Geometry Unit 8 CIRCLES
GeometryUnit 8CIRCLESContent adapted from Department of Education State Task at www.doe.k12.ga.us.Page 0
Learning Task: Proving Circles Are SimilarMaterials needed: Compass, ruler, and calculator.1.a. Using your compass, draw a circle with any radius.b. Label the center of the circle and select any point on the circle. Connect the two points. Thissegment is known as the radius.c. Measure the length of your radius. Use the formula ๐ถ 2๐๐ to calculate the circumferenceof your circle. Leave your answers in terms of ๐.d. Write your answer from part (c) as a fraction over the diameter of the circle. Simplify. (Hint:Diameter 2 times the radius)e. Compare your answer with your group members. What do you notice?f. What does this mean in regards to all circles?Page 1
Learning Task: Lines and Line Segments of a Circle Graphic OrganizerVocabulary erSecantTangentPage 2
Example: Using the diagram to the right, givean example of each of the following. Be sureto use proper notation.F1. CenterG2. Chord (other than the diameter)3. Diameter4. Radius5. Tangent6. Point of Tangency7. SecantSkills Practice: Using the diagram, give an example of each of the following. Be sure to use propernotation.1. Circle2. Secant3. TangentJ4. Radius5. DiameterHAGS6. Point of Tangency7. ChordFDPage 3
Learning Task: TangentsBC is tangent to A at point B.What type of segment is AB ?AFind m ABC BCWhat is the relationship between the radius and the tangent line that intersect at the point oftangency?1. If KD is tangent to circle S, find the length of SD .K24D7S2. Determine if EC is tangent to circle D. Explain your answer.E11C4345DPage 4
3. Is segment MH tangent to circle E? Justify your answer.E5M1312HHow are the two tangent segments to a circle that intersect at a point outside the circle related?HFind the values of x, y, and z in the diagram.15AzxyB36GWhat do you notice about the lengths of BH and BG?Since BH and BG are tangent to the circle, what can you conclude about the relationship of twotangents from the circle intersecting at one point outside the circle?THEOREM: Two tangent segments areif they intersect the same point outside the circle.Page 5
Guided Practice: Apply theorems relating to tangents of a circle1. JM and MK are tangent to circle L. Find the value of x.x 8JLM44K2.NA and NO are tangent to circle G. Find the value of x.A7x - 20GN2x - 5O3. Quadrilateral POST is circumscribed about circle Y. OR 13 in. and ST 12 in. Find theperimeter of POST.4. Ray k is tangent to circle R. What is the value of y?Page 6
Skills Practice1. In the diagram below, AB BD 5 and AD 7. Is BD tangent to2. AB is tangent toa.C at A and DB is tangent tob.3. AB and AD are tangent toa.4. AB is tangent toa.C at D. Find the value of x.c.C. Find the value of x.b.C. Find the value of r.b.C ? Explain.c.c.Page 7
5. Tell whether AB is tangent toa.6. AB and AD are tangent toC. Explain your reasoning.b.C.a. Name all congruent segmentsb. Name all congruent angles.c. Name two congruent triangles.7. MULTIPLE CHOICE: In the diagram below, EF and EG are tangent toA.B.C.D.C. What is the value of x?-4-114Page 8
Learning Task: Central Angles and Arcs of a Circle Graphic OrganizerACPBDVocabulary WordDefinitionExample (using correct notation)Central AngleArcSemicircleMinor ArcMajor ArcCongruent ArcsCentral Angle measurePage 9
Example: Use the figure to the right to identify and name the following:1. Two different central angles2. A minor arc3. A major arc4. A diameter5. A semicircleGuided Practice: In problems 1-4, NP is a diameter. Find the indicated measures.ฬ1. ๐๐ฬ2. ๐๐๐ฬ3. ๐๐๐ฬ4. ๐๐Use the figure to the right to find the measure of the indicate arcs.ฬ5. ๐บ๐ธฬ6. ๐บ๐ธ๐นฬ7. ๐บ๐นฬ8. ๐น๐ป๐ธIn problems 9-11, use Circle P to find the value of x and then find the arc measures. Pictures are notdrawn to scale.9.DA11.92 CP8x 45 C10.BAPA(3x 1) 120 PB140 BCPage 10
Inscribed Angles1. ADC is called an inscribed angle. Explain why.2. AC is called the intercepted arc of ADC . Explain why.3. Given the following measures, complete the theorem below.m ADC 35 mAC 70 Inscribed Angle TheoremThe measure of an inscribed angle is the measure of its intercepted arc.In other words, the intercepted arc is the measure of the inscribed angle.Guided Examples: Find the measure of the missing variable.1.2.4. Find the measure of angle LPN.3. Find the measure of arc XZTheoremIf two inscribed angles of a circle interceptthe same arc, then the angles are .Page 11
Skills PracticeUse the theorems to solve each question.1. a 2. b 4. h Hint: draw a radius5. d e 6. m n 3. c 6. f g 8. What is the sum of a, b, c, d, and e?10. What is wrong with this picture?Page 12
Learning Task: Theorems with Chords and ArcsExampleTheoremFind the value of KM.If two chords are congruent then their arcsare congruentAre JK and ML congruent?Two chords are congruent if they areequidistant from the center of the circleFind the measure of YX.Two chords are congruent if and only if theyare equidistant from the center of the circle.Is QS a diameter? Why or why not?To be a diameter the chord must be aperpendicular bisector of another chord.Pythagorean Theorem.A chord in a circle is 18 cm long and is 5 cm from thecenter of the circle. What is the length of the radius ofthe circle?Page 13
Skills PracticeAnswer the following problems using the theorems from the previous page.1. Find the measure of arc YZ if the measure of arc XW 95 2. Are segments TQ and UQ congruent?3. Find the measure of GF.4. Is segment QS a diameter? Explain your reasoning.5. The chord of a circle is 15 inches and it is drawn 4 inches from the center of the circle. What is thelength of the radius of the circle?Page 14
Additional Skills PracticeSolve for the missing variables.Page 15
13. y Page 16
Warm-upSolve the following equations for x.11. 46 (18 x 2)212. 14 x 3 (90)2Learning Task : Central Angles and Arcs of a Circle Graphic OrganizerLocation of the VertexInside the Circle At the center PictureTheoremNot at the centerOutside the CircleOn the circlePage 17
Guided Practice:1. Find the m ABD, the inscribed angle ofC.2. Find the m ABD, the inscribed angle ofC,if mBED 300 .3. Find the m ABD , the inscribed angle of C.4. Find the measure of ๐ด๐ต๐ถ.DA2x76CCA3xBB1305. Solve for x.6. Solve for x.2x Ax 160 H56 M120 TPage 18
7. Find the measure of angle M.8. Find m(arcB ) .M30 81 9. Find m h .b53 10. Find m(arcY ).89 95 hy145 131 11. Find m 112. m(arcW ).55ow120oPage 19
Skills PracticeFind the indicate arc/angle or missing variable.1. mDE 124 Find m DFE2. Find mGH.GDH70 mDE 124 ind m DFEFIE4. mMN (4 x 8) mMN (4x 8) Solvefor x.for x.Solve3. Find m JKLFind m JKL.JM85 KLN5.6.OL135 50 200 x 100 x MNPage 20
7.8.2x y 56 45 112 x 9.10.59o33oww110 Ayy x30 Page 21
Warm-UpIn problems 1-4, solve for x.1. 122 x 2 2022. 14(2x 1) 10(3x )3. 21(6 x 2) 30(4 x 2)4. 62 4(4 x )5. Recall the following definitions:A is a line that intersects a circle in exactly one point.A intersects a circle in two points.6. Is it possible for a line to intersect a circle in 3 points? 4 points? Explain why or why not.7. When a secant line intersects a circle in two points, it creates a chord. As you have already learned,a chord is a segment whose endpoints lie on the circle. How does a chord differ from a secantline?Page 22
Learning Task: Segment Lengths of a CircleCase 1 - Two Chords Intersecting Inside a CircleExample 1: In the circle below, AE 4 EC 12 BE 8 ED 6Calculate:AE EC BE ED What do you notice about the products of the lengths?Example 2: Find the value of x.Case 2 - Two Secants Intersecting Outside the CircleExample 1:Given: FG 5GH 3IJ 6Calculate:GFFI 4HIFG FH FI FJ What do you notice about the lengths being multiplied?JPage 23
Example 2: Find the value of x.x2412Example 3: Apply your knowledge of two secants intersecting outside the circle to the following:108xGuided Practice: Find the missing indicated segment in each of the following examples.2. Find BE in the circle below.1. Find AE.2. Find BE.1. Find AE in the circle below.BAAB26CE2E64NC3RPage 24
3. Find NR.4. Find x.L6M93x18N6CR45. Find x.36.yx105Page 25
The Segment Theorems Graphic OrganizerLetโs summarize the theorems relating to tangents, chords, and secants. Use the information from theprevious task to complete the graphic organizer.PictureTypeTheoremExampleFind JKSolve for x.Find the value of x.Find the value of x.Find the value of a.Page 26
Skills PracticeSketch a picture for each problem, choose a theorem, set up an equation, and then solve.1. Chords AB and CD intersect inside a circle at point E. AE 2, CE 4, and ED 3. Find EB.2. A diameter of a circle is perpendicular to a chord whose length is 12 inches. If the length of theshorter segment of the diameter is 4 inches, what is the length of the longer segment of the diameter?3. Chords AB and CD intersect inside a circle at point E. AE 5, CE 10, EB x, and ED x-4.Find EB and ED.4. Two secant segments are drawn to a circle from a point outside the circle. The external segment ofthe first secant segment is 8 centimeters and its internal segment is 6 centimeters.If the entire length of the second secant segment is 28 centimeters, what is the length of its externalsegment?5. A tangent segment and a secant segment are drawn to a circle from a point outside the circle.The length of the tangent segment is 15 inches. The external segment of the secant segment measures5 inches. What is the measure of the internal secant segment?6. The diameter of a circle is 19 inches. If the diameter is extended 5 inches beyond the circle to pointC, how long is the tangent segment from point C to the circle?Page 27
Skills Practice (continued)Find the value of the missing variable.Find the value of x.1.2.2x694x6313. x x x x3.204.3x3084 x x Page 28
5.6.x643x5x 7.8x 8.x31554x 2038x 9.x 10.x26x x Page 29
Learning Task: Properties of Angles for a Quadrilateral Inscribed in a CircleRecall: Define a quadrilateral.Now, you will investigate the relationships among the angles of the quadrilateral inscribed in a circle.The following quadrilateral, ABCD, is inscribed in a circle.B1. What arc does D intercept?2. Highlight this arc using a colored pencil.3. What arc does B intercept?A4. Highlight this arc using a colored pencil.5. What is the sum of the two highlighted arcs?6. The measure of an inscribed angle is half themeasure of its intercepted arc; therefore theDm D m B .C7. What arc does A intercept?8. Highlight this arc using a colored pencil.9. What arc does C intercept?10. Highlight this arc using a colored pencil.11. What is the sum of the two highlighted arcs?12. The measure of an inscribed angle is half the measure of itโs intercepted arc; therefore them A m C 13. Since D & B and A & C are opposite angles of a quadrilateral inscribed in a circle wecan conjecture that:Opposite angles of a quadrilateral inscribed in a circle are .Page 30
Guided Practice: Find the value of the indicated variable in the inscribed quadrilateral.1.2.x 47 (22x 1) 88 3.(18x - 19) 4.8y (x - 24) (3y - 20) (2y 10) 3x 10y Find the value of x and y inthe inscribed quadrilateral.x y Page 31
Skills Practice1. Find the value of x and y of the quadrilateral inscribed in the circle.x (y 6)3xy 12x2y2. The quadrilateral ABCD is inscribed in the circle. Solve for the value of x and y.A3y D80 (4x) B(2y - 20) x y CAlso, find angles A, B, C, and D in the quadrilateral shown above. A B C D Page 32
3. Given MSW 8 x 15 and MIW 7 x 15 , find the value of x inscribed in the quadrilateral.Sx WThen, find the measure of MSW and MIW. MSW MIW MIArcs by Parallel Lines InvestigationNext, you will investigate arcs formed by parallel lines that intersect a circle. Aline that intersects a circle in two points called a secant. A secant contains achord of the circle and passes through the interior of a circle, while a tangentline does not. Note that a secant is a line while a chord is a segment.Step 1- On a piece of patty paper, construct a large circle. Lay yourstraightedge across the circle so its parallel edges pass through the circle.Draw secants AB and CD along both edges of the straightedge.Step 2- Fold your patty paper to compare AD and BC . What can you say aboutAD and BC ?Step 3- Repeat Steps 1 and 2, using either lines paper or another object with parallel edges toconstruct different parallel secants. Share your results with your partner. Then complete theconjecture below.Conjecture:Parallel lines intercept arcs on a circle.Page 33
Guided Practice1. AB CD , mAB 72 , and mAC 25 .2. LE NM , mLE 86 , and mMON 186 .Find m LEM.Find mCED.EBLADMCNEO3. AB CD . Solve for p and q.Page 34
Inscribed Angle Theorem The measure of an inscribed angle is _ _ the measure of its intercepted arc. In other words, the intercepted arc is _ the measure of the inscribed angle. Guided Examples: Find the measure of the missing variable. 1. 2. 3. Find the measure of arc XZ 4. Find the measure of angle LPN. Theorem
Two circles are congruent circles if and only if they have congruent radii. A B if AC BD . AC BD if A B. Concentric circles are coplanar circles with the same center. Two coplanar circles that intersect at exactly one point are called tangent circles . Internally tangent circles Externally tangent circles Pairs of Circles
Geometry Unit 10: Circles Name_ Geometry Unit 10: Circles Ms. Talhami 2 Helpful Vocabulary Word Definition/Explanation Examples/Helpful Tips Geometry Unit 10: Circles Ms. Talhami 3 Equation of a Circle Determine the center an
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 Slide 16 / 150
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 Slide 15 / 150
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 Slide 15 / 149
Coplanar Circles and Common Tangents In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common tangent. A
circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common internal tangent A common external tangent two circles. Notes: intersects the segment that joins the centers of the two circles .
Coplanar circles that have a common center are called concentric circles. 2 points of intersection 1 point of intersection (tangent circles) no points of intersection concentric circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of .
