Chords and Arcs. Objectives To use congruent chords, arcs, and central angles To use perpendicular bisectors to chords

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1 - hords and rcs ommon ore State Standards G-.. Identify and describe relationships among inscribed angles, radii, and chords. M, M bjectives To use congruent chords, arcs, and central angles To use perpendicular bisectors to chords How can you use congruent triangles to help with this one? }, and j. If =, what is the length of? How do you know? MTHMTIL RTIS In the Solve It, you found the leng th of a chord, which is a segment whose endpoints are on a circle. The diagram shows the chord Q and its related arc, Q. ssential Understanding You can use information about congruent parts of a circle (or congruent circles) to find information about other parts of the circle (or circles). Q Lesson L VocabularyV chord The following theorems and their converses confirm that if you know that chords, arcs, or central angles in a circle are congruent, then you know the other two parts are congruent. - and Its onverse central angles have congruent arcs. onverse arcs have congruent central angles. If, then. If, then. You will prove - and its converse in ercises and. Lesson - hords and rcs

2 - and Its onverse central angles have congruent chords. onverse chords have congruent central angles. - and Its onverse chords have congruent arcs. onverse arcs have congruent chords. If, then. If, then. You will prove - and its converse in ercises and. If, then. If, then. You will prove - and its converse in ercises and. roblem Using ongruent hords Why is it important that the circles are congruent? Two circles may have central angles with congruent chords, but the central angles will not be congruent unless the circles are congruent. In the diagram, }. Given what can you conclude? because, within congruent circles, congruent chords have congruent central angles (conv. of Thm. -). because, within congruent circles, congruent chords have congruent arcs (Thm. -). Got It?. Reasoning Use the diagram in roblem. Suppose you are given } } and. How can you show? rom this, what else can you conclude? - and Its onverse Within a circle or in congruent circles, chords equidistant from the center or centers are congruent. onverse chords are equidistant from the center (or centers). If =, then. If, then =. You will prove the converse of - in ercise. hapter ircles

3 of - Given: },, #, # rove: Statements Reason ) ) Radii of a circle are congruent. ), #, # ) Given ) and are right angles. ) ef. of perpendicular segments ) ) HL ) ) orres. parts of s are. ), ) Isosceles Triangle ) ) Transitive roperty of ongruence ) ) If two of a are to two of another, then the third are. ) ) central angles have chords. roblem What is the length of RS in }? inding the Length of a hord S The diagram indicates that Q = QR =. and R and RS are both units from the center.. Q R The length of chord RS Q = QR =. Given in the diagram Q + QR = R Segment ddition ostulate. +. = R Substitute. = R dd. RS = R hords equidistant from the center of a circle are congruent. RS = Substitute. Got It?. What is the value of? Justify your answer. R RS, since they are the same distance from the center of the circle. So finding R gives the length of RS. Lesson - hords and rcs

4 The onverse of the erpendicular isector from Lesson - has special applications to a circle and its diameters, chords, and arcs. - In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. If... Then... is a diameter and # and You will prove - in ercise. - In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord. If... is a diameter and Then... # - In a circle, the perpendicular bisector of a chord contains the center of the circle. If... is the perpendicular bisector of chord Then... contains the center of } You will prove - in ercise. of - Given: } with diameter bisecting at rove: # : = because the radii of a circle are congruent. = by the definition of bisect. Thus, and are both equidistant from and. y the onverse of the erpendicular isector, both and are on the perpendicular bisector of. Two points determine one line or segment, so is the perpendicular bisector of. Since is part of, #. hapter ircles

5 roblem Using iameters and hords rchaeology n archaeologist found pieces of a jar. She wants to find the radius of the rim of the jar to help guide her as she reassembles the pieces. What is the radius of the rim? How does the construction help find the center? The perpendicular bisectors contain diameters of the circle. Two diameters intersect at the circle s center. Step Trace a piece of the rim. raw two chords and construct perpendicular bisectors. Step The center is the intersection of the perpendicular bisectors. Use the center to find the radius. inch The radius is in. Got It?. Trace a coin. What is its radius? roblem inding Measures in a ircle ind two sides of a right triangle. The third side is either the answer or leads to an answer. lgebra What is the value of each variable to the nearest tenth? LN = () = diameter # to a chord bisects the chord. r = + Use the ythagorean. K r r. ind the positive square root of each side. cm L M N cm y # = = y + = diameter that bisects a chord that is not a diameter is # to the chord. raw an auiliary. The auiliary because they are radii of the same circle. Use the ythagorean. y = Solve for y. y. ind the positive square root of each side. Got It?. Reasoning In part (b), how does the auiliary make the problem simpler to solve? Lesson - hords and rcs

6 Lesson heck o you know HW? In }, m = What is m? How do you know?. What is true of and? Why?. Since =, what do you know about the distance of and from the center of }? o you UNRSTN? MTHMTIL RTIS. Vocabulary Is a radius a chord? Is a diameter a chord? plain your answers.. rror nalysis What is the error in the diagram? S Q R ractice and roblem-solving ercises MTHMTIL RTIS ractice In ercises and, the circles are congruent. What can you conclude? See roblem... X Z Y T H G J N K M L ind the value of. See roblem In the diagram at the right, GH and KM are perpendicular bisectors of the chords they intersect. What can you conclude about the center of the circle? Justify your answer. K H G M See roblems and.. In }, is a diameter of the circle and #. What conclusions can you make? hapter ircles

7 lgebra ind the value of to the nearest tenth..... pply. Geometry in imensions In the figure at the right, sphere with radius cm is intersected by a plane cm from center. ind the radius of the cross section }.. Geometry in imensions plane intersects a sphere that has radius in., forming the cross section } with radius in. How far is the plane from the center of the sphere?. Think bout a lan Two concentric circles have radii of cm and cm. segment tangent to the smaller circle is a chord of the larger circle. What is the length of the segment to the nearest tenth? How will you start the diagram? Where is the best place to position the radius of each circle?. rove -. Given: } with rove: cm cm. rove -. Given: } with rove:. rove -. Given: } with rove:. rove -. Given: } with diameter # at rove:, } and } are congruent. is a chord of both circles.. If = in. and = in., how long is a radius?. If = cm and a radius = cm, how long is?. If a radius = ft and = ft, how long is?. onstruction Use - to construct a regular octagon. Lesson - hords and rcs

8 . In the diagram at the right, the endpoints of the chord are the points where the line = intersects the circle + y =. What is the length of the chord? Round your answer to the nearest tenth.. onstruction Use a circular object such as a can or a saucer to draw a circle. onstruct the center of the circle.. Writing s - and - both begin with the phrase, within a circle or in congruent circles. plain why the word congruent is essential for both theorems. y ind m. (Hint: You will need to use trigonometry in ercise.).... rove -. Given: / is the # bisector of WY. rove: / contains the center of }X.. Given: } with # rove: hallenge W Z X Y rove each of the following.. onverse of -: arcs have congruent central angles.. onverse of -: chords have congruent central angles.. onverse of -: arcs have congruent chords.. onverse of -: Within a circle or congruent circles, congruent chords are equidistant from the center (or centers).. If two circles are concentric and a chord of the larger circle is tangent to the smaller circle, prove that the point of tangency is the midpoint of the chord. hapter ircles

9 Standardized Test rep ST/T. The diameter of a circle is cm and a chord of the same circle is cm. To the nearest tenth, what is the distance of the chord from the center of the circle?. cm. cm. cm. cm. The Smart all ompany makes plastic balls for small children. The diameter of a ball is cm. The cost for creating a ball is cents per square centimeter. Which value is the most reasonable estimate for the cost of making balls? $ $, $ $,. rom the top of a building you look down at an object on the ground. Your eyes are ft above the ground and the angle of depression is. Which distance is the best estimate of how far the object is from the base of the building? ft ft ft ft Short Response. bicycle tire has a diameter of in. How many revolutions of the tire are necessary to travel ft? Show your work. Mied Review ssume that the lines that appear to be tangent are tangent. is the center of each circle. ind the value of to the nearest tenth. See Lesson The legs of a right triangle are in. and in. long. The bisector of the right angle cuts the hypotenuse into two segments. What is the length of each segment, rounded to the nearest tenth? See Lesson -. Get Ready! To prepare for Lesson -, do ercises. Identify the following in } at the right.. a semicircle. a minor arc. a major arc See Lesson -. ind the measure of each arc in }.. ST. STQ. RT S T R Q Lesson - hords and rcs