Topic 7: Properties of Circles

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1 This Packet Belongs to (Student Name) Topic 7: Properties of Circles Unit 6 Properties of Circles Module 15: Angles and Segments in Circles 15.1 Central Angles and Inscribed Angles 15.2 Angles in Inscribed Quadrilaterals 15.3 Tangents and Circumscribed Angles 15.4 Segment Relationships in Circles 15.5 Angles Relationships in Circles Module 16: Arc Length and Sector Area 16.1 Justifying Circumference and Area of a Circle 16.2 Arc Length and Radian Measure 16.3 Sector Area Module 17: Equations of Circles and Parabolas 17.1 Equation of a Circle Objectives Identify and describe relationships among inscribed angles, radii, and chords. Vocabulary: Chord, central angle, inscribed angle, arc, minor arc, major arc, semicircle, adjacent arc, intercepted arc Assignments: 15.1 ONLINE or Honors: # s 1 2, 5, 7, 8, 10, 12, 14 15, 18, 21, 24, 25 Regular: # s 1 3, 5, 7, 10, 12, 14, 18, 24, 15.2 ONLIEN OR Honors: # s 2, 5 8, 10 13, 17 18, 19, Regular: # s 2, 5, 10 12, 17 18, 2 1

2 chord diameter center radius arc Parts of the Circle A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. Radius the distance from the center to a point on the circle Diameter the distance across the circle through its center Congruent circles circles that have the same radius. Chord a segment whose endpoints lie on a circle. Arc a continuous portion of a circle consisting of two endpoints and all points between them. 3 Central Angles A central angle is an angle whose vertex is the CENTER of the circle and measures less than 180. Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle) 4 2

CENTRAL ANGLES AND ARCS The measure of a central angle is equal to the measure of the intercepted arc. O Central Angle Y 110 Z Intercepted Arc Segment AD is a diameter.

3 CENTRAL ANGLES AND ARCS The measure of a central angle is equal to the measure of the intercepted arc. O Central Angle Y 110 Z Intercepted Arc Segment AD is a diameter. Find the values of x and y and z in the figure. B 25 A x y 55 O C z D x = 25 y = 100 z = 55 5 SUM OF CENTRAL ANGLES The sum of the measures for the central angles of a circle with no interior points in common is 360º. 4x + 3x + 3x x + 2x 14 = 360 D 14x 4 = x = 356 x = 26 Find the measure of each arc. C 360º Arc AB = 88 Arc BC = 52 Arc CD = 38 Arc DE = 104 Arc AE = 78 E A 2x 6 B 3

4 A Major Arc Minor Arc More than 180 Less than 180 ACB P AB To name: use 3 letters C B To name: use 2 letters <APB is a Central Angle Central Angle: An Angle whose vertex is at the center of the circle 7 Semicircle: An Arc that equals 180 E D To name: use 3 letters P EDF F EF is a diameter, so every diameter divides the circle in half, which divides it into arcs of

7 14 INSCRIBED ANGLES An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle Is NOT! Is SO! Is NOT! Is SO! 15 7

Inscribed Angle Y Find the value of x and y in the

8 16 INSCRIBED ANGLE THEOREM The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Inscribed Angle Y Find the value of x and y in the figure. P Z Intercepted Arc X = 20 Y = 60 S 50 T x R y 17 8

9 18 Corollary 1. If two inscribed angles intercept the same arc, then the angles are congruent.. P y Q Find the value of x and y in the figure. S 50 X = 50 T x R Y =

10 20 All inscribed angles with the same endpoints in a circle have congruent angles and a shared intercepted arc

P 180 S 90 R An angle inscribed in a semicircle is a right angle. 22 15.

11 P 180 S 90 R An angle inscribed in a semicircle is a right angle Classwork/Homework Starts on Page 786 Honors: # s 1 2, 5, 7, 8, 10, 12, 14 15, 18, 21, 24, 25 Regular: # s 1 3, 5, 7, 10, 12, 14, 18, 24, 23 11

12 Objectives Prove properties of angles for a quadrilateral inscribed in a circle. Vocabulary: Inscribed quadrilateral theorem ONLIEN OR Honors: # s 2, 5 8, 10 13, 17 18, 19, Assignments: Regular:# s 2, 5, 10 12, INSCRIBED QUADRILATERAL THEOREM If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. P Q m PSR + m PQR = 180 m SPQ + m SRQ = 180 S HINT: All 4 angles will always add up to 360 R HINT: Each of the 4 angles is an INSCRIBED ANGLE

14 L Example B, continued: 29 14

16 Aside from the fact that quadrilateral was spelled horribly wrong. 32 MIXED REVIEW

17 15.2 Classwork/Homework Starts on Page 799 Honors: # s 2, 5 8, 10 13, 17 18, 19, Regular: # s 2, 5, 10 12, 17 18, 34 Objectives Identify and describe relationships among inscribed angles, radii, tangents, and chords. Vocabulary: Tangent, point of tangency, circumscribed angle ONLINE OR Honors: 3, 5 9, 12 16, 18, 22 Regular: 3, 5, 7, 8, 14, 15, 16 Assignments: 35 17

18 TANGENTS Tangent a line in the plane of a circle that intersects the circle in exactly one point. That point of intersection where the tangent and the circle meet is called the point of tangency. T S point of tangency ST is a tangent 36 CAN YOU NAME IT? Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. a. AH tangent H b. EI c. DF diameter chord B C E F d. CE radius I G A D 37 18

19 TYPES OF TANGENTS Tangent circles coplanar circles that intersect in one point Common tangent a line or segment that is tangent to two coplanar circles Common internal tangent intersects the segment that joins the centers of the two circles Common external tangent does not intersect the segment that joins the centers of the two circles common internal tangent common external tangent 38 TYPES OF TANGENTS Tell whether the common tangents are internal or external. a. b. common internal tangents common external tangents 39 19

20 PERPENDICULAR TANGENT THEOREM AKA TANGENT-RADIUS THEOREM If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. l PERPENDICULAR TANGENT CONVERSE AKA CONVERSE OF THE TANGENT-RADIUS THEOREM P If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. Q 40 TANGENT PRACTICE Tell whether CE is tangent to D. Use the converse of the Pythagorean Theorem to see if the triangle is right ? 45 2 E C ? 2025 D CED is not right, so CE is not tangent to D

23 B A 92 x C mabc = mabc = 268 x = 1 (268-92) 2 x = 1 2 (176) x =

CONGRUENT TANGENT SEGMENT THEOREM If two segments from the

= 9 C B 11 A EG FG EG FG x = 3 48 CONGRUENT TANGENT

24 CONGRUENT TANGENT SEGMENT THEOREM If two segments from the same exterior point are tangent to a circle, then they are congruent. EG and FG are tangent to D. D x G is the exterior point AD = AB x = 11 x 2 = 9 C B 11 A EG FG EG FG x = 3 48 CONGRUENT TANGENT SEGMENT PRACTICE 1. Find x and y. 2. HJK is circumscribed about G. Find the perimeter of HJK if NK = JL ( ) + ( ) = = =

25 Classwork/Homework Starts on Page 809 Honors: 3, 5 9, 12 16, 18, 22 Regular: 3, 5, 7, 8, 14, 15,

26 Find the lengths of segments formed by lines that intersect circles. Tangent segment, chord Objectives Vocabulary: Assignments: Worksheet 52 Warm Up Solve for x x = BC and DC are tangent to A. Find BC

27 54 Example 1: Applying the Chord-Chord Product Theorem Find the value of x and the length of each chord. J EJ JF = GJ JH 10(7) = 14(x) 70 = 14x 5 = x EF = = 17 GH = = 19 DE EC = AE EB 8(x) = 6(5) 8x = 30 x = 3.75 AB = = 11 CD = =

28 56 USING CHORDS OF A CIRCLE If two arcs of one circle have the same measure, then they are congruent arcs. Congruent arcs also have the same length. Z 23 W Y? X R

29 USING CHORDS OF A CIRCLE When a minor arc and a chord share the same endpoints, we call the arc the ARC OF THE CHORD. 58 Theorems about Chords In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. EF AB ARC-CHORD THEOREM 59 29

30 ARC-CHORD THEOREM PRACTICE Find the measure of GJ G 3X + 11 = 2X + 48 X + 11 = 48 X = 37 2(37) + 48 = = 122 (2x+48) (3x+11) A J K GJ = 122 B 61 30

31 Find the measure of: ARC-CHORD THEOREM PRACTICE BC, DC, and BDC C D A (2x + 48) (3x + 11) B 62 CHORD-TO-CENTER THEOREM In a circle or congruent circles, two chords are congruent if and only if they are equidistant from the center. 10 X X = 10 Chords are congruent if they are equidistant from the center, they are also congruent if their arcs are the same size

DIAMETER-CHORD THEOREM 12 30 64 DIAMETER-CHORD THEOREM PRACTICE In the diagram, FK =

32 DIAMETER-CHORD THEOREM DIAMETER-CHORD THEOREM PRACTICE In the diagram, FK = 40, AC = 40, AE = 25. Find EG, GH, and EF. F H G K E EF = 25 EG = 15 GH = 10 A B D C 65 32

33 PERPENDICULAR BISECTOR CHORD THEOREM 66 PERPENDICULAR BISECTOR CHORD PRACTICE 1. Find the length of the radius of a circle if a chord is 10 long and 12 from the center X =

34 15.4 Classwork/Homework Starts on Page 823 Worksheet 72 Warm Up 1. Identify each line or segment that intersects F. chords: CD secant: AE tangent: AB Find each measure. 2. m NMP 3. m NLP

35 Find the measures of angles formed by lines that intersect circles. Tangent, chord Objectives Vocabulary: Assignments: Worksheet 74 INTERIOR ANGLES Angles that are formed by two intersecting chords. (Vertex IN the circle) The intersection of these two chords will form congruent vertical angles and linear pairs. A D C B 75 35

36 THE INTERSECTING CHORDS ANGLE MEASURE THEOREM AKA INTERIOR ANGLES THEOREM The measure of the angle formed by the two chords is equal to ½ the sum of the measures of the intercepted arcs. C A 1 D B m 1 1 (mac mbd) 2 76 THE INTERSECTING CHORDS ANGLE MEASURE THEOREM PRACTICE AKA INTERIOR ANGLES THEOREM D A y x B C x y x 1 (91 85) y

38 TANGENT CHORD THEOREM 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. B m 1 = 1 2 mab C m 2 = 1 2 mbca 2 1 A 80 TANGENT CHORD THEOREM PRACTICE Line m is tangent to the circle. Find mrst m R 102 S Line m is tangent to the circle. Find m 1 R 1 m T T

39 TANGENT CHORD THEOREM PRACTICE BC is tangent to the circle. Find m CBD. (9x+20) A C 5x B D 82 EXTERIOR ANGLE An angle formed by two tangents drawn from a point outside the circle. EXTERIOR ANGLE THEOREM The measure of the angle formed is equal to ½ the difference of the intercepted arcs. 1 m 1 (k j) 2 k j

40 EXTERIOR ANGLE THEOREM PRACTICE Find m ACB 265 B 95 C A <C = ½(265 95) <C = ½(170) m<c =

41 15.5 Classwork/Homework Starts on Page 836 Worksheet 86 41

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with endpoints on the circle. Diameter - A chord which passes through