lide 1 / 59 lide / 59 New Jersey enter for eaching and Learning Progressive Mathematics Initiative his material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. hese materials may not be used for any commercial purpose without the written permission of the owners. NJL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. eometry ircles 014-06-0 www.njctl.org lick to go to website: www.njctl.org lide / 59 able of ontents lide 4 / 59 lick on a topic to go to that section Parts of a ircle entral ngles & rcs rc Length & Radians hords, Inscribed ngles & riangles ircles and heir Parts angents & ecants Return to the table of contents egments & ircles Questions from Released PR amination lide 5 / 59 ircles are a type of igure lide 6 / 59 ircles figure lies in a plane and is contained by a boundary. uclid defined a circle and its center in this way: uclid defined figures in this way: efinition 15: circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. efinition 1: boundary is that which is an etremity of anything. efinition 14: figure is that which is contained by any boundary or boundaries. boundary divides a plane into those parts that are within the boundary and those parts that are outside it. hat which is within the boundary is the "figure." efinition 16: nd the point is called the center of the circle. his states that all the radii (plural of radius) drawn from the center of a circle are of equal length, which is a very important aspect of circles and their radii.
lide 7 / 59 lide 8 / 59 ircles and heir Parts nother way of saying this is that a circle is made up of all the points that are an equal distance from the center of the circle. ircles and heir Parts he symbol for a circle is and is named by a capital letter placed by the center of the circle. he below circle is named: ircle or radius center is a radius of radius (plural, radii) is a line segment drawn from the center of the circle to any point on the circle. circumference lide 9 / 59 lide 10 / 59 Radii radius (plural, radii) is a line segment drawn from the center of the circle to any point on its circumference. It follows from the definition of a circle that all the radii of a circle are congruent since they must all have equal length. iameters efinition 17: diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. n unlimited number of radii can be drawn in a circle. hat all radii of a circle are congruent will be important to solving problems. In this drawing, we know that line segments, and are all congruent. ince the diameter passes through the center of the circle and etends to the circumference on either side, it is twice the length of a radius of that circle. lide 11 / 59 lide 1 / 59 iameters here are an unlimited number of diameters which can be drawn within a circle. hey are all the same length, so they are all congruent. hords chord is a line segment whose endpoints lie on the circumference of the circle. o, a diameter is a special case of a chord. Why is a radius not a chord hat all diameters of a circle are congruent will be important to solving problems. In this drawing, we know that line segments, and are all congruent. ll the line segments in this drawing are chords.
lide 1 / 59 lide 14 / 59 emicircles hords here are an unlimited number of chords which can be drawn in a circle. efinition 18: semicircle is the figure contained by the diameter and the circumference cut off by it. nd the center of the semicircle is the same as that of the circle. hords are not necessarily the same length, so are not necessarily congruent. diameter hords can be of any length up to a maimum. semicircles What is the longest chord that can be drawn in a circle lide 15 / 59 iameters and Radii lide 16 / 59 1 diameter of a circle is the longest chord of the circle. he measure of the diameter, d, is twice the measure of the radius, r. rue In this case, = alse In general, d = r or r = 1/ d ample: In the diagram to the left, = 5. etermine &. lide 17 / 59 radius of a circle is a chord of a circle. lide 18 / 59 he length of the diameter of a circle is equal to twice the length of its radius. rue alse rue alse
lide 19 / 59 4 lide 0 / 59 If the radius of a circle measures.8 meters, what is the measure of the diameter 5 How many diameters can be drawn in a circle 1 4 infinitely many lide 1 / 59 lide / 59 rcs entral ngles & rcs rc or rc is the path between points and on the circumference of the circle. Return to the table of contents lide / 59 lide 4 / 59 rcs rc or rcs f course, you may have wondered why we went around the circle the way we did (in blue). his shorter path (blue) is called the minor arc. We use letters to denote minor arcs he longer path (green) is called the major arc. If we want to refer to a major arc, we will add another point along that path and include it in the name. or instance, we would name the green arc, rc to distinguish it from rc. rc or Now let's discuss some ways we can use arcs, and measure them.
lide 5 / 59 lide 6 / 59 entral ngles entral ngles central angle of a circle is any angle which has vertices consisting of the center of the circle and two points on the circumference. 700 o, the measure of the angle is the same as the measure of rc, also denoted by m. How many different central angles can you find in this diagram Name them. he measure of a central angle is equal to the measure of the arc it intercepts. In this case, the m is 700, since that is the m. hat also means the m is 900, since a full trip around the circle is 600. lide 7 / 59 lide 8 / 59 djacent rcs djacent rcs djacent arcs: two arcs of the same circle are adjacent if they have a common endpoint. rom the ngle ddition Postulate we know that m = m + m In this case, rc and rc are adjacent since they share the endpoint. It follows then that the measures of adjacent arcs can be added to find the measure of the arc formed by the adjacent arcs. In this case that: m = m + m which is the rc ddition Postulate lide 9 / 59 lide 0 / 59 6 ind the measure of rc RU. 7 ind the measure of rc R. 00 1000 600 R 900 800 V 00 U 1000 600 R 900 800 V U nswer nswer
lide 1 / 59 lide / 59 8 ind the measure of rc RV. 9 ind the measure of rc U. 900 1000 U 80 0 600 R 00 nswer 00 900 1000 600 V lide / 59 lide 4 / 59 10 Which type of arc is rc QR 11 Which type of arc is rc QR Minor rc Minor rc Major rc Major rc emicircle emicircle None of these None of these Q Q 10 600 10 600 0 0 800 800 R Note that you need to use the indicated degree measures as the drawing is not to scale. lide 5 / 59 R Note that you need to use the indicated degree measures as the drawing is not to scale. lide 6 / 59 1 Which type of arc is rc Q 1 Which type of arc is rc Minor rc Major rc U 800 R V nswer Minor rc Q 10 600 0 emicircle Major rc Q 10 600 0 emicircle 800 None of these 800 None of these Note that you need to use the indicated degree measures as the drawing is not to scale. R Note that you need to use the indicated degree measures as the drawing is not to scale. R
lide 7 / 59 lide 8 / 59 ongruent ircles and rcs 14 Which type of arc is rc R Minor rc ll circles are similar since they all have the identical shape Major rc emicircle ircles which have the same radius are congruent, since they will overlap at every point if their centers are lined up. None of these Q 10 600 0 800 Note that you need to use the indicated degree measures as the drawing is not to scale. R lide 9 / 59 ongruent ircles and rcs rcs are similar if they have the same measure rcs are congruent if they have the same measure and are either part of the same circle or of congruent circles. lide 40 / 59 ongruent ircles and rcs ince the measure of an arc is equal to that of the central angle which intercepts it, arcs within the same circle which are intercepted by central angles of the same measure, are congruent. 550 550 rc and rc are congruent because they are in the same circle and have the same measure. lide 41 / 59 ongruent ircles and rcs rcs are similar if they have the same measure rcs are congruent if they have the same measure and are either part of the same circle or of congruent circles. 550 550 rc and rc are congruent because they are in the same circle and have the same measure. lide 4 / 59 ongruent ircles and rcs he two circles below are also called concentric circles, since they share the same center, but have different radii lengths. oncentric circles are similar, but not congruent. R R U rc R and rc U are similar since they have the same measure. ut they are not congruent because they are arcs of circles that are not congruent. U
lide 4 / 59 lide 44 / 59 15 oth ircle and ircle have a radius of.0 m. re rc and rc similar 16 oth ircle and ircle have a radius of.0 m. re rc and rc congruent Yes Yes No No 750 750 750 lide 45 / 59 17 # 750 lide 46 / 59 18 LP MN rue rue alse alse M 1800 L 700 400 850 P N lide 47 / 59 19 ircle P has a radius of and has a measureof 90. What is the length of 6 9 lide 48 / 59 0 wo concentric circles always have congruent radii. rue alse P Note that you need to use the indicated degree measures as the drawing is not to scale.
lide 49 / 59 lide 50 / 59 1 If two circles have the same center, they are congruent. anny cuts a pie into 6 congruent pieces. What isthe measure of the central angle of each piece rue alse lide 51 / 59 lide 5 / 59 # rc Length & Radians efore we etend our thinking of central angles and arc measures to arc lengths, it's worth reflecting on the number π, which will be central to our work. his number was a devastating discovery to reek mathematicians. In fact, the reason that he lements was written without relying on numbers, was because numbers were considered unreliable to the reeks after π was discovered. Return to the table of contents lide 5 / 59 lide 54 / 59 # Until then, Pythagoras, and his followers believed that "ll was Number." ut when they sought to find the number that is ratio of the circumference to the diameter of a circle, they found that there wasn't one. he closer they looked, the more impossible it became to find a number solution to the simple epression of /d: the circumference divided by the diameter of a circle. Until then, Pythagoras, and his followers believed that "ll was Number." ut when they sought to find the number that is ratio of the http://bobchoat.files.wordpress.com/01/06/pi-day004.jpg circumference to the diameter of a circle, they found that there wasn't one. he closer they looked, the more impossible it became to find a number that solves the simple equation of /d. http://bobchoat.files.wordpress.com/01/06/pi-day004.jpg
lide 55 / 59 lide 56 / 59 # # π is an eample of an irrational number. number that is not the ratio of two integers. o, no matter how far you take it, it keeps going without settling down. We are comfortable with irrational numbers now, but the reeks weren't. Now we know that there are many more irrational numbers than rational numbers. In mathematics, it's best to just leave answers with the symbol π. In science, engineering and other fields which need a rational answer, and where π shows up a lot, the value of π is just estimated with the number of digits necessary for the problem. or most problems,.14 is close enough. Rational numbers are like islands in a sea of irrational numbers. or others, you might use.14159...but you will rarely need more than that. ut, we are more familiar with those islands as that's where we grew up. or this course, just leave your answers with the number π as part of your answer. lide 57 / 59 lide 58 / 59 rc Lengths rc Lengths he relationship between the circumference of a circle to its diameter is = πd ince d = r, this is usually epressed as 700 ut, if we are also told that the radius of the circle is 0 cm, we can determine the length of rc, also denoted as. = πr We know that a full trip around a circle is equal to 600, so if we know the angle of an arc and the radius, we can determine the length of the arc. hat's how far you'd have to travel along that arc to get from Point to Point. lide 59 / 59 lide 60 / 59 rc Lengths rc Lengths We could do this by first figuring out the circumference using 70 0 0 cm lternatively, we could just set this up as a ratio and solve it in one step. = πr = π(0 cm) = 40π cm hen figuring what percentage of the circumference rc is by the ratio of ()/ = 700/600 = 0.1944 ince 600 is the entire circumference. In the figure off to the left, we know that the measure of rc is equal to that of the entral ngle...700. 70 0 0 cm rc length = entral angle ircumference 60 = 70 πr 60 70 (πr) = 60 o, is 70 (π)(0) = 60 (0.1944)(40π cm) = 7.8π cm (about 4 cm) = 7.8π cm
lide 61 / 59 lide 6 / 59 ample rc Lengths or any arc, you can find its length by multiplying the circumference of the circle (πr) by the angle of arc divided by 60. # In ircle, the central angle is 600 and the radius is 8 cm. ind the length of rc In this case, the measure of the arc is θ, since it is equal to the central angle. hen, πr = = 8 cm θ 600 600 θ πr 600 lide 6 / 59 In circle where is a diameter, find the length of rc. he length of diameter is 15 cm. lide 64 / 59 4 In circle where is a diameter, find the length of rc. he length of is 15 cm. 150 150 15 cm 15 cm lide 65 / 59 5 In circle where is a diameter, find the length of rc. he length of is 15 cm. lide 66 / 59 6 In circle can it be assumed that is a diameter Yes No 150 15 cm 150
lide 67 / 59 lide 68 / 59 8 ind the circumference of circle. 7 ind the length of 450 cm 750 6.8 cm lide 69 / 59 lide 70 / 59 Radians 9 In circle, Lines WY & XZ are diameters and have lengths of 6. he m XY = 1400, what is YZ π 6π 4 π 4π nother way of measuring angles, as an alternative to degrees, is radians. Where degrees are arbitrary (Why are there 60 degrees in circle), radians are very natural. Radians are the ratio of an arc length divided by the radius of the arc. raw a Picture. Hint: click to reveal lide 71 / 59 lide 7 / 59 Radians o, the radian measure of angle is just the length of rc divided by the length of radius ( or ). he circumference of a circle is given by = πr. Radians o, the radian measure of a full trip around a circle is πr/r = π. here are no units for radians, since the lengths cancel out, but you can write rads or radians just to indicate what you are doing. ince there are no units, these angle measures are much easier to use when you study trigonometry, physics and calculus. ll scientific calculators allow you to use degrees or radians. Just make sure it is set to the correct one when you are entering angle measurements.
lide 7 / 59 lide 74 / 59 Radians Radians ince a full trip around a circle is 600, and is also π radians, they must be equal. lternatively, you could write a proportion and use the crossproduct property to find your missing measurement. o, to convert from one to the other, just multiply by the appropriate conversion factor. π = 600 600 π =1 degrees radians = π 600 ince π = 600, each of these fractions is just equal to 1. oth methods will be shown in the net eample. Use whichever method is easier for you. Multiplying anything by them doesn't change it's value, since multiplying by 1 has no effect. lide 75 / 59 lide 76 / 59 Radians Radians π = 600 600 π =1 or instance, m# = 150. It is also equal to m# = (150) π 0 60 If you are using the proportion and the cross-product property, you could set up & solve the problem with the works shown below. degrees radians = π 600 15 = π 60 m# = 0.69 π radians lide 77 / 59 0 How many radians is 180 degrees (Round π to.14 in your answer.) 50π = 60 = 5π = 0.69π radians 6 lide 78 / 59 1 How many radians is 90 degrees (Round π to.14 in your answer.)
lide 79 / 59 How many radians is 140 (Round π to.14 in your answer.) lide 80 / 59 How many degrees is π radians lide 81 / 59 4 How many degrees is π radians lide 8 / 59 5 How many degrees is 1.0 radian lide 8 / 59 6 How many degrees is 1.6 radians lide 84 / 59 7 ircle has a radius of. What is the arc length for # π/ π π π/4 10 45 0 PR Released Question - Response ormat
lide 85 / 59 lide 86 / 59 8 ircle has a radius of. 9 ircle has a radius of. What is the arc length for # What is the arc length for # π/ π/ π π π/4 π π 10 45 0 π/4 10 0 PR Released Question - Response ormat PR Released Question - Response ormat lide 87 / 59 lide 88 / 59 Question /7 40 ircle has a radius of. opic: ngles, rcs and rc Lengths What is the arc length for # π/ 10 45 0 π 45 π π/4 10 45 0 PR Released Question - Response ormat PR Released Question lide 89 / 59 lide 90 / 59 lick on the link below and complete the labs before the hords lesson. hords, Inscribed ngles & riangles Return to the table of contents Lab - Properties of hords
lide 91 / 59 lide 9 / 59 he rc of the hord efinition: When a chord and a minor arc have the same endpoints, we call the arc he rc of the hord. hord isector heorem If a diameter or radius of a circle is perpendicular to a chord, then the diameter or radius bisects the chord and its arc. P Q PQ is the arc of PQ is a diameter of the circle and is perpendicular to chord herefore, X X & X Recall the definition of a chord: a line segment with endpoints on the circle. lide 9 / 59 lide 94 / 59 Proof of hord isector heorem X Proof of hord isector heorem iven: is a diameter which is perpendicular to Prove: bisects and its minor arc X s a first step, let's draw radii from the enter to the points and to construct Δ. lide 95 / 59 41 ill in the statement for step #. X and X are complementary X and X are right angles Now, we can use the triangles which have been formed in our proof. X tatement 1 X X # X # ΔX # ΔX Reason Right angles are formed by perpendicular lines 4 Refleive Property of # 5 Hypotenuse - Leg heorem # 6 X # X and X # X 7 # 8 bisects and Minor rc [his object is a pull tab] is a diameter which is perpendicular iven to nswer Proof of hord isector heorem lide 96 / 59 rcs intercepted by # central s of a circle are #
lide 97 / 59 lide 98 / 59 4 ill in the reason for step #. 4 ill in the statement for step #4. X and X are complementary X and X are right angles X # X # ΔX # ΔX efinition of supplementary orresponding parts of # Δs are # ide-ngle-ide heorem nswer X efinition of bisector Reason 4 Refleive Property of # 5 Hypotenuse - Leg heorem # 8 bisects and Minor rc Right angles are formed by perpendicular lines 4 Refleive Property of # 5 Hypotenuse - Leg heorem 6 X # X and X # X rcs intercepted by # central s of a circle are # 7 # Reason # rcs intercepted by # central s of a circle are # 8 bisects and Minor rc lide 99 / 59 45 ill in the reason for step #6. Radii of circles are # X # X # ΔX # ΔX X tatement [his object is a pull tab] Reason is a diameter which is perpendicular 1 iven to Right angles are formed by perpendicular lines Right angles are formed by perpendicular lines # 4 Refleive Property of # 4 Refleive Property of # 5 Hypotenuse - Leg heorem 5 Hypotenuse - Leg heorem 6 X # X and X # X 6 X # X and X # X rcs intercepted by # central s of a circle are # 7 # 8 bisects and Minor rc 8 bisects and Minor rc lide 101 / 59 Proof of hord isector heorem iven: is a diameter which is perpendicular to Prove: bisects and its minor arc X efinition of bisector 1 Reason # Right angles are formed by perpendicular lines 4 Refleive Property of # 5 Hypotenuse - Leg heorem 7 # 8 bisects and Minor rc X tatement 6 X # X and X # X [his object is a pull tab] is a diameter which is perpendicular iven to nswer orresponding parts of # Δs are # ide-ngle-ide heorem tatement lide 10 / 59 46 ill in the reason for step #8. efinition of supplementary rcs intercepted by # central s of a circle are # 7 # Radii of circles are # [his object is a pull tab] Reason # X efinition of bisector is a diameter which is perpendicular iven to orresponding parts of # Δs are # ide-ngle-ide heorem efinition of supplementary nswer X and X are complementary X and X are right angles tatement lide 100 / 59 44 ill in the statement for step #5. 1 7 # [his object is a pull tab] is a diameter which is perpendicular iven to 6 X # X and X # X tatement 1 Right angles are formed by perpendicular lines [his object is a pull tab] X is a diameter which is perpendicular 1 iven to nswer tatement nswer Radii of circles are # rcs intercepted by # central s of a circle are # 1 Reason is a diameter which is perpendicular iven to X and X are right angles Right angles are formed by perpendicular lines # Radii of a circle are # 4 X # X Refleive Property of # 5 ΔX # ΔX Hypotenuse - Leg heorem 6 X # X and X # X orresponding parts of # Δs are # 7 # rcs intercepted by # central s of a circle are # 8 bisects and Minor rc efinition of bisector
lide 10 / 59 lide 104 / 59 onverse of hord isector heorem In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter. X he chord is the perpendicular bisector of the chord. Proof of onverse of hord isector heorem In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter. herefore, is a diameter of the circle and passes through the center of the circle. X lide 105 / 59 Proof of rcs and hords heorem In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. his follows from the fact that the measure of an arc is equal to that of the central angle which intercepts it. iff *iff stands for "if and only if" ince all the radii,,, and are congruent and the sides and are congruent, the triangles and are congruent by ide-ide-ide. hat means that the central angles are congruent which means that the arcs are congruent since arcs intercepted by congruent central angles are congruent. lide 107 / 59 lide 108 / 59 IIN R If XY YZ, then point Y and any line segment, or ray, that contains Y, bisects XYZ XMPL Y,, and (9)0 his just follows from the definition of a bisector as dividing something into two pieces of equal measure. Z ind: X onstruct ΔX and ΔX and by proving them congruent, you show that and are radii, which means that chord passes through the center, and is a diameter. lide 106 / 59 rcs and hords heorem In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. his proof is very much like that of the original theorem of which this is the converse. (80 - )0
lide 109 / 59 heorem In the same, or congruent, circles, two chords are congruent if and only if they are equidistant from the center. lide 110 / 59 heorem In the same, or congruent, circles, two chords are congruent if and only if they are equidistant from the center. iff iff Proving this just requires creating Δ and Δ and proving that they are congruent, which means that their altitudes are equal, which is their distance from the center of the circle. Here Δ and Δ and their congruent sides are shown. lide 111 / 59 ample iven circle, QR = = 16. ind U. lide 11 / 59 47 In circle R, and m = 108. ind m. R U Q R V 1080 5-9 lide 11 / 59 48 iven circle below, the length of is: 5 10 15 0 10 lide 114 / 59 49 iven: circle P, PV = PW, QR = + 6, and = - 1. ind the length of QR. 1 7 0 8 R V. Q P W
lide 115 / 59 lide 116 / 59 Inscribed ngles 50 H is a diameter of the circle. rue alse M Inscribed angles are angles whose vertices lie on the circle and whose sides are chords of the circle. 5 ngle is an inscribed angle and rc is its intercepted arc. H lide 117 / 59 lide 118 / 59 Inscribed ngle heorem Inscribed ngles he measure of an inscribed angle is equal to half the measure of the intercepted arc or central angle he minor arc that lies in the interior of the inscribed angle and has endpoints which are vertices of the angle is called the intercepted arc. o prove this, we need to find the measure of relative to the measure of the intercepted rc rc is the intercepted arc of inscribed angle. Lab - Inscribed ngles lide 119 / 59 lide 10 / 59 Inscribed ngle heorem Inscribed ngle heorem o prove the theorem, we first draw the diameter which creates Δ and Δ he measure of, shown with green lines, is the same as the measure of the minor arc, shown in blue.
lide 11 / 59 lide 1 / 59 5 What is true of all radii of the same circle 51 What are,, and chords radii nothing special diameters the are perpendicular they are parallel not special they are of equal length lide 1 / 59 lide 14 / 59 54 Which is true about base angles of isosceles Δs 5 What types of triangles are and scalene they are equal right isosceles not special they are double they are half nothing special Why lide 15 / 59 Inscribed ngle heorem qual angles are marked by and y. lide 16 / 59 55 What types of angle are angles a and b of and interior eterior not special right y y y a b y
lide 17 / 59 lide 18 / 59 57 o what is the measure of the sum of a + b equal 56 Which of the below is true a = a=b a + b = arc a + b = arc b = y =y a + b = arc a+b=+y a b y y a b y y lide 19 / 59 lide 10 / 59 Inscribed ngle heorem lso, a+ b = +y = (+y) y a nd +y = m o, (m ) = m b nd m = 1/(m ) y he measure of the inscribed angle is equal to half the measure of the central angle and, therefore, half the measure of the intercepted arc. his was proven if the center is within the angle, but is true for all cases. Inscribed ngle heorem It's not essential to go through the proof of the other two cases, but if you have time, it's good practice. irst, if the inscribed angle does not include the center. lso, it is equal to half the measure of the central angle intercepting that arc. lide 1 / 59 Inscribed ngle heorem lide 11 / 59 he measure of an inscribed angle, such as, is equal to half the measure of the intercepted arc, which is. o, a+ b = Inscribed ngle heorem y ab y ince these triangles overlap so much one side of the inscribed angle,, and its associated isosceles triangle and eternal angle are shown in orange. he other leg and its associated isosceles triangle and eternal angle are shown in dark blue. he proof then follows the same form.
lide 1 / 59 lide 14 / 59 Inscribed ngle heorem Inscribed ngle heorem y ab y m = -y m = a-b m = -y = (-y) o, m = 1/(m ) he inscribed angle equals half the measure of the intercepted arc lide 15 / 59 lide 16 / 59 ample Inscribed ngle heorem m = ind m, mq and mqr. a = m = a = R Q 500 o, m = 1/(m ) a he last case is if the a leg of the inscribed angle passes through the center. a = b = y he inscribed angle equals half the measure of the intercepted arc 480 P lide 17 / 59 lide 18 / 59 heorem heorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent. his follows directly from our prior proofs showing that the measure of an inscribed angle is equal to half that of the intercepted arc. y the transitive property of equality, if two inscribed angles intercept the same are they must be equal. In this case, m = m since the both intercept rc. In a circle, parallel chords intercept congruent arcs. Note that this does N mean that rc is equal to rc. In circle, if hord is parallel to hord then rc is equal to rc.
lide 19 / 59 lide 140 / 59 58 iven the figure below, which pairs of angles are congruent R U R U R U R R 59 ind m Y R X Y U P Z lide 141 / 59 60 In a circle, two parallel chords on opposite sides of the center have arcs which measure 1000 and 100. lide 14 / 59 61 iven circle, find the value of. ind the measure of one of the arcs included between the chords. 00 lide 14 / 59 lide 144 / 59 ry his 6 iven circle, find the value of. In this circle: mpq = 5 mq = 11 m = 88 1000 ind m 1, m, m & m 4 Q 50 P 1 4
lide 145 / 59 lide 146 / 59 Inscribed riangles orollary to Inscribed ngle heorem triangle is inscribed if all its vertices lie on a circle. If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle... inscribed triangle. is a diameter of the circle if ΔL is a right Δ, and # L is its right angle, follows directly from the inscribed angle theorem.. ince # L has a measure of 900, it must intercept an arc whose measure is 1800, this is half the circle, so must be a diameter L m# L = 90 lide 147 / 59 lide 148 / 59 6 In the diagram, is a central angle and m = 600. What is m 150 00 600. 100 64 What is the value of 5 10 1 15 lide 149 / 59 65 If m # = 440, find m#. he figure shows Δ inscribed in circle. PR Released Question - Response ormat (1 + 40)0 (8 + 10)0 lide 150 / 59 Question 1/5 opic: Inscribed ngles
lide 151 / 59 lide 15 / 59 66 ind the value of. 67 he length of is because... 10 10 less than 10 P greater than 10 10 P not to scale not to scale he figure shows a circle with center P, a diameter, and inscribed Δ. P = 10. Let m# = 0 and m# = (+54)0. he figure shows a circle with center P, a diameter, and inscribed Δ. P = 10. Let m# = 0 and m# = (+54)0. PR Released Question - Response ormat PR Released Question - Response ormat lide 15 / 59 lide 154 / 59 Question 17/5 68 he length of is... because. opic: Inscribed ngles ΔP is equilateral m#p < 600 m#p > 600 10 P not to scale he figure shows a circle with center P, a diameter, and inscribed Δ. P = 10. Let m# = 0 and m# = (+54)0. PR Released Question - Response ormat a. 10 d. P is equilateral b. less than 10 e. m P < 60 c. greater than 10 f. m P > 60 lide 155 / 59 Point is the center of a circle, and is a diameter of the circle. Point is a point on the circle different from and. 69 he statement > is: lide 156 / 59 Point is the center of a circle, and is a diameter of the circle. Point is a point on the circle different from and. 70 he statement m# = 1/ (m#) is: lways rue lways rue ometimes rue ometimes rue Never rue Never rue PR Released Question - Response ormat PR Released Question - Response ormat
lide 157 / 59 lide 158 / 59 Point is the center of a circle, and is a diameter of the circle. Point is a point on the circle different from and. Point is the center of a circle, and is a diameter of the circle. Point is a point on the circle different from and. 7 he statement that m# = (m#) is: 71 he statement that m# = 900 is: lways rue lways rue ometimes rue ometimes rue Never rue Never rue PR Released Question - Response ormat PR Released Question - Response ormat lide 159 / 59 lide 160 / 59 Point is the center of a circle, and is a diameter of the circle. Point is a point on the circle different from and. Question /5 opic: Inscribed ngles 7 If m# = 00, what is m#) 00 400 700 1400 PR Released Question - Response ormat lide 161 / 59 lide 16 / 59 angents angents & ecants I Return to the table of contents H tangent is a line, ray or segment which touches a circle at just one point. ll three types of tangent lines are shown on this drawing. he point where the line touches the circle is called the "point of tangency." In this case, those points are, and H.
lide 16 / 59 lide 164 / 59 N angents ecants Note that for a ray or segment to be a tangent, it must not touch the circle in more than one point even if it were etended. he segment and ray shown to the left are N tangents because if they were etended they would touch the circle in more than one point. lide 165 / 59 I secant is a line, ray or segment which touches a circle at two points. ll three types of secant lines are shown on this drawing. lide 166 / 59 Intersections of ircles Intersections of ircles oplanar circles can intersect at zero, one, or two points. elow are shown three ways in which a pair of circles can have no points of contact. ircles which are sideby-side may not intersect. angent ircles intersect at one point. he two types of tangent circles are shown below. ircles within one another may not intersect.. Remember that: ircles within one another and have a common center are "concentric." lide 167 / 59 Intersections of ircles hese circles intersect at two points. an two distinct circles intersect at three points lide 168 / 59 ommon angents wo circles can have between zero and four common tangents. wo completely separate circles have four common tangents.
lide 169 / 59 lide 170 / 59 ommon angents ommon angents wo circles can have between zero and four common tangents. wo circles can have between zero and four common tangents. wo eternally tangent circles have three common tangents. wo overlapping circles have two common tangents. lide 171 / 59 ommon angents lide 17 / 59 74 How many common tangent lines do the circles have wo circles can have between zero and four common tangents. wo non-tangent circles within one another have no common tangents. lide 17 / 59 75 How many common tangent lines do the circles have lide 174 / 59 76 How many common tangent lines do the circles have
lide 175 / 59 lide 176 / 59 77 How many common tangent lines do the circles have 78 Which term best describes Line H Radius iameter hord ecant angent ommon angent Point of angency enter lide 177 / 59 lide 178 / 59 79 Which term best describes Line 80 Which term best describes Line H H Radius iameter hord ecant angent ommon angent Point of angency enter Radius iameter hord ecant angent ommon angent Point of angency enter lide 179 / 59 lide 180 / 59 81 Which term best describes Points and 8 Which term best describes Line H H Radius iameter hord ecant angent ommon angent Point of angency enter Radius iameter hord ecant angent ommon angent Point of angency enter
lide 181 / 59 lide 18 / 59 8 Which term best describes Line 84 Which term best describes Point H H Radius iameter hord ecant angent ommon angent Point of angency enter Radius iameter hord ecant angent ommon angent Point of angency enter lide 184 / 59 angents and Radii are Perpendicular 85 Which term best describes egment Radius iameter hord ecant angent ommon angent Point of angency enter lide 18 / 59 H tangent and radius that intersect on a circle are perpendicular. his is an essential result for mathematics and physics. It appears to be true, but let's prove it. lide 185 / 59 lide 186 / 59 Proof that angents and Radii are Perpendicular Proof that angents and Radii are Perpendicular Let's use an indirect proof. If is a leg of the right triangle and is the hypotenuse, then must be longer than. We'll make an assumption and see if it leads to a contradiction. ut, etends from the center of the circle to beyond the circle. Let's assume that another point on is where a line from Point is perpendicular to the. Let's name that point, so the line perpendicular to is. hen must be a right triangle with and being the legs. hen must be the hypotenuse. nd only etends from the center of the circle to the circumference of the circle. o, must be shorter than. his is a contradiction, which proves that our original assumption was incorrect.
lide 187 / 59 lide 188 / 59 Proof that angents and Radii are Perpendicular Using an Intersecting angents & Radius to olve Problems ur assumption was that was not perpendicular to. Whenever you are given, or can draw, a circle and a tangent, you can construct a radius to the point of tangency. If that is false, then it must be that is perpendicular to, which is what set out to prove. he tangent and radius will form a right angle. his is often very helpful and what is needed to solve a problem. Lab - angent Lines lide 189 / 59 lide 190 / 59 Using an Intersecting angent & Radius to olve Problems Velocity ravity ometimes, that takes the form of creating a right triangle, with all the information that is provided. Using an Intersecting angent & Radius to olve Problems his property of a tangent and intersecting radius shows up often in physics. r, the maimum torque that can be applied on a wheel by a road when braking, or by a tire on a road while accelerating. or instance, it eplains why the force of gravity pulling the moon into its circular orbit is perpendicular to the direction of its orbital velocity. he force acts along the radius and the motion is tangent to the circular orbit. lide 191 / 59 Using an Intersecting angent & Radius to olve Problems lide 19 / 59 heorem angent segments from a common eternal point are congruent. R and are right angles. R r, the transfer of force due to a bicycle chain. P is congruent to itself by the Refleive Property. P PR and P are congruent because they are radii. ΔPR and ΔP are congruent due to the Hypotenuse-Leg heorem. herefore, R and are congruent since they are corresponding parts of congruent triangles.
lide 19 / 59 lide 194 / 59 ample 86 is a radius of circle. Is tangent to circle iven: R is tangent to circle at and R is tangent to circle at. ind. Yes No 60 5 67 8 R + 4 lide 195 / 59 lide 196 / 59 87 is a point of tangency. ind the radius r of circle. 1.7 60 14 r.5 5 r 48 cm 89,, and are tangents to circle. = 5, = 8, and = 4. ind the perimeter of triangle. R - 8 lide 198 / 59 ngles Intercepted by angents and ecants angents and secants can form other angle relationships in circle. Recall the measure of an inscribed angle is 1/ its intercepted arc. his can be etended to any angle that has its verte on the circle. his includes angles formed by two secants, a secant and a tangent, a tangent and a chord, and two tangents. 6 cm lide 197 / 59 88 In circle, is tangent at and is tangent at. ind.
lide 199 / 59 lide 00 / 59 heorem: angent and a hord heorem If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is half the difference of their intercepted arcs. his is just a special case of the rule that the measure of an inscribed angle is equal to half the measure of its intercepted arc. M his results in three rules for calculating angles, but they are very similar to one another, so not hard to remember. In this case, m 1 = 1 mrm 1 R specially, if you see one right after the other. m = 1 mmr lide 01 / 59 lide 0 / 59 he Intercepted rc of a angent and a ecant he Intercepted rc of wo angents m 1 = P 1 (m - m) Q 1 m = 1 (mpqm - mpm) M lide 0 / 59 he Intercepted rc of wo ecants lide 04 / 59 he Intercepted rc of wo hords If two chords intersect inside a circle, then the measure of each angle is the average of the two intercepted arcs. X M m = W 1 (mxy - mwz) 1 m 1 = 1 1 (m + mmh) Z Y H his is equally true for each pair of vertical angles. he other pair of angles is on the net slide.
lide 05 / 59 lide 06 / 59 hords vs. ecants and/or angents he Intercepted rc of wo hords If two chords intersect inside a circle, then the measure of each angle is the average of the two intercepted arcs. way to remember whether to add or subtract the two arcs that the segments intersect is to visualize the segments rotating on their intersection point to see which operation sign they are close to resembling. M X M m = 1 (mm + mh) W H H Z In the diagram above, if you manipulate the segments at the intersection point, they make an addition sign, so add the arcs together before taking 1/. lide 07 / 59 90 ind the value of. Y In the diagram above, if you manipulate the rays at their intersection point, they overlap each other, making a subtraction sign, so subtract the arcs before taking 1/. lide 08 / 59 91 ind the value of. 1 00 0 760 1780 0 1 5 60 lide 09 / 59 9 ind the value of. lide 10 / 59 9 ind the value of. 780 H 0 40 ( - 0 ) ( + 06 ) 40
lide 11 / 59 lide 1 / 59 95 ind m#1. 94 ind m. 6 00 1 650 lide 1 / 59 96 ind the value of. 0 1. lide 14 / 59 ample o find the angle, you need the measure of both intercepted arcs. irst, find the measure of the minor arc m. hen we can calculate the measure of the angle 0. 5 4 50 470 0 lide 15 / 59 97 ind the value of. lide 16 / 59 98 ind the value of. tudents type their answers here tudents type their answers here 00 0 0 1 0 00
lide 17 / 59 lide 18 / 59 99 ind the value of 100 ind the value of. tudents type their answers here 5 00 tudents type their answers here 0 1 00 ( 5 + 10 0 ) lide 19 / 59 lide 0 / 59 Released PR am Question 101 ind the value of. he following question from the released PR eam uses what we just learned and combines it with what we learned earlier to create a challenging question. ( - 00 ) 00 Please try it on your own. hen we'll go through the process we used to solve it. lide 1 / 59 Question 5/5 lide / 59 opic: angents Question 5/5 opic: angents his is a great problem and draws on a lot of what we've learned. ake a shot at it in your groups. hen we'll work on it step by step together by asking questions that break the problem into pieces. P = = 6
lide / 59 10 What have we learned that will help solve this problem lide 4 / 59 Question 5/5 opic: angents tangent and intersecting radius are perpendicular ll radii of a circle are congruent imilar triangles have proportional corresponding parts ll of the above irst, draw a radius from the center of each circle to the point where the given tangent meets the circle, points and P. P = = 6 P = = 6 lide 5 / 59 Question 5/5 opic: angents lide 6 / 59 10 What can you see from this drawing ΔP and ΔM are right triangles Now draw ΔP and ΔM. ΔP and ΔM are similar he corresponding sides of P and M are proportional ll of the above P = = 6 P = = 6 lide 7 / 59 104 What are the lengths of P and M lide 8 / 59 105 What is the ratio of proportionality, k, between ΔP and ΔM P = 6 and M = 6 P = and M = P = 6 and M = hey can't be found from this P = = 6 P = = 6 6 6
lide 9 / 59 lide 0 / 59 106 Write an epression for the length of P. 107 Write an epression for the length of M. P = 1 M = 1 P = 1 + M = + P = 6 + It can't be found from this P = 6 + It can't be found from this P = = 6 P = = 6 6 6 6 6 lide 1 / 59 lide / 59 108 Write a proportion relating M and P. 6 = 1 + + = 1 + 6 + 6 = 1 + It can't be found from this 109 What is the length of M P = = 6 P = = 6 6 6 6 6 110 Write an equation to find. = + 9 lide / 59 lide 4 / 59 111 What is the length of 9 = - 9 = + It can't be found from this P = = 6 P = = 6 6 6 9
lide 5 / 59 lide 6 / 59 Question 5/5 11 What would be a good way to check your answer. opic: angents olve for and see if it is equal to olve for P and see if it is equal to olve for and see if it is double olve for and see if it is half P = = 6 6 6 lide 7 / 59 lide 8 / 59 heorem If two chords intersect inside a circle, then the products of the measures of the segments of the chords are equal. Return to the table of contents lide 9 / 59 11 ind the value of. 114 ind the length of ML. M 4 5 lide 40 / 59 5 + L J K +4 +1 nswer egments & ircles
lide 41 / 59 lide 4 / 59 116 ind the value of. 115 ind the length of JK. M L J 16 K + 9 18 +4 +1 lide 4 / 59 118 ind the value of. 117 ind the value of. - 4 5 6 lide 44 / 59 + + 1 1 9 + 6 + 6 lide 45 / 59 ecant egment heorem If two secant segments are drawn to a circle from an eternal point, then the product of the measures of one secant segment and its eternal secant segment equals the product of the measures of the other secant segment and its eternal secant segment. lide 46 / 59 119 ind the value of. 9 6 5
lide 47 / 59 lide 48 / 59 10 ind the value of. 11 ind the value of. 5 + + 1 4 + 4-1 - lide 49 / 59 angent-ecant heorem lide 50 / 59 1 ind the length of R. Q If a tangent segment and a secant segment are drawn to a circle from an eternal point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its eternal secant segment. 16 R 8 lide 51 / 59 1 ind the value of. lide 5 / 59 14 ind the value of. 4 1 1
lide 5 / 59 lide 54 / 59 Question /7 opic: ngles, rcs and rc Lengths Questions from Released PR amination 10 45 0 Return to the table of contents lide 55 / 59 Question 1/5 opic: Inscribed ngles lide 56 / 59 Question 5/5 lide 57 / 59 Question 17/5 opic: Inscribed ngles a. 10 d. P is equilateral b. less than 10 e. m P < 60 c. greater than 10 f. m P > 60 opic: angents lide 58 / 59 Question /5 opic: Inscribed ngles
lide 59 / 59