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This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL 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. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

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Geometry Circles www.njctl.org 2014-03-31

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Table of Contents Parts of a Circle Angles & Arcs Chords, Inscribed Angles & Polygons Segments & Circles Equations of a Circle Click on a topic to go to that section Tangents & Secants Area of a Sector

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Parts of a Circle Return to the table of contents

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A 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. center

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The symbol for a circle is and is named by a capital letter placed by the center of the circle.. A B (circle A or. A)A) is a radius of. A A 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.. is a radius of A)A). (circle A or A.

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A M C R T is the diameter of circle A is a chord of circle A A chord is a segment that has its endpoints on the circle. A diameter is a chord that goes through the center of the circle. All diameters of a circle are congruent. What are the radii in this diagram? [This object is a pull tab] Answer &

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The relationship between the diameter and the radius A The measure of the diameter, d, is twice the measure of the radius, r. Therefore, or M C T If then what is the length of, In. A what is the length of [This object is a pull tab] Answer AC = 5 TC = 10

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1A diameter of a circle is the longest chord of the circle. True False [This object is a pull tab] Answer True

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2A radius of a circle is a chord of a circle. True False [This object is a pull tab] Answer False

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3Two radii of a circle always equal the length of a diameter of a circle. True False [This object is a pull tab] Answer True

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4If the radius of a circle measures 3.8 meters, what is the measure of the diameter? [This object is a pull tab] Answer 7.6 m

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5How many diameters can be drawn in a circle? A 1 B 2 C 4 D infinitely many [This object is a pull tab] Answer D

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A secant of a circle is a line that intersects the circle at two points. A B D E k l line l is a secant of this circle. A tangent is a line in the plane of a circle that intersects the circle at exactly one point (the point of tangency). line k is a tangent D is the point of tangency. tangent ray,, and the tangent segment,, are also called tangents. They must be part of a tangent line. Note: This is not a tangent ray.

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COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points tangent circles 1 point concentric circles..... no points

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A Common Tangent is a line, ray, or segment that is tangent to 2 coplanar circles. Internally tangent (tangent line passes between them) Externally tangent (tangent line does not pass between them)

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6How many common tangent lines do the circles have? [This object is a pull tab] Answer 4

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7How many common tangent lines do the circles have? [This object is a pull tab] Answer 1

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8How many common tangent lines do the circles have? [This object is a pull tab] Answer 2

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9How many common tangent lines do the circles have? [This object is a pull tab] Answer 0

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Using the diagram below, match the notation with the term that best describes it: A C D E F G...... B. center radiuschord diameter secant tangent point of tangency common tangent [This object is a pull tab] Answer Center Common Tangent Chord Secant Tangent Point of Tangency Radius Diameter

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Angles & Arcs Return to the table of contents

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An ARC is an unbroken piece of a circle with endpoints on the circle... A B Arc of the circle or AB Arcs are measured in two ways: 1) As the measure of the central angle in degrees 2) As the length of the arc itself in linear units (Recall that the measure of the whole circle is 360o.)

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A central angle is an angle whose vertex is the center of the circle. M A T H S.. In, is the central angle. Name another central angle. [This object is a pull tab] Answer

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M A T H S.. minor arc MA If is less than 1800, then the points on that lie in the interior of form the minor arc with endpoints M and H. Name another minor arc. MA Highlight [This object is a pull tab] Answer

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M A T H S.. major arc Points M and A and all points of exterior to form a major arc, Major arcs are the "long way" around the circle. Major arcs are greater than 180o. Highlight Major arcs are named by their endpoints and a point on the arc. Name another major arc. MSA [This object is a pull tab] Answer

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M A T H S.. minor arc A semicircle is an arc whose endpoints are the endpoints of the diameter. MAT is a semicircle. Highlight the semicircle. Semicircles are named by their endpoints and a point on the arc. Name another semicircle. [This object is a pull tab] Answer

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The measure of a minor arc is the measure of its central angle. The measure of the major arc is 3600 minus the measure of the central angle. Measurement By A Central Angle A B D. 400 G 3600 - 400 = 3200

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The Length of the Arc Itself (AKA - Arc Length) Arc length is a portion of the circumference of a circle. Arc Length Corollary - 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 3600. C A T r arc length of = 3600 CT arc length of = 3600. or

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C A T 8 cm 600 EXAMPLE In, the central angle is 600 and the radius is 8 cm. Find the length of A CT [This object is a pull tab] Answer CT arc length of = 3600. = 600 3600. 8.38 cm

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EXAMPLE S A Y 4.19 in 400 A In, the central angle is 400 and the length of is 4.19 in. Find the circumference of A.A. SY A.A. In, the central angle is 400 and the length of is 4.19 in. Find the circumference of SY A [This object is a pull tab] Answer arc length of = 3600 SY = 3600 400 4.19 = 9 1 = 37.71 in

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10In circle C where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

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11In circle C, where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

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12In circle C, where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

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13In circle C can it be assumed that AB is a diameter? Yes No 1350 A C B D [This object is a pull tab] Answer Yes

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14Find the length of 450 A C 3 cm B [This object is a pull tab] Answer

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15Find the circumference of circle T. T 750 6.82 cm [This object is a pull tab] Answer

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1400 16In circle T, WY & XZ are diameters. WY = XZ = 6. If XY =, what is the length of YZ? A B C D T W Y X Z [This object is a pull tab] Answer A

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Adjacent arcs: two arcs of the same circle are adjacent if they have a common endpoint. Just as with adjacent angles, measures of adjacent arcs can be added to find the measure of the arc formed by the adjacent arcs. ADJACENT ARCS... C A T + =

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EXAMPLE A result of a survey about the ages of people in a city are shown. Find the indicated measures. >65 45-64 15-17 17-44 S U V R 300 900 800 600 1000 T 1. 2. 3. 4. [This object is a pull tab] Answer = 600 + 800 = 1400 1000 + 300 = 1300 = 600 + 800 + 900 = 2300 = 3600 - 900 = 2700

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Match the type of arc and it's measure to the given arcs below: 1200 800 600 T S R Q minor arcmajor arc semicircle 120024001800 1600 800 [This object is a pull tab] Teacher Notes Arc labels and measurements in the box are infinitely cloned so they can be pulled up and matched with the arc.

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CONGRUENT CIRCLES & ARCS Two circles are congruent if they have the same radius. Two arcs are congruent if they have the same measure and they are arcs of the same circle or congruent circles. C D E F 550 R S T 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.

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17 True False 1800 700 400 A B C D [This object is a pull tab] Answer True

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18 True False 850 M N L P [This object is a pull tab] Answer False

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900 19Circle P has a radius of 3 and has a measure of. What is the length of ? A B C D P A B [This object is a pull tab] Answer A

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20Two concentric circles always have congruent radii. True False [This object is a pull tab] Answer False

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21If two circles have the same center, they are congruent. True False [This object is a pull tab] Answer False

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22Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece? [This object is a pull tab] Answer

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Chords, Inscribed Angles & Polygons Return to the table of contents

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is the arc of When a minor arc and a chord have the same endpoints, we call the arc The Arc of the Chord.. C P Q **Recall the definition of a chord - a segment with endpoints on the circle.

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THEOREM: In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter. T P S Q E is the perpendicular bisector of. Therefore, is a diameter of the circle. Likewise, the perpendicular bisector of a chord of a circle passes through the center of a circle.

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THEOREM: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. A C E S X. is a diameter of the circle and is perpendicular to chord Therefore,

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THEOREM: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. A B C D iff *iff stands for "if and only if"

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If, then point Y and any line segment, or ray, that contains Y, bisects BISECTING ARCS C X Z Y

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Find:,, and EXAMPLE A B C D E. (9x)0 (80 - x)0 and,, Find: [This object is a pull tab] Answer = 9(8) = 720 = 80 - 8 = 720 = 720 + 720 = 1440 9x = 80 - x 10x = 80 x = 8

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THEOREM: In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center.. C G D E A F B iff

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EXAMPLE Given circle C, QR = ST = 16. Find CU.. Q R S T U V 2x 5x - 9 C Since the chords QR & ST are congruent, they are equidistant from C. Therefore, [This object is a pull tab] Answer 2x = 5x - 9 9 = 3xCU = 2(3) = 6 3 = x

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23In circle R, and. Find A B C D R. 1080 [This object is a pull tab] Answer 1080

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24Given circle C below, the length of is: A 5 B 10 C 15 D 20 D B F C. 10 A [This object is a pull tab] Answer D

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25Given: circle P, PV = PW, QR = 2x + 6, and ST = 3x - 1. Find the length of QR. A 1 B 7 C 20 D 8 R S Q T P W. V [This object is a pull tab] Answer C

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26AH is a diameter of the circle. True False A S H M 3 3 5 T [This object is a pull tab] Answer False

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INSCRIBED ANGLES D O G Inscribed angles are angles whose vertices are in on the circle and whose sides are chords of the circle. The 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.

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THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc. C A T

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EXAMPLE Q R T S P. 500 480 Findand [This object is a pull tab] Answer

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THEOREM: If two inscribed angles of a circle intercept the same arc, then the angles are congruent. D C B A since they both intercept

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In a circle, parallel chords intercept congruent arcs. O B. A D C In circle O, if, then In circle O, if

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27Given circle C below, find D E C A B. 1000 350 [This object is a pull tab] Answer 500

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28Given circle C below, find D E C A B. 1000 350 [This object is a pull tab] Answer 1100

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29 Given the figure below, which pairs of angles are congruent? A B C D R S U T [This object is a pull tab] Answer A

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30Find X Y Z P. [This object is a pull tab] Answer 900

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31In a circle, two parallel chords on opposite sides of the center have arcs which measure 1000 and 1200. Find the measure of one of the arcs included between the chords. [This object is a pull tab] Answer 700

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32Given circle O, find the value of x.. O A B C D x 300 [This object is a pull tab] Answer 1200

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33Given circle O, find the value of x.. O A B C D x 1000 350 [This object is a pull tab] Answer 1200

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In the circle below, andFind, and Try This P S 1 2 3 4 Q T [This object is a pull tab] Answer

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INSCRIBED POLYGONS A polygon is inscribed if all its vertices lie on a circle.... inscribed triangle.... inscribed quadrilateral

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THEOREM: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. A L G x. iff AC is a diameter of the circle.

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THEOREM: A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. N E R A C. N, E, A, and R lie on circle C iff

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EXAMPLE Find the value of each variable: 2a 4b 2b L K J M [This object is a pull tab] Answer 2a + 2a = 180 4a = 180 a = 450 4b + 2b = 180 6b = 180 b = 300

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34The value of x is A B C D 1500 980 1120 1800 C B A D x y 680 820 [This object is a pull tab] Answer B

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35In the diagram, is a central angle and. What is ? 150 300 600 1200 A B C D. B A D C 600 300 150 [This object is a pull tab] Answer B

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36What is the value of x? A 5 B 10 C 13 D 15 E F G (12x + 40)0 (8x + 10)0 [This object is a pull tab] Answer A

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Tangents & Secants Return to the table of contents

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**Recall the definition of a tangent line: A line that intersects the circle in exactly one point. THEOREM: 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)... X B l l Line is tangent to circle X iff would be the point of tangency. BB Line is tangent to circle X iff would be the point of tangency. l l

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Verify A Line is Tangent to a Circle. T P S 35 37 12 } Given:is a radius of circle P Is tangent to circle P? [This object is a pull tab] Answer Since 352 + 122 = 372, triangle PST is a right triangle. Therefore, ST is perpendicular to radius TP at its endpoint on circle P. So, ST is tangent to circle P at T.

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Finding the Radius of a Circle. A C B r r 50 ft 80 ft If B is a point of tangency, find the radius of circle C. [This object is a pull tab] Answer AC2 + BC2 = AB2 802 + r2 = (50 + r)2 6400 + r2 = r2 + 100r + 2500 6400 = 100r + 2500 3900 = 100r 39 = r So, r = 39 ft.

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THEOREM: Tangent segments from a common external point are congruent. R A T P. If AR and AT are tangent segments, then

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EXAMPLE Given: RS is tangent to circle C at S and RT is tangent to circle C at T. Find x. S R T C. 28 3x + 4 [This object is a pull tab] Answer 3x + 4 = 28 3x = 24 x = 8

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37 AB is a radius of circle A. Is BC tangent to circle A? Yes No. B C A 60 25 67 } [This object is a pull tab] Answer No

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38S is a point of tangency. Find the radius r of circle T. A31.7 B60 C14 D3.5. T S R r r 48 cm 36 cm [This object is a pull tab] Answer C

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39 In circle C, DA is tangent at A and DB is tangent at B. Find x. A D B C. 25 3x - 8 [This object is a pull tab] Answer

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40AB, BC, and CA are tangents to circle O. AD = 5, AC= 8, and BE = 4. Find the perimeter of triangle ABC.. B E F A C D O [This object is a pull tab] Answer

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Tangents and secants can form other angle relationships in circle. Recall the measure of an inscribed angle is 1/2 its intercepted arc. This can be extended to any angle that has its vertex on the circle. This includes angles formed by two secants, a secant and a tangent, a tangent and a chord, and two tangents.

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A Tangent and a 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.... A M R 2 1

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A Tangent and a Secant, Two Tangents, and Two Secants THEOREM: 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 B C 1 a tangent and a secant P Q M 2. two tangentstwo secants W X Y Z 3

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THEOREM: 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. M A H T 1 2

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EXAMPLE Find the value of x. D C A B x0 760 1780 [This object is a pull tab] Answer

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EXAMPLE Find the value of x. 1300 x0 1560 [This object is a pull tab] Answer x = 1/2 (1300 + 1560) x = 1430

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41Find the value of x. C H D F x0 780 420 E [This object is a pull tab] Answer

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42Find the value of x. 340 (x + 6)0 (3x - 2)0 [This object is a pull tab] Answer

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43Find A B 650 [This object is a pull tab] Answer

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44Find 1 2600 [This object is a pull tab] Answer

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45Find the value of x. x 122.50 450 [This object is a pull tab] Answer

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2470 A B x0 To find the angle, you need the measure of both intercepted arcs. First, find the measure of the minor arc. Then we can calculate the measure of the angle. x0 [This object is a pull tab] Answer First find the minor arc.

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46Find the value of x. 2200 x0 [This object is a pull tab] Answer First find the minor arc.

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47Find the value of x. x0 1000 [This object is a pull tab] Answer First find the major arc.

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48Find the value of x x0 500 [This object is a pull tab] Answer Find the major arc.

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49Find the value of x. 1200 (5x + 10)0 [This object is a pull tab] Answer Find the major arc.

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50Find the value of x. (2x - 30)0 300 x [This object is a pull tab] Answer

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Segments & Circles Return to the table of contents

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THEOREM: If two chords intersect inside a circle, then the products of the measures of the segments of the chords are equal. A C D B E

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EXAMPLE Find the value of x. 5 5 x 4 [This object is a pull tab] Answer

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EXAMPLE Find ML & JK. x + 2 x + 4 x x + 1 MK J L [This object is a pull tab] Answer ML = (2 + 2) +( 2 + 1) = 7 JK = 2 + (2 + 4) = 8

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51Find the value of x. 18 9 16 x [This object is a pull tab] Answer

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52Find the value of x. A-2 B 4 C 5 D 6 x 2 2x + 6 x [This object is a pull tab] Answer D

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THEOREM: 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. A B E C D

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EXAMPLE Find the value of x. 9 6 x 5 [This object is a pull tab] Answer

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53Find the value of x. 3 x + 2 x + 1 x - 1 [This object is a pull tab] Answer

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54Find the value of x. x + 4 x - 2 5 4 [This object is a pull tab] Answer

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THEOREM: 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. A E C D

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EXAMPLE Find RS. R S Q T 16 x 8 [This object is a pull tab] Answer Since we are dealing with measurement, we only want the positive answer:

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55Find the value of x. 1 x 3 [This object is a pull tab] Answer

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56Find the value of x. x 12 24 [This object is a pull tab] Answer

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Equations of a Circle Return to the table of contents

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y x r (x, y) Let (x, y) be any point on a circle with center at the origin and radius, r. By the Pythagorean Theorem, x2 + y2 = r2 This is the equation of a circle with center at the origin.

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EXAMPLE Write the equation of the circle. 4 [This object is a pull tab] Answer x2 + y2 = (4)2 x2 + y2 = 16

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For circles whose center is not at the origin, we use the distance formula to derive the equation.. (x, y) (h, k) r This is the standard equation of a circle.

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EXAMPLE Write the standard equation of a circle with center (-2, 3) & radius 3.8. [This object is a pull tab] Answer

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EXAMPLE The point (-5, 6) is on a circle with center (-1, 3). Write the standard equation of the circle. [This object is a pull tab] Answer Then substitute the center and radius into the standard equation of a circle: First, we need to find the length of the radius:

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EXAMPLE The 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. Then draw the circle.

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57What is the standard equation of the circle below? A B C D x2 + y2 = 400 (x - 10)2 + (y - 10)2 = 400 (x + 10)2 + (y - 10)2 = 400 (x - 10)2 + (y + 10)2 = 400 10 [This object is a pull tab] Answer A

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58What is the standard equation of the circle? A B C D (x - 4)2 + (y - 3)2 = 9 (x + 4)2 + (y + 3)2 = 9 (x + 4)2 + (y + 3)2 = 81 (x - 4)2 + (y - 3)2 = 81 [This object is a pull tab] Answer D

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59What is the center of (x - 4)2 + (y - 2)2 = 64? A (0,0) B (4,2) C (-4, -2) D(4, -2) [This object is a pull tab] Answer B

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60What is the radius of (x - 4)2 + (y - 2)2 = 64? [This object is a pull tab] Answer r=8

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61The standard equation of a circle is (x - 2)2 + (y + 1)2 = 16. What is the diameter of the circle? A2 B4 C8 D16 [This object is a pull tab] Answer C

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62Which point does not lie on the circle described by the equation (x + 2)2 + (y - 4)2 = 25? A(-2, -1) B(1, 8) C(3, 4) D(0, 5) [This object is a pull tab] Answer D

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Return to the table of contents Area of a Sector

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A sector of a circle is the portion of the circle enclosed by two radii and the arc that connects them. A B C Minor Sector Major Sector

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63Which arc borders the minor sector? A B A B C D [This object is a pull tab] Answer A

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64Which arc borders the major sector? A B W X Y Z [This object is a pull tab] Answer B

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Lets think about the formula... T 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 When the central angle is in degrees, the fraction of the circle is out of the total 360 degrees.

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Finding the Area of a Sector 1. Use the formula:when θ is in degrees 450 A B C r=3 [This object is a pull tab] Answer

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Example: Find the Area of the major sector. C A T 8 cm 600 [This object is a pull tab] Answer cm2

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65Find the area of the minor sector of the circle. Round your answer to the nearest hundredth. C A T 5.5 cm 300 [This object is a pull tab] Answer cm2

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66Find the Area of the major sector for the circle. Round your answer to the nearest thousandth. C A T 12 cm 850 [This object is a pull tab] Answer cm2

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67What is the central angle for the major sector of the circle? C A G 15 cm 1200 [This object is a pull tab] Answer

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68Find the area of the major sector. Round to the nearest hundredth. C A G 15 cm 1200 [This object is a pull tab] Answer cm2

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69The sum of the major and minor sectors' areas is equal to the total area of the circle. True False [This object is a pull tab] Answer True

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70A group of 10 students orders pizza. They 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? [This object is a pull tab] Answer Each student gets 4 pieces

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71You have a circular sprinkler in your yard. The sprinkler has a radius of 25 ft. How many square feet does the sprinkler water if it only rotates 120 degrees? [This object is a pull tab] Answer

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