Hampton High School Geometry Lesson Plan Tangents of Circles Michelle Bousquet

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1 Hampton High School Geometry Lesson Plan Tangents of Circles Michelle Bousquet Context: This lesson is designed for a 90 minute Geometry class at Hampton High School. The class contains a maximum of 19 students. This is the first of five lessons on tangents, chords, secants, and the equation of a circle. The class has already had five lessons, including a test, on angles and arc measures in a circle. Objective: The students will correctly answer the questions on a worksheet pertaining to tangents to a circle as checked by the teacher by the end of the class period. If there is time, the students will correct solve an application problem using tangents to a circle as checked by the teacher by the end of the class period. SOL: G.11 The student will use angles, arcs, chords, tangents, and secants to: a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles Materials/Resources: All worksheets provided Content Instructional Strategies: 1. Warm up: Attached. The warm up is a review of the Pythagorean Theorem, which will be used during class. While the students are completing the warm-up, take attendance. After going over the warm up, briefly introduce this new part of the Circle Unit. The students will not be learning about angle measures pertaining to circles, but rather they will be learning about lengths. 15 minutes 2. Introduce concepts of tangent line, secant line, and chord by drawing a circle on the board. 5 minutes 3. Real world connection to what we will learn: Hammer Throw. Before starting the video, describe how the athletes will be using tangent lines to a circle to throw the hammer, a metal ball that usually weighs about 16 pounds. 5 minutes. Begin Circles: Tangents Worksheet. 10 minutes 5. Pass out the Tangents to Circles Worksheet (2 pages) by Kuta Software and go through exercises 1-12 as examples with the class. 20 minutes 6. Tangents to a Circle worksheet (1 page). The students will work individually or with their groups to complete the worksheet. These problems are more advanced than the examples and will likely require hints or assistance. As groups complete the worksheet, have them raise their hands to check the answers with the teacher.

2 FOR ADVANCED STUDENTS, STUDENTS THAT FINISH EARLY, OR IF THERE IS EXTRA TIME: In book pg 557, #29, minutes 7. Review what the class learned and answer any questions. 5 minutes Evaluation: By accurately completing the Tangents to a Circle practice worksheet as checked by the teacher, the teacher will have a good idea as to what concepts the students are still struggling with. Also, the teacher will evaluate student progress by accurate problem solving of the connection to real life. Differentiation and Adaptations: If there is not as much time as anticipated, the class can complete the last worksheet for homework or the teacher can do the questions with the class. If there is extra time or there are advanced students, the students will complete the astronomy question provided. Sources: Warm Up questions: Boyd, C. J., Cummins, J., Malloy, C., Carter, J., & Flores, A. (2005). Geometry (pp. 551). Columbus, OH: The McGraw-Hill Companies, Inc. Tangents to Circles Worksheet:

3 Kuta Software LLC. (2012, December 8). Tangents to circles. Retrieved from Tangents%20to%20Circles.pdf 1-2 Tangents to a Circle Worksheet: Glencoe/McGraw-Hill. (2001). Geometry concepts and applications. Retrieved from f Astronomy Question: Boyd, C. J., Cummins, J., Malloy, C., Carter, J., & Flores, A. (2005). Geometry (pp. 557). Columbus, OH: The McGraw-Hill Companies, Inc.

4 Name: Date: Warm Up Determine whether each figure is a right triangle in. in. 5 in m 10 m 3 m 5 ft ft 28 ft

5 Name: Date: Circles: Tangents Tangent Lines Definition: A line in the plane of a circle that intersects the circle at point. Rule 1: Tangents from the same point are. Example: In the diagram above, find the value of x. Rule 2: A radius and a tangent intersect at a. Example: In the diagram above, find the length of PK.

6 Just as circles can be circumscribed around a polygon, polygons can be about a circle. We can also say that the circle is in the polygon. All sides of the circumscribed polygon are to the circle.

7 Kuta Software - Infinite Geometry Tangents to Circles Name Date Period Determine if line AB is tangent to the circle. 1) 16 B 8 2) A 6.6 B A ) A ) B A 15.2 Find the segment length indicated. Assume that lines which appear to be tangent are tangent. B 5) 8.5 6)?? ) 8) ?? c F250R1f2j mkbubtmay cswolfqtdwva7rye2 nlalacw.h G CAElhl9 erjibgshwtqsg qrfegswenrlvve7d1.d 3 SMFakd8eV xwpietmhv uianufmibniwt3ej ygtexoemlemtlrkys.9 Worksheet by Kuta Software LLC

8 Find the perimeter of each polygon. Assume that lines which appear to be tangent are tangent. 9) 10) ) 12) Find the angle measure indicated. Assume that lines which appear to be tangent are tangent. 13)? 1) 63? 15) 16) ?? -2- d K2m0l1B25 TK5uQtqaO qsqogftnwvavrwen QLSLoCE.M o maulbl irli7gwhbtks2 Dr3eOs9efr0v9ezd7.R h omgaldmec zwoi0toht signlf5iunsi5tgey AGWe6olm3eNt3rlyv.7 Worksheet by Kuta Software LLC

9 1-2 NAME DATE PERIOD Practice Student Edition Pages Tangents to a Circle For each Q, find the value of x. Assume segments that appear to be tangent are tangent Glencoe/McGraw-Hill 80 Geometry: Concepts and Applications

10 Warm Up Key: 1. No 2. No 3. Yes Fill In The Blank Tangent lines: Exactly one; congruent; x = 6; right angle; PK = 10; circumscribed; inscribed; tangent Tangents to Circles Worksheet from Kuta Software-Infinite Geometry 1. yes 2. no 3. yes. yes Tangents to a Circle 1. x=12 2. x=5 3. x=7. x= x= x=.8

11 Hampton High School Geometry Lesson Plan Intersecting Chords and Secants of Circles Michelle Bousquet Context: This lesson is designed for a 90 minute Geometry class at Hampton High School. The class contains a maximum of 19 students. This is the second of five lessons on tangents, chords, secants, and the equation of a circle. Objective: The students will correctly identify problems as intersecting chords, secant-secant, or secant-tangent and then use the corresponding equation to accurately solve problems on a worksheet as checked by the teacher at the end of the class period. SOL: G.11 The student will use angles, arcs, chords, tangents, and secants to: a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles Materials/Resources: Worksheets attached Content Instructional Strategies: 1. Warm Up: The students will complete a warm-up containing two questions from previous Geometry SOL tests. The questions pertain to the tangent line concepts learned in the previous class period. The warm-up worksheet is attached. While the students are completing the warm-up, take attendance. Then, go over the warm-up with the class. 15 minutes 2. Reference chords the class briefly learned one class period before. Pass out the Circles: Intersecting Chords and Secants worksheet. Go through the Intersecting Chords section with the class. 15 minutes 3. Review concept of secant the class briefly learned the class period before. Fill in the blank on the worksheet pertaining to secants. Real world connection to what we will learn about secants: Secant lines make rainbows! When it rains, raindrops bend sunlight as it passes through them. The different angles formed make different colors of the rainbow! 5 minutes. Complete the Secant section of the worksheet with the class. Since these students struggle with algebra, make sure to show each step slowly and carefully. 20 minutes 5. Pass out the Segment Lengths in Circles worksheet. Before starting the worksheet, ask the students to work in groups to go through each question to determine if it is intersecting chords, secant-secant, or secant-tangent. More advanced students can begin the worksheet without doing this step. Then, the groups can work through the problems using the correct formula. When they are finished, they will raise their hands to get their answers checked. If the faster students finish early, have them write a couple sentences

12 describing the difference between tangent and secant lines. 30 minutes 6. Review what the class learned and answer any questions. 5 minutes Evaluation: The students will work individually or in groups to complete the Segment Lengths in Circles worksheet. The teacher will first assess whether the students identify the correct type of problem: intersecting chords, secant-secant, or secant-tangent. Then, the teacher will assess if the students can apply the formulas and use algebra to solve the problems. Differentiation and Adaptations: If there is not enough time, the students can finish the worksheet for homework. Advanced students do not need to identify what type of question each problem is before solving. The students can work individually or in groups. Struggling students will be encouraged to work with their groups. If there is extra time, begin the next lesson, or have students write a couple sentences to describe the differences between secant lines and tangent lines. Sources: Warm Up #31: Commonwealth of Virginia Department of Education. (2003). End of course geometry. Retrieved from Warm Up #32: Commonwealth of Virginia Department of Education (2002). End of course geometry. Retrieved from Segment Lengths in Circles Worksheet Kuta Software LLC. (2012, December 8). Segment lengths in circles. Retrieved from Segment%20Lengths%20in%20Circles.pdf

13 Name: Date: Warm Up

14 Name: Date: Intersecting Chords Chords: Intersecting Chords and Secants Chord: For a given circle, a segment with on the circle. A B E C In the circle above, ( )( ) = ( )( ) Example: D

15 Secants Definition: Any line that intersects a circle in exactly points. Secant-Secant Rule: (Outside)(Whole Secant) = (Outside)(Whole Secant) =. Whole Secant = +. Example 1: Example 2:

16 Secant-Tangent Rule: (Outside)(Whole Secant)=(Tangent) = Whole Secant = + Example 1:! 2 Example 2:

17 Kuta Software - Infinite Geometry Segment Lengths in Circles Name Date Period Solve for x. Assume that lines which appear tangent are tangent. 1) 15 2) 9 5 x x 3 3) x 3 x 6 5 ) x 6 5) 6) x x 5 x 8 7) 6 8) 6x 8x 5 x X m2l0q1m2b MKGuUtEax YSioaf3tYw3abrfeb jlmlkci.q S LAFlclS NrpigohLtqsj 0rheysbeHr3vse9dw.G L BMIa7dVea Dwbixtyhw zixntfxilnaict7eo ygfexowmdeatgrpyl.e Worksheet by Kuta Software LLC

18 Find the measure of the line segment indicated. Assume that lines which appear tangent are tangent. 9) Find UW 10) Find KM T U x + 2 C 2x + 5 V M W N 2 3 L 2x + 5 K W 11) Find NM 12) Find NL L x 3 M K 16 N x + 3 S N x 8 M x 5 K 3 L 13) Find CE 1) Find CA S 10 2x 17 C A D x L 1 C x + 1 B F 8 9 D E 15) Find HG 16) Find WS G 12 F x + 8 H 15 E R S 6x W x P Q -2- W h2b0k1b2x ekluxtuag 6SdoafJttwOa2rNe3 xlglhcs.u f LAilG l Ur6iugghwtvsx gr NeRsQehrsv9eYd8.f z zmeatd2e6 wwpiythhg DIbnDfRiUnmimtaeh ag8eoolmcettdrpyq.9 Worksheet by Kuta Software LLC

19 Warm-Up Key 31. C 32. H Circles: Intersecting Chords and Secants Worksheet Key Intersecting Chords Fill in the blank: Both endpoints; (AE)(EC)=(DE)(EB) Example: x=2 Secants Secant-Secant Fill in the blank: two; OW=OW; outside; inside Example 1: x=5 Example 2: x=3 Secant-Tangent Fill in the blank: OW=T^2; outside; inside Example 1: x=28.05 Example 2: a=6 Segment Lengths in Circles Worksheet

20 Hampton High School Geometry Lesson Plan Equation of a Circle Michelle Bousquet Context: This lesson is designed for a 90 minute Geometry class at Hampton High School. The class contains a maximum of 19 students. This is the third of five lessons on tangents, chords, secants, and the equation of a circle. Objective: The students will discover the equation of a circle using an online inquiry-based activity as guided by a worksheet. The students will then work individually or in groups to accurately apply the equation to a worksheet as checked by the teacher at the end of class. SOL: G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. Materials/Resources: Attached Worksheets Laptops- 1 laptop per student Content Instructional Strategies: 1. Warm Up: The students will complete a warm-up containing three questions from previous Geometry SOL tests. The questions pertain to intersecting chords, secant-secant lines through a circle, and secant-tangent lines on a circle. The warm-up worksheet is attached. While the students are completing the warm-up, take attendance. Then, the teacher will go over the warm-up with the class. 15 minutes 2. Pass out laptops and have students log in using student ID number and password. Have the students go to 5 minutes 3. Pass out the Equations of Circles worksheets and have the students use the website to discover the equations of circles. 20 minutes. Using direct instruction, finish the Equations of Circles worksheet and all examples with the class. If the students seem to understand the concept well, have them work in groups for some of the examples. 25 minutes 5. Pass out the Equations of Circles worksheet by Kuta Software. Have the students work in groups to complete #1-6 and #9-11. The faster students can also do #7-8 and #12-1. Students will raise their hands to get the teacher to check answers before class is over. 20 minutes 6. Review what the class learned and answer any questions. 5 minutes Evaluation: The teacher will evaluate student understanding by first by observing which students correctly discover the formula for the equation of a circle in the inquiry-based online

21 activity. Then, the students will be assessed on their ability to accurately complete the Equations of Circles worksheet by Kuta Software. Differentiation and Adaptations: If there is not enough time in the class period, the last worksheet can be assigned as homework or can be reviewed the next class. The teacher can also work out the exercises on the board. If there is extra time, the teacher can begin the next lesson. For individual students who finish before the rest of the class, there are additional problems on the Kuta worksheet that they should be assigned. Sources: Warm Up #32-1: Commonwealth of Virginia Department of Education. (200). End of course geometry. Retrieved from Warm Up #32-2: Commonwealth of Virginia Department of Education. (2006). End of course geometry. Retrieved from Warm Up #31 Commonwealth of Virginia Department of Education. (2001). End of course geometry. Retrieved from Equations of Circles Kuta Software Worksheet Kuta Software LLC. (2012, December 8). Equations of circles. Retrieved from Equations%20of%20Circles.pdf

22 Name: Date: Warm Up

23

24 Name: Date: Equations of Circles 1. Go to 2. Scroll down to the graph of a circle. 3. Place the center of the circle on (0,0). Put the red dot on the circle on point (-1,0).. What is the radius of this circle? 5. Look at the equation above the graph. What does it say? 6. Now, put the center of the circle on (2,1). Put the other red dot on (0,1). 7. What is the radius of this circle? 8. Look at the equation above the graph. What does it say? 9. Now, put the center of the circle on (-1,3). Put the other red dot on (-,3). 10. What is the radius of this circle? 11. Look at the equation above the graph. What does it say? 12. Do you see any relationships between the center point and the radius in the equations?

25 Equation of a Circle (h,k): r: Equation: Example 1: Write the equation of a circle with diameter whose endpoints are (20,-10) and (8, 6). Example 2: Write the equation of a circle with a diameter whose endpoints are (6, 2) and (0, -6).

26 Example 3: Write the equation of a circle with the center at (-2, ) and a radius of 3cm. Example : Find the center and the radius if the equation of a circle is (x " 3) 2 + (y + 5) 2 = 25.!

27 Kuta Software - Infinite Geometry Equations of Circles Name Date Period Identify the center and radius of each. Then sketch the graph. 1) (x 1) 2 + (y + 3) 2 = y ) (x 2) 2 + (y + 1) 2 = 16 y x x ) (x 1) 2 + (y + ) 2 = 9 y ) x 2 + (y 3) 2 = 1 y x x b O2B0j12k hkxustxa WShoAf7tTw3afrXeW 9LhLHCV.x 6 EAflrlk lrliuguhats8 jraetsee5rjvvexde.h r nmzasdev jwwiwtyhn bi8nuf6infiktjei NGAe0oVmfe5torFyo.3 Worksheet by Kuta Software LLC

28 5) y 2 + x 20 2y = x 2 y ) 9 = y 2 x y x x ) 9 = 2y y 2 6x x 2 y ) 16 + x 2 + y 2 8x 6y = 0 y x x Use the information provided to write the equation of each circle. 9) Center: (13, 13) Radius: 10) Center: ( 13, 16) Point on Circle: ( 10, 16) 11) Ends of a diameter: (18, 13) and (, 3) 12) Center: (10, 1) Tangent to x = 13 13) Center lies in the first quadrant Tangent to x = 8, y = 3, and x = 1 1) Center: (0, 13) Area: 25π -2- G2n0G1L2s SKfu5tqaa 2Skoofpt6wwaarlev ZLFL1Cl.F u DAQl l nrdingchktosf rteysvecr2v1e1du.n c BMFa2d1ek gwpirt8hr EIhnbfpilnniJt5eV pgreroemeeitqrhyt.6 Worksheet by Kuta Software LLC

29 Warm Up Key 32. F 32. H 31. B!!!! Equations of Circles Key. r=1 5. x 2 + y 2 = 1 7. r=2 8. x 2 + (y "1) 2 = 10. r=3 11. (x + ) 2 + (y " 3) 2 = 9 Equation of a Circle Center Radius (x " h) 2 + (y " k) 2 = r 2 Example 1: (x "1) 2 + (y + 2) 2 = 100 Example 2: (x " 3) 2 + (y + 2) 2 = 25 Example 3: (x + 2) 2 + (y " ) 2 = 9 Example : Center: (3, -5) Radius: 5! Equations! of Circles Kuta Software Worksheet Copy! of Answers on Next Page

30 Kuta Software - Infinite Geometry Equations of Circles Name Date Period Identify the center and radius of each. Then sketch the graph. 1) (x 1) 2 + (y + 3) 2 = y 8 Center: (1, 3) Radius: ) (x 2) 2 + (y + 1) 2 = 16 y 8 Center: (2, 1) Radius: x x ) (x 1) 2 + (y + ) 2 = 9 y 8 Center: (1, ) Radius: ) x 2 + (y 3) 2 = 1 y 8 Center: (0, 3) Radius: x x H 52y0Z1e23 LKAuYtNav XSEodfWt7wdaRrheo KLYLNCL.S W 7AIlblT irhixg3hstwsx RrleTsuewrvNevdo.E c WMqaldmem Pwqi5tshG 2IsnpfBiCnii1tYea CGTeeoGmjegtDrnyo.b Worksheet by Kuta Software LLC

31 5) y 2 + x 20 2y = x 2 y 8 Center: ( 2, 1) Radius: ) 9 = y 2 x 2 y 8 Center: (0, 0) Radius: x x ) 9 = 2y y 2 6x x 2 y 8 Center: ( 3, 1) Radius: ) 16 + x 2 + y 2 8x 6y = 0 y 8 Center: (, 3) Radius: x x Use the information provided to write the equation of each circle. 9) Center: (13, 13) Radius: 10) Center: ( 13, 16) Point on Circle: ( 10, 16) (x 13) 2 + (y + 13) 2 = 16 11) Ends of a diameter: (18, 13) and (, 3) (x 11) 2 + (y + 8) 2 = 7 13) Center lies in the first quadrant Tangent to x = 8, y = 3, and x = 1 (x 11) 2 + (y 6) 2 = 9 (x + 13) 2 + (y + 16) 2 = 9 12) Center: (10, 1) Tangent to x = 13 (x 10) 2 + (y + 1) 2 = 9 1) Center: (0, 13) Area: 25π x 2 + (y 13) 2 = 25 Create your own worksheets like this one with Infinite Geometry. Free trial available at KutaSoftware.com -2- B Q29081D23 XKLutaN esxoefntawwaxrkea HLOLnCa.F P ha8lflc HrZitgdh2tDs2 Nr2eRs7e6rNvdebdV.X R FM3awdwe1 yww itth1 BIMnxfZijnRiWtDek CGdepoamDe9tFr Qyf.d Worksheet by Kuta Software LLC

32 Hampton High School Geometry Lesson Plan Review of Tangents, Intersecting Chords, Secants, and Equation of a Circle Michelle Bousquet Context: This lesson is designed for a 90 minute Geometry class at Hampton High School. The class contains a maximum of 19 students. This is the fourth of five lessons on tangents, chords, secants, and the equation of a circle. The class will be reviewing all the concepts they have learned in this Unit to prepare for the test. Objective: The students will understand and accurately solve all problems pertaining to segment and line lengths in or on circles as displayed in a review game for the test by the end of the class period. The teacher will assess each group s answers and review any problems before moving on to the next set. SOLs: G.11 The student will use angles, arcs, chords, tangents, and secants to: a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. Materials/Resources: Attached Worksheets White boards: 1 for each group of three students Dry erase markers: at least 1 for each group of three students Calculators Content Instructional Strategies: 1. Warm Up: The students will complete a warm-up containing two questions pertaining to equations of circles. The warm-up worksheet is attached. While the students are completing the warm-up, the teacher will take attendance. Then, the teacher will go over the warm-up with the class. 20 minutes 2. Working in their assigned groups, the class will play a review game to prepare for the test next class. Each group member will be assigned a number: 1, 2, or 3. The teacher will display a slide with three problems on it. Each student will solve the problem corresponding to his or her number on a sheet a paper. Then, the group of students will check each other s work and write their three different answers on the white board. After five minutes on each set of problems, the groups will hold up their white boards and the teacher will check answers. Each group that gets all three questions correct will get a point. The teacher will briefly go through each problem after the answers have been checked. The teacher will continue showing slides. At the end of the game, whichever group has the most points will get 5 points added to their test next class. The slides are

33 attached. 55 minutes 3. Review what the class learned and answer any questions. 15 minutes Evaluation: The students will be evaluated by the success of their groups. Each student, by the end of the game will have answered one question pertaining to each of seven categories: Tangent lines, finding the perimeter, intersecting chords, secant-secant, secant-tangent, using an equation of a circle, and graphing a circle. Differentiation and Adaptations: Since the students will be working in groups, the struggling with students can work with their other group members to help figure out the problems. The advanced students can check every group member s work. If there is not enough time, the teacher can end the game early. If there is extra time, the students can study for the test. Sources: Exercises for game: Glencoe/McGraw-Hill. (2003). Geometry: Chapter 10 resource masters. Retrieved from

34 Name: Date: Warm Up 1. 2.

35 Slide 1: 1. Find x Slide 2: Find x

36 3. Find x Slide 3 1. Find the perimeter:

37 2. 3. Graph the following equation: Slide : 1. Find x 2. Find x

38 3. Find the perimeter. Slide 5: 1. Find x.

39 2. Find y. 3. Find x. Slide 6: 1.

40 2. Find the perimeter of Triangle ABC. 3. Find x. Slide 7:

41 1. Graph the following equation: 2. Write the equation of the circle: 3. Find x.

42 Warm Up Key 1. a. (x + 2) 2 + (y " 6) 2 = 25 b. ex: (3,6), (-7,6) 2. a. (3, -) b. r=! c. d=8 d. ex: (7, -), (-1, -) Review Game Key: Slide 1: 1. x=2 2. Center at (0,1) and r=3 3. (-11,13) Slide 2: 1. x=-10, 2. x= x=18.03 Slide 3: x=11 3. Center at (-2,0), radius= Slide : 1. x=1 2. x= Slide 5: 1. x=30 2. y= x=7.35 Slide 6: 1. r= x=2.33 Slide 7: 1. Center at (2,1) and radius=3 2. (x " 2) 2 + (y + ) 2 = 1 3. x=16!