 SpringBoard Mathematics with Meaning TM Geometry 284 SpringBoard Mathematics with Meaning Geometry
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