Geometry. Circles. Slide 1 / 150. Slide 2 / 150. Slide 3 / 150. Table of Contents

Transcription

1 New Jersey enter for eaching and Learning Slide 1 / 150 Progressive Mathematics Initiative his material is made freely available at wwwnjctlorg 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 lick to go to website: wwwnjctlorg Geometry Slide 2 / 150 ircles wwwnjctlorg able of ontents Parts of a ircle ngles & rcs hords, Inscribed ngles & Polygons angents & Secants Segments & ircles Equations of a ircle rea of a Sector lick on a topic to go to that section Slide 3 / 150

2 Slide 4 / 150 Parts of a ircle Return to the table of contents circle is the set of all points in a plane that are a fixed distance from a given point in the plane called the center Slide 5 / 150 center Slide 6 / 150 he symbol for a circle is and is named by a capital letter placed by the center of the circle (circle or ) is a radius of radius (plural, radii) is a line segment drawn from the center of the circle to any point on the circle It follows from the definition of a circle that all radii of a circle are congruent

3 M R is a chord of circle chord is a segment that has its endpoints on the circle Slide 7 / 150 is the diameter of circle diameter is a chord that goes through the center of the circle ll diameters of a circle are congruent What are the radii in this diagram? he relationship between the diameter and radius the he measure of the diameter, d, is twice the measure of the radius, r Slide 8 / 150 M herefore, or In If, then what is the length of what is the length of 1 diameter of a circle is the longest chord of the circle rue Slide 9 / 150 False

4 2 radius of a circle is a chord of a circle Slide 10 / 150 rue False 3 wo radii of a circle always equal the length of a diameter of a circle Slide 11 / 150 rue False 4 If the radius of a circle measures 38 meters, what is the measure of the diameter? Slide 12 / 150

5 5 How many diameters can be drawn in a circle? Slide 13 / infinitely many secant of a circle is a line that intersects the circle at two points line l is a secant of this circle Slide 14 / 150 l tangent is a line in the plane of a circle that intersects the circle at exactly one point (the point of tangency) E k line k is a tangent is the point of tangency tangent ray,, and the tangent segment,, are also called tangents hey must be part of a tangent line Note: his is not a tangent ray OPLNR IRLES are two circles in the same plane which intersect at 2 points, 1 point, or no points Slide 15 / 150 oplanar circles that intersects in 1 point are called tangent circles oplanar circles that have a common center are called concentric 2 points tangent circles 1 point no points concentric circles

6 ommon angent is a line, ray, or segment that is tangent to 2 coplanar circles Slide 16 / 150 Internally tangent (tangent line passes between them) Externally tangent (tangent line does not pass between them) 6 How many common tangent lines do the circles have? Slide 17 / How many common tangent lines do the circles have? Slide 18 / 150

7 8 How many common tangent lines do the circles have? Slide 19 / How many common tangent lines do the circles have? Slide 20 / 150 Using the diagram below, match the notation with the term that best describes it: F G E Slide 21 / 150 center chord secant radius diameter tangent common tangent point of tangency

8 Slide 22 / 150 ngles & rcs Return to the table of contents n R is an unbroken piece of a circle with endpoints on the circle rc of the circle or Slide 23 / 150 rcs are measured in two ways: 1) s the measure of the central angle in degrees 2) s the length of the arc itself in linear units (Recall that the measure of the whole circle is 360 o ) S H central angle is an angle whose vertex is the center of the circle M In, is the central angle Name another central angle Slide 24 / 150

9 If is less than 180 0, then the points on that lie in the interior of form the minor arc with endpoints M and H Slide 25 / 150 S H M minor arc M Highlight M Name another minor arc Slide 26 / 150 major arc S H M Points M and and all points of exterior to form a major arc, MS Major arcs are the "long way" around the circle Major arcs are greater than 180 o Highlight Major arcs are named by their endpoints and a point on the arc Name another major arc MS Slide 27 / 150 S H M minor arc semicircle is an arc whose endpoints are the endpoints of the diameter M is a semicircle Highlight the semicircle Semicircles are named by their endpoints and a point on the arc Name another semicircle

10 Measurement y entral ngle Slide 28 / 150 he measure of a minor arc is the measure of its central angle he measure of the major arc is minus the measure of the central angle 40 0 G = he Length of the rc Itself (K - rc Length) rc length is a portion of the circumference of a circle Slide 29 / 150 rc Length orollary - In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to r arc length of = 36 0 or arc length of = 36 0 EXMPLE Slide 30 / 150 In, the central angle is 60 0 and the radius is 8 cm Find the length of 8 cm 60 0

11 EXMPLE In, the central angle is 40 0 and the length of SY is 419 in Find the circumference of Slide 31 / 150 S 419 in 40 0 Y 10 In circle where is a diameter, find Slide 32 / in 11 In circle, where is a diameter, find Slide 33 / in

12 12 In circle, where is a diameter, find Slide 34 / in 13 In circle can it be assumed that is a diameter? Slide 35 / 150 Yes No Find the length of Slide 36 / cm

13 15 Find the circumference of circle Slide 37 / cm 16 In circle, WY & XZ are diameters WY = XZ = 6 Slide 38 / 150 If XY = 14 0, what is the length of YZ? W X Z Y JEN RS djacent arcs: two arcs of the same circle are adjacent if they have a common endpoint Slide 39 / 150 Just as with adjacent angles, measures of adjacent arcs can be added to find the measure of the arc formed by the adjacent arcs = +

14 EXMPLE result of a survey about the ages of people in a city are shown Find the indicated measures 1 S >65 Slide 40 / U 4 R V Match the type of arc and it's measure to the given arcs below: Slide 41 / 150 Q S R eacher Notes minor arc major arc semicircle ONGRUEN IRLES & RS wo circles are congruent if they have the same radius wo arcs are congruent if they have the same measure and they are arcs of the same circle or congruent circles Slide 42 / 150 E R F S U because they are in the same circle and & have the same measure, but are not congruent because they are arcs of circles that are not congruent

15 17 Slide 43 / 150 rue False M Slide 44 / 150 L rue False 85 0 P N 19 ircle P has a radius of 3 and has a measure of 90 0 What is the length of? Slide 45 / 150 P

16 20 wo concentric circles always have congruent radii Slide 46 / 150 rue False 21 If two circles have the same center, they are congruent Slide 47 / 150 rue False 22 anny cuts a pie into 6 congruent pieces What is the measure of the central angle of each piece? Slide 48 / 150

17 Slide 49 / 150 hords, Inscribed ngles & Polygons Return to the table of contents Slide 50 / 150 lick on the link below and complete the labs before the hords lesson Lab - Properties of hords When a minor arc and a chord have the same endpoints, we call the arc he rc of the hord P Slide 51 / 150 Q is the arc of **Recall the definition of a chord - a segment with endpoints on the circle

18 HEOREM: Slide 52 / 150 In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter is the perpendicular bisector of herefore, is a diameter of the circle S Q E Likewise, the perpendicular bisector of a chord of a circle passes through the center of a circle P HEOREM: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc Slide 53 / 150 is a diameter of the circle and is perpendicular to chord X S herefore, E HEOREM: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent Slide 54 / 150 iff *iff stands for "if and only if"

19 ISEING RS Slide 55 / 150 X Y If, then point Y and any line segment, or ray, that contains Y, bisects Z EXMPLE Find:,, and (9x) 0 Slide 56 / 150 E (80 - x) 0 HEOREM: In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center Slide 57 / 150 E G iff F

20 Given circle, QR = S = 16 Find U R Q S U 2x 5x - 9 V EXMPLE Since the chords QR & S are congruent, they are equidistant from herefore, Slide 58 / In circle R, and Find R Slide 59 / Given circle below, the length of is: Slide 60 / F

21 25 Given: circle P, PV = PW, QR = 2x + 6, and S = 3x - 1 Find the length of QR Slide 61 / Q V P W R S 26 H is a diameter of the circle Slide 62 / 150 rue False M 3 3 S 5 H INSRIE NGLES Slide 63 / 150 Inscribed angles are angles whose vertices are in on the circle and whose sides are chords of the circle O G he arc that lies in the interior of an inscribed angle, and has endpoints on the angle, is called the intercepted arc is an inscribed angle and is its intercepted arc lick on the link below and complete the lab Lab - Inscribed ngles

22 HEOREM: he measure of an inscribed angle is half the measure of its intercepted arc Slide 64 / 150 EXMPLE Slide 65 / 150 Find and Q 500 P R 48 0 S HEOREM: If two inscribed angles of a circle intercept the same arc, then the angles are congruent Slide 66 / 150 since they both intercept

23 In a circle, parallel chords intercept congruent arcs Slide 67 / 150 O In circle O, if, then 27 Given circle below, find Slide 68 / 150 E Given circle below, find Slide 69 / 150 E

24 29 Given the figure below, which pairs of angles are congruent? R S U Slide 70 / Find Slide 71 / 150 X Y P Z 31 In a circle, two parallel chords on opposite sides of the center have arcs which measure and Find the measure of one of the arcs included between the chords Slide 72 / 150

25 32 Given circle O, find the value of x x Slide 73 / Ȯ 33 Given circle O, find the value of x Slide 74 / O x ry his Slide 75 / 150 In the circle below, and Find, and Q P S

26 INSRIE POLYGONS Slide 76 / 150 polygon is inscribed if all its vertices lie on a circle inscribed triangle inscribed quadrilateral HEOREM: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle Slide 77 / 150 L x iff is a diameter of the circle G HEOREM: quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary E Slide 78 / 150 N N, E,, and R lie on circle iff R

27 EXMPLE Slide 79 / 150 Find the value of each variable: L 2a 2b M K 4b 2a J 34 he value of x is Slide 80 / x y 35 In the diagram, is a central angle and What is? Slide 81 /

28 36 What is the value of x? E Slide 82 / (12x + 40) F 0 (8x + 10) G Slide 83 / 150 angents & Secants Return to the table of contents **Recall the definition of a tangent line: line that intersects the circle in exactly one point Slide 84 / 150 HEOREM: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle (the point of tangency) l Line is tangent to circle X iff would be the point of tangency lick on the link below and complete the lab l X l Lab - angent Lines

29 Verify Line is angent to a ircle Slide 85 / 150 S } P Given: is a radius of circle P Is tangent to circle P? Finding the Radius of a ircle If is a point of tangency, find the radius of circle Slide 86 / ft 50 ft r r HEOREM: angent segments from a common external point are congruent R P Slide 87 / 150 If R and are tangent segments, then

30 EXMPLE Slide 88 / 150 Given: RS is tangent to circle at S and R is tangent to circle at Find x S 28 3x + 4 R 37 is a radius of circle Is tangent to circle? Slide 89 / 150 Yes No }67 38 S is a point of tangency Find the radius r of circle Slide 90 / r S r 48 cm 36 cm R

31 39 In circle, is tangent at and is tangent at Find x Slide 91 / x ,, and are tangents to circle O = 5, = 8, and E = 4 Find the perimeter of triangle Slide 92 / 150 E O F Slide 93 / 150 angents and secants can form other angle relationships in circle Recall the measure of an inscribed angle is 1/2 its intercepted arc his can be extended to any angle that has its vertex on the circle his includes angles formed by two secants, a secant and a tangent, a tangent and a chord, and two tangents

32 angent and a hord Slide 94 / 150 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 M 2 1 R angent and a Secant, wo angents, and wo Secants Slide 95 / 150 HEOREM: 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 its intercepted arcs a tangent and a secant 1 two tangents P 2 M Q two secants X W 3 Y Z HEOREM: If two chords intersect inside a circle, then the measure of each angle is half the sum of the intercepted arcs by the angle and vertical angle Slide 96 / 150 M 1 2 H

33 Find the value of x EXMPLE Slide 97 / 150 x EXMPLE Slide 98 / 150 Find the value of x 13 0 x Find the value of x Slide 99 / E H 42 0 x 0 F

34 42 Find the value of x Slide 100 / (3x - 2) (x + 6) Find Slide 101 / Find Slide 102 /

35 45 Find the value of x Slide 103 / 150 x o find the angle, you need the measure of both intercepted arcs First, find the measure of the minor arc hen we can calculate the measure of the angle x 0 Slide 104 / x 0 46 Find the value of x Slide 105 / 150 Students type their answers here 22 0 x 0

36 47 Find the value of x Students type their answers here Slide 106 / 150 x Find the value of x Students type their answers here Slide 107 / x 0 49 Find the value of x Students type their answers here Slide 108 / 150 (5x + 10)

37 50 Find the value of x Slide 109 / (2x - 30) 30 0 x Slide 110 / 150 Segments & ircles Return to the table of contents HEOREM: If two chords intersect inside a circle, then the products of the measures of the segments of the chords are equal Slide 111 / 150 E

38 Find the value of x EXMPLE Slide 112 / x Find ML & JK EXMPLE Slide 113 / 150 M K J x x + 2 L x + 4 x Find the value of x Slide 114 / 150 x

39 52 Find the value of x Slide 115 / x x 2x + 6 HEOREM: If two secant segments are drawn to a circle from an external point, then the product of the measures of one secant segment and its external secant segment equals the product of the measures of the other secant segment and its external secant segment Slide 116 / 150 E EXMPLE Slide 117 / 150 Find the value of x 9 6 x 5

40 53 Find the value of x Slide 118 / x + 1 x + 2 x Find the value of x Slide 119 / x - 2 x + 4 HEOREM: If a tangent segment and a secant segment are drawn to a circle from an external 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 external secant segment Slide 120 / 150 E

41 Find RS Q EXMPLE Slide 121 / R x S 8 55 Find the value of x Slide 122 / x 56 Find the value of x Slide 123 / x

42 Slide 124 / 150 Equations of a ircle Return to the table of contents r (x, y) y Let (x, y) be any point on a circle with center at the origin and radius, r y the Pythagorean heorem, Slide 125 / 150 x x 2 + y 2 = r 2 his is the equation of a circle with center at the origin EXMPLE Write the equation of the circle Slide 126 / 150 4

43 For circles whose center is not at the origin, we use the distance formula to derive the equation Slide 127 / 150 (x, y) r (h, k) his is the standard equation of a circle EXMPLE Slide 128 / 150 Write the standard equation of a circle with center (-2, 3) & radius 38 EXMPLE Slide 129 / 150 he point (-5, 6) is on a circle with center (-1, 3) Write the standard equation of the circle

44 EXMPLE Slide 130 / 150 he equation of a circle is (x - 4) 2 + (y + 2) 2 = 36 Graph the circle We know the center of the circle is (4, -2) and the radius is 6 First plot the center and move 6 places in each direction hen draw the circle 57 What is the standard equation of the circle below? Slide 131 / 150 x 2 + y 2 = 400 (x - 10) 2 + (y - 10) 2 = 400 (x + 10) 2 + (y - 10) 2 = 400 (x - 10) 2 + (y + 10) 2 = What is the standard equation of the circle? (x - 4) 2 + (y - 3) 2 = 81 Slide 132 / 150 (x - 4) 2 + (y - 3) 2 = 9 (x + 4) 2 + (y + 3) 2 = 81 (x + 4) 2 + (y + 3) 2 = 9

45 59 What is the center of (x - 4) 2 + (y - 2) 2 = 64? Slide 133 / 150 (0,0) (4,2) (-4, -2) (4, -2) 60 What is the radius of (x - 4) 2 + (y - 2) 2 = 64? Slide 134 / he standard equation of a circle is (x - 2) 2 + (y + 1) 2 = 16 What is the diameter of the circle? Slide 135 /

46 62 Which point does not lie on the circle described by the equation (x + 2) 2 + (y - 4) 2 = 25? Slide 136 / 150 (-2, -1) (1, 8) (3, 4) (0, 5) Slide 137 / 150 rea of a Sector Return to the table of contents Slide 138 / 150 sector of a circle is the portion of the circle enclosed by two radii and the arc that connects them Minor Sector Major Sector

47 63 Which arc borders the minor sector? Slide 139 / Which arc borders the major sector? W X Y Z Slide 140 / 150 Lets think about the formula he area of a circle is found by We want to find the area of part of the circle, so the formula for the area of a sector is the fraction of the circle multiplied by the area of the circle Slide 141 / 150 When the central angle is in degrees, the fraction of the circle is out of the total 360 degrees

48 Finding the rea of a Sector 1 Use the formula: when θ is in degrees Slide 142 / 150 r= Example: Find the rea of the major sector Slide 143 / cm Find the area of the minor sector of the circle Round your answer to the nearest hundredth Slide 144 / cm 30 0

49 66 Find the rea of the major sector for the circle Round your answer to the nearest thousandth Slide 145 / cm What is the central angle for the major sector of the circle? Slide 146 / cm 12 0 G 68 Find the area of the major sector Round to the nearest hundredth Slide 147 / cm 12 0 G

50 69 he sum of the major and minor sectors' areas is equal to the total area of the circle Slide 148 / 150 rue False 70 group of 10 students orders pizza hey order 5 12" pizzas, that contain 8 slices each If they split the pizzas equally, how many square inches of pizza does each student get? Slide 149 / You have a circular sprinkler in your yard he sprinkler has a radius of 25 ft How many square feet does the sprinkler water if it only rotates 120 degrees? Slide 150 / 150