Honors Geometry Chapter 6 Tentative Syllabus

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1 Honors Geometry hapter 6 Tentative Syllabus YOU R XPT TO HK YOUR NSWRS FOR OMING TO LSS. Red ay Red ate lue ay lue ate Topic Homework (due next class period) Fri Nov 20 Mon Nov Tangent Properties 6.2 hord Properties In-class: 6.2 Wkbk p.40, # Wkbk: p.39, #1 6 p.320, #1 13, Tue Nov 24 Mon Nov rcs and ngles 6.4 Other ngles In-class: 6.3 Wkbk p.41 ircle ngle hase #1 (Notes p.20) p.327, #1 16, 22 24, 26 ircle ngle hase #2 & #3 (Notes p.21 and 22) Tue ec 1 Wed ec ircumference & iameter 6.6 round the World In-lass: 6.5 Wkbk p.43 p.337, #1 13 p.342, #2, 4, 6 Review for Final xam #1 ( U on exam day) Thurs ec 3 Fri ec rc Length In-class: 6.7 Wkbk p.45 p , 17 ircle ngle hase #4 (Notes p.23) Mon ec 7 Tue ec Indirect Proof Notes Review: hp 6 Practice Test p , 25 28, 31, 32, 35, 38, 46, 47, 52, 53, 62, 63 Wed ec 9 Thurs ec 10 hapter 6 Test Indirect Proof Practice Final xam Review #2 (ue day of Final xam) Fri ec 11 Mon ec 14 Final xam Review Study for Final xam Tue ec 15 Wed ec 16 njoy Winter reak! reated by W.L. ass and used with permission, p.1

2 Section Indiana Standard Learning Target Gl1 efine, identify, and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, and congruent concentric circles. Gl3 Identify and describe relationships among inscribed angles, radii, and chords, including the following: the relationship exists between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; and the radius of a circle is perpendicular to a tangent where the radius intersects the circle. Gl6 onstruct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Review and use basic properties of a circle and circle vocabulary. iscover and use properties of tangents of circles. onstruct a tangent line (Investigation 2) iscover and use properties of chords in a circle. (p.322 #23) 6.3 Gl1 efine, identify, and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, and congruent concentric circles. Gl4 Solve real-world and other mathematical problems that involve finding measures of circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed, and intersections of secants and tangents). 6.4 GLP4 evelop geometric proofs, including direct proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry, using two-column, paragraphs, and flow charts formats. 6.5 Gl1 efine, identify, and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, and congruent concentric circles. 6.6 Gl4 Solve real-world and other mathematical problems that involve finding measures of circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed, and intersections of secants and tangents). GQP5 educe formulas relating lengths and sides, perimeters, and areas of regular polygons. Understand how limiting cases of such formulas lead to expressions for the circumference and the area of a circle. 6.7 Gl4, Gl5 onstruct a circle that passes through three given points not on a line and justify the process used. iscover and use relationships between an inscribed angle of a circle and its intercepted arc. onstruct a cyclic quadrilateral. Know and use that the opposite angles of a cyclic quadrilateral are supplementary. (Investigation 4) Prove and use circle conjectures. alculate pi, the ratio of the circumference of a circle to its diameter. pply circle properties and the circumference of a circle to solve problems. (p.343 #9) iscover and use a formula for finding the length of an arc of a circle. (p.354 #24) reated by W.L. ass and used with permission, p.2

3 6.1 and 6.2 Notes Tangent and hord Properties ig ideas about TNGNTS! 1. tangent and a radius are perpendicular to each other at the point of tangency. Label the picture based on the property. 3. Recall: The measure of an arc is equal to the measure of its central angle. If m 115 then m = 4. Recall: ll the radii (plural of radius) are congruent to each other in a circle. Label the picture based on the property. 2. Tangent segments from the same point outside a circle are congruent. Label the picture based on the property. pply the tangent theorems ssume that lines that appear tangent are tangent x x 65 x x 106 x 83 x reated by W.L. ass and used with permission, p.3

4 7. One more application a.) Write an equation for the tangent line. Find slope of the radius Find slope of the tangent line (Hint: the radius and tangent are perpendicular) Use the point-slope formula to write the equation for the tangent line. y y m x x 1 1 Rewrite the equation in slope-intercept form b.) YOU TRY Write an equation for the tangent line. little more vocabulary Internally and xternally Tangent ircles are considered internally or externally tangent when they are to the same at the same. reated by W.L. ass and used with permission, p.4

5 ig ideas about HORS 1. VOULRY entral angle v. Inscribed ngle. entral ngle n angle that has its vertex on the of the circle Inscribed ngle n angle that has its vertex the circle N the sides of the angle are of the circle. Use the definitions above and the diagram at the below name the following K O N H U M H O R T entral ngles: Inscribed ngles: 2. If two chords in a circle are congruent, then their central angles are congruent. Use your compass to show GH raw in radii F, F, FG, FH F GFH by Therefore, F GFH by 3. If two chords in a circle are congruent, then their arcs are congruent. From the proof above in ig Idea #2, we saw that F GFH Recall: The measure of an arc is to the measure of its central angle. Therefore, if the central angles are congruent, then GH Label the picture based on the ig Ideas 2 & 3. F G H reated by W.L. ass and used with permission, p.5

6 4. The perpendicular from the center of a circle to a chord bisects the chord. onstruct a perpendicular from F to and label the intersection M. Use your compass to show M M onstruct a perpendicular from F to GH and label the intersection P Use your compass to show GP PH 5. If two chords in a circle are congruent, then they are equidistant from the center. Recall: istance from a point to line/segment must be measured on a line. Since we ve already constructed the perpendicular in ig Idea #4 Use your compass to show FM FP. Label the picture based on ig Ideas 4 & 5. F G H 6. The perpendicular bisector of a chord is also the diameter of the circle. onstruct the perpendicular bisector of NO Name the bisector m. onstruct the perpendicular bisector of PQ Name the bisector p. Label the intersection of m and p point F. Summary: oth and contain diameters of the circle. N Point is the of the circle! O P Q reated by W.L. ass and used with permission, p.6

7 6.3 Notes rcs and ngles Review/Warm-up Write the equation for the tangent line shown below. 6.3 IG IS! 1. VOULRY entral angle v. Inscribed ngle. entral ngle n angle that has its vertex on the of the circle Inscribed ngle n angle that has its vertex the circle N the sides of the angle are of the circle. 2. The measure of an inscribed angle is one-half the measure of its intercepted arc. Use a piece of patty paper to trace 2m N raw central angle OIP 2 m N to m I ompare Recall: The measure of an arc is to the measure of its central angle. Summary: mi 1 2 mi 2 mn Measure of Inscribed ngle = Measure of Intercepted rc = O N I P reated by W.L. ass and used with permission, p.7

8 3. Inscribed angles that intercept the same arc (or congruent arcs) are congruent. The intercepted arc for O is The intercepted arc for P is Use a piece of patty paper to show O P What about N and R? N R 4. Parallel lines (segments) intercept congruent arcs. If ST // WU, then SW Why? raw transversal SU. TSU WUS because Look again at ig Idea #3... ongruent rcs ongruent angles I S T O P W U 5. ngles inscribed in a semicircle are right angles. raw inscribed YWZ What s the measure of its intercepted arc? Therefore mw raw inscribed ZXY What s the measure of its intercepted arc? Therefore mz Y W Z What kind of triangles are YWZ and ZXY?. re they congruent to each other?. re they isosceles?. X reated by W.L. ass and used with permission, p.8

9 6. The opposite angles in a cyclic quadrilateral are supplementary. efinition: yclic quadrilateral N xamine mn m R What do you notice? Therefore I P N and are supplementary Use a piece of patty paper to trace mo m P O What do you notice? Therefore O and are supplementary R pply what you have learned Solve for the variable in each diagram. 1. x x 28 x x x y 8. y x x x Hint: Write the formula first (2x + 11) x (5x + 4) 208 (4x - 3) reated by W.L. ass and used with permission, p.9

10 h 6 xploration Other ngle Measures Match the conjecture to the appropriate diagram below then write an equation to find in marked angle. 1. Intersecting Secants onjecture: The measure of an angle formed by 2 secants that intersect the circle is 2. Intersecting hords onjecture: The measure of angle formed by 2 intersecting (or 2 secants intersecting inside) is 3. Tangent-Secant onjecture: The measure of an angle formed by an intersecting tangent and a secant to a circle is 4. Intersecting Tangents onjecture: The measure of angle formed by 2 intersecting to a circle is 5. Tangent hord onjecture: The measure of an angle formed by the intersection of a tangent and a chord at the point of tangency is. Z. T. 1 1 P S 1 T T reated by W.L. ass and used with permission, p.10

11 pply what you know 1. Finding ngle and rc Measures a.) Given: m = 54, mark this in the picture. b.) m = c.) Find the intercepted arc for and m = d.) Find the intercepted arc for and m = e.) What do you notice about and? 2. Finding the Measure of an ngle Formed by Two hords a.) Solve for x. b.) Solve for x. 24 x x is not the center! is not the center! reated by W.L. ass and used with permission, p.11

12 3. Finding angles and arcs when intersection is OUTSI the circle. a..) Solve for x. Mark the intercepted arcs x b.) Solve for x. Mark the intercepted arcs 50 x 114 c.) Solve for x. Mark the intercepted arcs m =, x 56 d.) Solve for x. Mark the intercepted arcs m =, 36 x e.) Solve for x. Mark the intercepted arcs m =, x 125 f.) Solve for x. Mark the intercepted arcs m =, 6.5 and 6.6 Notes 52 x reated by W.L. ass and used with permission, p.12

13 ircumference/iameter, round the World 6.5 ircumference onjecture If is the circumference and d is the diameter of a circle, then there is a number such that If r is the radius then. d and 1. If 7 m, find the radius. 3. If 36 m, find the radius. 2. If d 8.5 m, find the circumference (no decimals). 4. If d 6.25 m, find the circumference (no decimals) 6.6 Practice 5. satellite in a nearly circular orbit is 2000 km above the arth s surface. The radius of the arth is approximately 6400 km. If the satellite completes its orbit in 12 hours, calculate the speed of the satellite in km per hour. 6. 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? More than a billion? (1 km = 1000m) 7. Goldi s Pizza Palace is known throughout the city. The small aby ear pizza has a 6-inch radius and sells for $9.75. The savory medium Mama ear pizza sells for $12.00 and has an 8 inch radius. The large Papa ear pizza is a hefty 20 inches in diameter and sells for $ The edge is stuffed with cheese and it s the best part of the pizza. What size has the most pizza edge per dollar? What is the circumference of this pizza? 6.7 Notes rc Length reated by W.L. ass and used with permission, p.13

14 efinition 6.7 Finding rc Length v. rc Measure The of an arc is equal to Proportion Find the indicated measures. 1. U 50 9 H mgh = mhug = length of HUG = G reated by W.L. ass and used with permission, p.14

15 2. U myp = myup = length of YP = Y P 3. WORKING KWRS Given: P = 16π R 70 P mt = mrt = iameter = T reated by W.L. ass and used with permission, p.15

16 13.5 Notes Indirect Proof General steps for an indirect proof.. 1. Given: 1 is not to 2 Prove: p is not // n p n 2 1 Statement a.) b.) c.) d.) Reason a.) b.) c.) d.) 2. Given: m1 67 X Prove: X is not to 1 Statement a.) b.) c.) d.) e.) Reason a.) b.) c.) d.) e.) reated by W.L. ass and used with permission, p.16

17 3. Given: 1 is not to 2 Prove: is not isosceles with vertex angle Statement Reason 1 2 a.) b.) c.) d.) a.) b.) c.) d.) 4. Given: OJ OK Prove: J is not to K O does not bisect JOK O 1 2 J K Statement Reason a.) b.) c.) d.) a.) b.) c.) d.) e.) OJ e.) f.) g.) h.) f.) g.) h.) reated by W.L. ass and used with permission, p.17

18 ircle ngle hase # Given: is a diameter For each problem, determine the measure of the arc or angle. e careful to look where your vertex is. One mistake can lead to many! Write your answers on the blanks. Write your answers on the blanks provided. 1. m 2. m 3. m 4. m 6. m 7. m 8. m 9. m 5. m reated by W.L. ass and used with permission, p.18

19 ircle ngle hase #2 #9 #1 #2 #5 #8 #6 # #4 #7 # 10 # 11 # 12 # 13 #3 Given: is a diameter For each problem, determine the measure of the arc or angle. e careful to look where your vertex is. One mistake can lead to many! Write your answers on the blanks provided reated by W.L. ass and used with permission, p.19

20 ircle ngle hase #3 #4 #8 #1 #7 #3 #2 #9 # 15 # # 11 # 10 # 13 # #6 #5 Given: // For each problem, determine the measure of the arc or angle. e careful to look where your vertex is. One mistake can lead to many! Write your answers on the blanks provided reated by W.L. ass and used with permission, p.20

21 ircle ngle hase #4 #1 #11 #8 #5 #9 #12 #2 44 #3 #13 #14 #10 18 #4 42 #6 #7 For each problem, determine the measure of the arc or angle. e careful to look where your vertex is. One mistake can lead to many! Write your answers on the blanks provided reated by W.L. ass and used with permission, p.21

22 1. HPTR 6 IRL SUMMRY Other things to remember ongruent chords ongruent arcs How to find the center of a circle with only 3 points How to write the equation of a tangent line. ircumference Formula rc Length reated by W.L. ass and used with permission, p.22

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