6.2 PLANNING. Chord Properties. Investigation 1 Defining Angles in a Circle

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1 LESSN 6.2 You will do foolish things, but do them with enthusiasm. SINIE GRIELL LETTE Step 1 central Step 1 angle has its verte at the center of the circle. Step 2 n Step 2 inscribed angle has its verte on the circle and its sides are chords. NTM STNRS NTENT Number lgebra Geometr Measurement ata/robabilit hord roperties In the last lesson ou discovered some properties of a tangent, a line that intersects the circle onl once. In this lesson ou will investigate properties of a chord, a line segment whose endpoints lie on the circle. In a person with correct vision, light ras from distant objects are focused to a point on the retina. If the ee represents a circle, then the path of the light from the lens to the retina represents a chord. The angle formed b two of these chords to the same point on the retina represents an inscribed angle. First ou will define two tpes of angles in a circle. Investigation 1 efining ngles in a ircle Write a good definition of each boldfaced term. iscuss our definitions with others in our group. gree on a common set of definitions as a class and add them to our definition list. In our notebook, draw and label a figure to illustrate each term. entral ngle.,, and are central angles of circle. Inscribed ngle. E,, and E are inscribed angles. RESS roblem Solving Reasoning ommunication onnections Representation R R, S, RST, ST, and SR are not central angles of circle W X S R, STU, and VWX are not inscribed angles. LESSN JETIVES iscover properties of chords to a circle ractice construction skills V T T U S R LESSN 6.2 LNNING LESSN UTLINE ne da: 25 min Investigation 10 min Sharing 5min losing 5min Eercises MTERILS construction tools protractors Gear Fragment (W) for ne Step Sketchpad activit hord roperties, optional Sketchpad demonstration Intersecting Tangents onjecture, optional TEHING In this lesson students discover some properties relating central angles, chords, and arcs of circles. egin with the one-step investigation, or ask groups to work through Investigations 1 through 4. You can replace or etend Investigations 1 and 2 with the namic Geometr Eploration at You might have students use patt paper for Investigations 2 through 4. You can have students construct circles on patt paper b tracing circular objects. Guiding Investigation 1 ne Step Hand out the Gear Fragment worksheet and pose this problem: In repairing a machine, ou find a fragment of a circular gear. To replace the gear, ou need to know its diameter. How can ou find the gear s LESSN 6.2 hord roperties 317

2 diameter You ma need to review the term diameter right awa. s ou circulate, wonder aloud as needed whether the center is somehow related to the chords of the circle, pointing out what ou mean b chord. Step 2 If groups are having difficult defining inscribed angle, [sk] Where have ou heard the word inscribed before [ triangle inscribed in a circle (whose center is the circumcenter of the triangle) has all three vertices on the circle, and its sides are chords of the circle. n inscribed triangle has three inscribed angles.] Guiding Investigation 2 [lert] If ou find that all or most students in a group are using a central angle that measures 60 (because the didn t change the compass setting after drawing the circle), point out that the don t have enough information for good inductive reasoning; then have them all change their settings from the circle s radius and begin again. [lert] Students ma have trouble constructing congruent chords. hallenge them to find the endpoints without using a ruler. The might mark two points on the circle and then use a constant compass opening to locate two other points on the circle the same distance apart. Step 3 If their paper is too thick to see through when folded, students ma want to cop their figure to patt paper. Students ma find this investigation and the net two investigations fairl straightforward and ma begin writing conjec- You will need a compass a straightedge a protractor patt paper (optional) Step 1 Step 2 Step 3 Fold it so that Step 3 coincides with and coincides with. Step 4 ull the cord! on t I need to construct it first tures without going through the steps or comparing results with others in their group. Remind them that the process is more important than the results. If students finish earl, ou might challenge them with the one-step problem. Investigation 2 hords and Their entral ngles Net ou will discover some properties of chords and central angles. You will also see a relationship between chords and arcs. onstruct a large circle. Label the center. onstruct two congruent chords in our circle. Label the chords and, then construct radii,,, and. With our protractor, measure and. How do the compare Share our results with others in our group. Then cop and complete the conjecture. hord entral ngles onjecture If two chords in a circle are congruent, then the determine two central angles that are congruent How can ou fold our circle construction to check the conjecture Recall that the measure of an arc is defined as the measure of its central angle. If two central angles are congruent, their intercepted arcs must be congruent. ombine this fact with the hord entral ngles onjecture to complete the net conjecture. hord rcs onjecture intercepted arcs If two chords in a circle are congruent, then their are congruent HTER 6 iscovering and roving ircle roperties

3 You will need a compass a straightedge patt paper (optional) Step 1 Step 2 Step 3 Investigation 3 hords and the enter of the ircle In this investigation ou will discover relationships about a chord and the center of its circle. onstruct a large circle and mark the center. onstruct two nonparallel congruent chords. Then construct the perpendiculars from the center to each chord. How does the perpendicular from the center of a circle to a chord divide the chord op and complete the conjecture. erpendicular to a hord onjecture The perpendicular from the center of a circle to a chord is the of the chord. bisector Let s continue this investigation to discover a relationship between the length of congruent chords and their distances from the center of the circle. ompare the distances (measured along the perpendicular) from the center to the chords. re the results the same if ou change the sie of the circle and the length of the chords State our observations as our net conjecture. -58 hord istance to enter onjecture Two congruent chords in a circle are equidistant from the center of the circle. -57 Guiding Investigation 3 Step 1 Students can use the same two congruent chords and the same drawing for Investigations 2 and 3. [lert] Students ma need some review in how to construct perpendiculars. Using compass constructions to complete all four investigations ma be too time-consuming, but this construction and the net can be completed quickl using patt-paper constructions. The perpendicular through the center of the circle to the chord can be folded. Guiding Investigation 4 The perpendicular bisectors of the chords can be constructed b simpl folding the chord in half. The erpendicular isector of a hord onjecture is the converse of the erpendicular to a hord onjecture. You will need a compass a straightedge patt paper (optional) Step 1 Investigation 4 erpendicular isector of a hord Net, ou will discover a propert of perpendicular bisectors of chords. onstruct a large circle and mark the center. onstruct two nonparallel chords that are not diameters. Then construct the perpendicular bisector of each chord and etend the bisectors until the intersect. Sharing Ideas (continued) measure can have ver different lengths. sk whether having the same measure is enough to make two arcs congruent, as is the case for line segments and angles. [sk] What s needed to ensure that two arcs have the same sie and shape [ongruent arcs must be on the same or congruent circles.] (Some students ma think that an arc of a larger circle will have the same sie and shape as an arc of a smaller circle. To the etent possible, let other students convince them that the curvature will be different.) [sk] ongruent chords are equidistant from the center. an we sa anthing about distance from the center if one chord is longer than the other [In the same circle, shorter chords are farther from the center.] SHRING IES s usual, for presentations select groups that have a variet of statements of the conjectures. s students present, encourage them to put the ideas in their own words, not just those of the conjectures as presented in the student book. Help the class reach consensus on the wording to record in their notebooks. [sk] How would ou define congruent arcs lthough the measure of an arc was first defined in hapter 1, some students ma still be wondering wh arcs are measured in degrees based on their central angle. rawing a central angle of 90 ma help some students relate the central angle to a quarter of a circle. ut one source of difficult ma be a natural tendenc to think that measure means sie in some direct wa, and the measure of an arc doesn t give its length.in fact,in different circles, arcs with the same LESSN 6.2 hord roperties 319

4 ssessing rogress You can assess students understanding of (and use of the vocabular for) radius, chord, central angle, and inscribed angle and their skill at constructing a circle, measuring an angle with a protractor, and comparing segments with a compass. You might also see how well the understand the difference between drawing and constructing. losing the Lesson Summarie that the major conjectures of this lesson are about congruent chords of a circle: The determine congruent central angles, the intercept congruent arcs, and the are equidistant from the center. nother pair of conjectures about chords forms a biconditional: line through the center of a circle is perpendicular to a chord if and onl if it bisects the chord. If students are still shak about these ideas, ou might want to use one of Eercises 1 6 as an eample. UILING UNERSTNING The eercises focus on appling the conjectures about chords. Encourage students to sketch pictures on their own papers. The can then mark and label all information accordingl. SSIGNING HMEWRK Essential 1 14, 17, 18 erformance assessment 17, 18 ortfolio 20 Journal 13, 14, 16 Group 15, 21 Review lgebra review 15, 21, 26 MTERILS circular objects and patt paper (Eercise 17) Eercises 18 and 19 (T), optional kemath.com/g EXERISES Solve Eercises State which conjectures or definitions ou used w 165 Step 2 What do ou notice about the point of intersection ompare our results with the results of others near ou. op and complete the conjecture. erpendicular isector of a hord onjecture The perpendicular bisector of a chord. passes through the center of the circle [ For interactive versions of these investigations, see the namic Geometr Eploration hord roperties at ] With the perpendicular bisector of a chord, ou can find the center of an circle, and therefore the verte of the central angle to an arc. ll ou have to do is construct the perpendicular bisectors of nonparallel chords definition of measure of an arc hord rcs onj. hord entral ngles onj is a diameter. Find 6. GIN is a kite. 8 cm m and m. Find w,, and. w cm 115 G 65 I N w hord istance to enter onj cm 4 cm 8. m hord rcs onjecture 8 cm 3 cm Find w,,, and. 6 cm w 110 What is the perimeter of cm w 120 F T 110 E 48 erpendicular to a hord onj. definition of arc measure Helping with the Eercises 5. m 68 ; m 34 (ecause is isosceles, m m, m m 68, and therefore m 34.) Eercise 7 [lert] Some students ma neglect to add on the length. Eercise 9 s needed, [sk] What do the measures of all the arcs add up to [360 ] 70 You will need onstruction tools for Eercises , hord rcs onjecture; 96, hord entral ngles onjecture; 42, Isosceles Triangle onjecture and Triangle Sum onjecture. w HTER 6 iscovering and roving ircle roperties

5 10., mi eveloping roof What s wrong 12. eveloping roof What s wrong Find,, and. with this picture with this picture I 37 cm 18 cm The length of the chord is greater than the length of the diameter. 13. raw a circle and two chords of unequal length. Which is closer to the center of the circle, the longer chord or the shorter chord Eplain. 14. raw two circles with different radii. In each circle, draw a chord so that the chords have the same length. raw the central angle determined b each chord. Which central angle is larger Eplain. 15. olgon MN is a rectangle inscribed in a circle centered at the origin. Find the coordinates of points M, N, and. M(4, 3), N(4, 3), (4, 3) 16. onstruction onstruct a triangle. Using the sides of the triangle as chords, construct a circle passing through all three vertices. Eplain. Wh does this seem familiar 17. onstruction Trace a circle onto a blank sheet of paper without using our compass. Locate the center of the circle using a compass and straightedge. Trace another circle onto patt paper and find the center b folding. 18. onstruction dventurer akota avis digs up a piece of a circular ceramic plate. Suppose he believes that some ancient plates with this particular design have a diameter of 15 cm. He wants to calculate the diameter of the original plate to see if the piece he found is part of such a plate. He has onl this piece of the circular plate, shown at right, to make his calculations. Trace the outer edge of the plate onto a sheet of paper. Help him find the diameter. M (, 3) The perpendicular bisector of the segment does not pass through the center of the circle. (4, 3) N (, ) (, ) Eercise 10 [lert] Students ma miss the fact that the two radii in are congruent, making it an isosceles triangle , 48, 66 ; orresponding ngles onjecture, Isosceles Triangle onjecture, Linear air onjecture Eercise 12 If students are having difficult, [sk] What does the perpendicular bisector have to go through [the center of the circle] 13. The longer chord is closer to the center; the longest chord, which is the diameter, passes through the center. 14. The central angle of the smaller circle is larger, because the chord is closer to the center. 19. onstruction The satellite photo at right shows onl a portion of a lunar crater. How can cartographers use the photo to find its center Trace the crater and locate its center. Using the scale shown, find its radius. To learn more about satellite photos, go to The can draw two chords and locate the intersection of their perpendicular bisectors. The radius is just over 5 km km 5 cm 5 cm Eercise 16 s needed, remind students of the meaning of a triangle inscribed in a circle (or a circle circumscribed around a triangle). 16. The center of the circle is the circumcenter of the triangle. ossible construction: Eercise 17 Have available round objects larger than coins for tracing. 17. possible construction: Eercises If students are having difficult, wonder aloud whether there s a conjecture that ends with something about the center of a circle. [erpendicular isector of a hord] cm LESSN 6.2 hord roperties 321

6 20. eveloping roof omplete the flowchart proof shown, which proves that if two chords of a circle are congruent, then the determine two congruent central angles. Given: ircle with chords Show: Flowchart roof 1 Given 2 4 ll radii of a circle SSS ongruence are congruent onjecture 3 ll radii of a circle are congruent 21. ircle has center (0, 0) and passes through points (3, 4) and (4, 3). Find an equation to show that the perpendicular bisector of passes through the center of the circle. Eplain our reasoning. 1 ; 7 (0, 0) is a point on this line. 5 T (3, 4) (4, 3) Eercise 23 You might use the Sketchpad demonstration Intersecting Tangents onjecture to preview or replace this eercise. hapter 5 Review 22. eveloping roof Identif each of these statements as true or false. If the statement is true, eplain wh. If it is false, give a countereample. a. If the diagonals of a quadrilateral are congruent, but onl one is the perpendicular bisector of the other, then the quadrilateral is a kite. true b. If the quadrilateral has eactl one line of reflectional smmetr, then the quadrilateral is a kite. false, isosceles trapeoid c. If the diagonals of a quadrilateral are congruent and bisect each other, then it is a square. false, rectangle 23. Mini-Investigation Use what ou learned in the last lesson about the angle formed b a tangent and a radius to find the missing arc measure or angle measure in each diagram. Eamine these cases to find a relationship between the measure of the angle formed b two tangents to a circle,, and the measure of the intercepted arc,. Then cop and complete the conjecture below onjecture: The measure of the angle formed b two intersecting tangents to a circle is (Intersecting Tangents onjecture). 180 minus the measure of the intercepted arc 22a. X ossible eplanation: If is the perpendicular bisector of, then ever point on is equidistant from endpoints and. Therefore and. ecause is not the perpendicular bisector of, is not equidistant from and. Likewise, is not equidistant from and. So, and are not congruent, and and are not congruent. Thus has eactl two pairs of consecutive congruent sides, so it is a kite. 322 HTER 6 iscovering and roving ircle roperties