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Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2G3a Time allotted for this Lesson: 10 Hours Essential Question: LESSON 3 – Segments of Circles How do we use triangle similarity to understand properties of intersecting chords, tangents, and secants of circles? Activating Strategies: (Learners Mentally Active) Session 1 Cut terms from the page “Activating Strategy - Parts of a Circle”. Give each student a card with a term for a part of a circle. In collaborative pairs, students will discuss the part of the circle that is on his/her card. Draw a large circle on board. Each student will draw, label, and identify the part of the circle that is on his/her card. Session 2 Complete and answer questions for “Activating Strategy – Similar Triangles”. Subsequent Sessions Use the activating strategy “Chords and Similar Triangles” with collaborative pairs. Think-Pair-Share . Acceleration/Previewing: (Key Vocabulary) Chord, secant, tangent, point of tangency, central angle, inscribed angle, intercepted arc, similar triangles, AA similarity Teaching Strategies: (Collaborative Pairs; Distributed Guided Practice; Distributed Summarizing; Graphic Organizers) Session 1 Students will work in collaborative pairs. Using a ruler, students will complete “Chords of a Circles” worksheet. Whole class will discuss results. Practice problems. (supplemental material) – radius drawn perpendicular to chord, radius bisects chord and arc, equal chords intercept equal arcs. Students will work in collaborative pairs. Using a ruler, students will complete “Tangents of Circles” worksheet. Whole class will discuss results. Practice problems. (supplemental material) – tangents to circles Session 2 Discuss similar triangles from activating strategy. Ask pairs to go to board and draw 2 non-similar triangles with 2 pairs of congruent angles. As a whole group, discuss how this is not possible because If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar (AA similarity). Math 2 Unit 3 Lesson 3 Segments of Circles Page 1 In pairs, students should complete GO “What relationship exists among segments of two intersecting chords in a circle?” As a whole group, discuss results. Practice problems. (supplemental material) Session 3 In whole group, discuss activating strategy. In collaborative pairs, students will complete GO “What relationship exists among the segments of two secants drawn from a common point outside the circle?” and “What relationship exists among the segments of a secant and a tangent drawn from a common point outside the circle?” In whole group, discuss results. Distributed Guided Practice/Summarizing Prompts: (Prompts Designed to Initiate Periodic Practice or Summarizing) “If two angles in one triangle are congruent to two angles in another triangle, what can be determined about the third angles? Are the two triangles congruent? Are the two triangles similar?” “What is the relationship between two inscribed angles that intercept the same arc?” “What relationship exists among segments of two intersecting chords in a circle?” “What relationship exists among the segments of two secants drawn from a common point outside the circle?” “What relationship exists among the segments of a secant and a tangent drawn from a common point outside the circle?” Extending/Refining Strategies: Students will create a foldable graphic organizer that describes angles and segments that are on, inside, and outside circle. Fold paper hotdog style and divide into three parts: On the circle, Inside the circle, and Outside the circle. On inside of foldable for each category, put angle information on top half and segment information on bottom half. Using the foldable GO, students will work practice problems. (supplemental material) Summarizing Strategies: Learners Summarize & Answer Essential Question Session 2 As the students enter the room, they will be given worksheet “Summarizing - Similar Triangles?”. Students will decide if each set of triangles is similar. Patty paper may be used to help students. Additional Tasks/Activities on pages 13 – 15 may be used as needed. Math 2 Unit 3 Lesson 3 Segments of Circles Page 2 Activating Strategy - Parts of a Circle CENTER MINOR ARC MAJOR ARC CHORD INSCRIBED ANGLE CENTRAL ANGLE TANGENT SECANT RADIUS DIAMETER SEMICIRCLE Math 2 Unit 3 Lesson 3 Segments of Circles Page 3 Activating Strategy – Similar Triangles Use the triangles at the right to answer each question. ABC DEF. 1. Measure each segment and angle of the triangles. E AB = _____ DE = _____ BC = _____ EF = _____ B AC = _____ DF = _____ mA = _____ mD = _____ mB = _____ mE = _____ mC = _____ mF = _____ F C A 2. What is the scale factor of ABC to DEF? 3. AB 4. What angle relationship do you observe? = _ = DF D BC Math 2 Unit 3 Lesson 3 Segments of Circles Page 4 Activating Strategy Inscribed Angles and Similar Triangles A B 1 E 2 D C Using the figure above, complete the following: 1. B 1 arc ______ 2 C 1 arc ______ 2 What can you conclude about B and C ?__________________________________ 2. What relation exists between 1 and 2 ? Explain how you know.________________ _________________________________________________________________________ 3. From the results in steps 1 and 2, what must be true about A and D ? Explain your answer.__________________________________________________________________ _________________________________________________________________________ 4. What conclusion can be reached about ∆ ABE and ∆ DCE? Explain your reasoning. __________________________________________________________________________ Math 2 Unit 3 Lesson 3 Segments of Circles Page 5 Chords of a Circle O 1. Draw the following chords: AB, CD, EF, and GH. 2. Measure the length of each chord. AB= _____, CD = _____, EF = _____, GH = _____ 3. Measure the distance from the chord to the center of the circle. The distance from the center of the circle to: AB = _____, CD = _____, EF = _____, GH = _____ 4. Mary made the following conjecture: If two chords are the same distance from the center of the circle, the chords are congruent. a. Do you agree or disagree? b. Support your answer mathematically. c. State the converse of this conjecture. d. Explain whether or not the converse is true. Math 2 Unit 3 Lesson 3 Segments of Circles Page 6 Tangents of Circles 1. Measure the lengths of the following line segments. AB = ____ and BC = ____ DE = ____ and EF = ____ GH = ____ and HI = ____ 2. What conjectures can you make? Math 2 Unit 3 Lesson 3 Segments of Circles Page 7 3. Use a protractor to find the measure of the following angles. mJKL = ____ mMNO = ____ mPQR = ____ 4. What conjectures can you make? 5. Using the figure at the right, which line segments are the same length? Math 2 Unit 3 Lesson 3 Segments of Circles Page 8 What relationship exists among segments of two intersecting chords in a circle? C A Draw chords AC and BD. Can you show CAB BDC? _____ X Why? ________________________ Can you show ACD ABD? _____ Why? ________________________ Is ACX ~ DBX? _____ Why? ________________________ D Complete the following proportion. B AX ? ? XB So AX XB ? ? When two chords intersect, the ___________ of the segments of one chord equals the __________ of the segments of the other chord. Try this one. 3 4 n n = ____ 10 Math 2 Unit 3 Lesson 3 Segments of Circles Page 9 What relationship exists among the segments of two secants drawn from a common point outside the circle? B Draw chords XC and BY. Can you show ABY ACX? _____ X Why? ____________________ Is ABY ~ ACX? _____ A Why? ____________________ Y Complete the following proportion. C AX AC ? ? SO AX ? ? AC When two secants are drawn from a common point, the ___________ of the outside part and the whole secant equals the __________ of the outside part and the whole other secant. Try this one. n = ____ n 4 2 8 Math 2 Unit 3 Lesson 3 Segments of Circles Page 10 What relationship exists among the segments of a secant and a tangent drawn from a common point outside the circle? Draw chords BC and BD. B Can you show ABD ACB? _____ Why? ____________________ A Is ABD ~ ACB? _____ Why? ____________________ Complete the following proportion. D AD ? ? AC C SO ?2 AD AC When a secant and a tangent are drawn from a common point, the ___________ of the tangent equals the __________ of the outside part and the whole secant. Try this one. n 4 12 Math 2 Unit 3 Lesson 3 n = _____ Segments of Circles Page 11 Summarizing - Similar or Not? 1. Is ABC ~ DEF? 2. Is ABD ~ ACE? 3. Is ABC ~ DFE? B A 60° D 30° 30° E 90° C F 4. Is ABD ~ ACB? Math 2 Unit 3 Lesson 3 Segments of Circles Page 12 Additional Tasks/Activities Lines and Line Segments of a Circle Learning Task http://www.geogebra.org/en/wiki/index.php/Circles_%28Angles%29 :refer to the perpendicular and center of circle investigations ONLY Alternate version of center of circles: Finding the Center Again Show the students a broken plate or some circular object that has only part showing. Ask them if they have any ideas about how to find the entire circle. Anthropologists find artifacts that are only parts of the complete item and must work to discover what the item might be – depending on its size. Do astronomers see an entire crate on the moon or must they use mathematics to determine the size of the crater? After the discussion the following activity can be done with MIRAs. Have the students construct a circle on a sheet of paper and construct two nonparallel and noncongruent chords. Using a MIRA construct the perpendicular bisector of each chord. Do these two perpendicular bisectors intersect? _______________ Do you notice anything about the intersection? __________________ Compare with your neighbors and try to fill in the blanks of the following statements. The perpendicular bisectors of chords of a circle _________________________________________. The perpendicular from the center of a circle to a chord is the ___________________ of the chord. (And the _____________ of the arc.) If two chords of a circle are congruent then they determine two central angles of the circle that are ______________. The following link investigates the length of intersecting chords: http://www.geogebra.org/en/upload/files/UC_MAT/chords_in_a_circle.html The following link investigates the length of secant segments and tangent segments: http://www.geogebra.org/en/upload/files/UC_MAT/chords_outside_a_circle.html Math 2 Unit 3 Lesson 3 Segments of Circles Page 13 After the students have written the relationships for the sides, they should think about how to prove what they have learned through experimentation. This theorem states that a×b is always equal to c×d no matter where the chords are placed. By adding two segments in the picture, you can create two triangles: c a b d Figure 1 Challenge the students to prove this relationship is true by proving these triangles are similar. What condition for the similarity do you use to prove it? By establishing proportional equations for corresponding sides of the similar triangles, you can make the same conclusion as the theorem. Again, using the using the same principle, have the students prove that the secants formula they discovered works: B A P C D Figure 2 Prove this theorem by adding segments BC and AD to the above figure. Find pairs of angles of the same size, and determine which triangles are similar to each other. Write a proportion using ratios of the corresponding sides of these triangles Math 2 Unit 3 Lesson 3 Segments of Circles Page 14 Another Way to Find the Center and Facts about Tangents Learning Task Draw a circle using a compass and mark the center O Put a pencil point on the circle and then put the straight edge up next to the pencil so that it touches the circle in one and only one point. Label that point P (point of tangency). Use the straight edge to draw the tangent line through P and mark point A on the line. Construct the radius OP . Measure angle APO using a protractor. Repeat the above activity using the same circle. Put another point S on the circle. This time construct a line through the radius OS and either construct a perpendicular at S with compass and straight edge or use a MIRA to construct the perpendicular at S OS . Mark point B on the perpendicular line. Are both APO and BSO both right angles? _______________ Check with your neighbors to see if they got the same results and complete the following statements. A tangent to a circle ____________________ to the radius drawn to the point of tangency. B D A C BA DB and AC DC With this information, what can you conclude about two tangents drawn from the same point outside the circle? What do these tangent conjectures have to do with space travel? Math 2 Unit 3 Lesson 3 Segments of Circles Page 15 Summary of Angles and Segments: Location of Vertex Relationship of Angle and Arcs On the center of circle On the circle Outside the circle Inside the circle but not the center of the circle Segments Relationship of Lengths of Segments Intersecting chords Chord and Tangent at Point of Tangency Two secants Two tangents Intersecting secant and tangent Math 2 Unit 3 Lesson 3 Segments of Circles Page 16