10.1 Lines And Segments That Intersect Circles
10.1Lines and Segments ThatIntersect CirclesEssential QuestionWhat are the definitions of the lines andsegments that intersect a circle?Lines and Line Segments That Intersect CirclesngtachordtenWork with a partner. The drawing atthe right shows five lines or segments thatintersect a circle. Use the relationships shownto write a definition for each type of line orsegment. Then use the Internet or some otherresource to verify your ecaRadius:Diameter:Using String to Draw a CircleWork with a partner. Use two pencils, a piece of string, and a piece of paper.a. Tie the two ends of the piece of string loosely around the two pencils.b. Anchor one pencil on the paper at the center of the circle. Use the other pencilto draw a circle around the anchor point while using slight pressure to keep thestring taut. Do not let the string wind around either pencil.REASONINGABSTRACTLYTo be proficient in math,you need to know andflexibly use differentproperties of operationsand objects.c. Explain how the distance between the two pencil points as you draw the circleis related to two of the lines or line segments you defined in Exploration 1.Communicate Your Answer3. What are the definitions of the lines and segments that intersect a circle?4. Of the five types of lines and segments in Exploration 1, which one is a subsetof another? Explain.5. Explain how to draw a circle with a diameter of 8 inches.Section 10.1hs geo pe 1001.indd 529Lines and Segments That Intersect Circles5291/19/15 2:31 PM
10.1 LessonWhat You Will LearnIdentify special segments and lines.Draw and identify common tangents.Core VocabulVocabularylarryUse properties of tangents.circle, p. 530center, p. 530radius, p. 530chord, p. 530diameter, p. 530secant, p. 530tangent, p. 530point of tangency, p. 530tangent circles, p. 531concentric circles, p. 531common tangent, p. 531Identifying Special Segments and LinesA circle is the set of all points in a plane that are equidistant froma given point called the center of the circle. A circle with center Pis called “circle P” and can be written as P.Pcircle P, or PCore ConceptLines and Segments That Intersect CirclesA segment whose endpoints are the center andany point on a circle is a radius.chordcenterA chord is a segment whose endpoints are ona circle. A diameter is a chord that contains thecenter of the circle.READINGThe words “radius” and“diameter” refer to lengthsas well as segments. For agiven circle, think of a radiusand a diameter as segmentsand the radius and thediameter as lengths.radiusdiameterA secant is a line that intersects a circle intwo points.A tangent is a line in the plane of a circle thatintersects the circle in exactly one point, theAB andpoint of tangency. The tangent ray ⃗— are also called tangents.the tangent segment ABsecantpoint oftangencytangent BAIdentifying Special Segments and LinesDACBGSTUDY TIPIn this book, assume that allsegments, rays, or lines thatappear to be tangent toa circle are tangents.ETell whether the line, ray, or segment is best described asa radius, chord, diameter, secant, or tangent of C.—a. AC—b. AB ⃗c. DEd. ⃖ ⃗AESOLUTION— is a radius because C is the center and A is a point on the circle.a. AC— is a diameter because it is a chord that contains the center C.b. ABc. ⃗DE is a tangent ray because it is contained in a line that intersects the circle inexactly one point.d. ⃖ ⃗AE is a secant because it is a line that intersects the circle in two points.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com— —1. In Example 1, what word best describes AG ? CB ?2. In Example 1, name a tangent and a tangent segment.530Chapter 10hs geo pe 1001.indd 530Circles1/19/15 2:31 PM
Drawing and Identifying Common TangentsCore ConceptCoplanar Circles and Common TangentsIn 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. Coplanarcircles that have a common center are called concentric circles.no points ofintersection1 point of intersection(tangent circles)2 points ofintersectionconcentriccirclesA line or segment that is tangent to two coplanar circles is called a commontangent. A common internal tangent intersects the segment that joins the centersof the two circles. A common external tangent does not intersect the segment thatjoins the centers of the two circles.Drawing and Identifying Common TangentsTell how many common tangents the circles have and draw them. Use blue to indicatecommon external tangents and red to indicate common internal tangents.a.b.c.SOLUTIONDraw the segment that joins the centers of the two circles. Then draw the commontangents. Use blue to indicate lines that do not intersect the segment joining the centersand red to indicate lines that intersect the segment joining the centers.a. 4 common tangentsb. 3 common tangentsMonitoring Progressc. 2 common tangentsHelp in English and Spanish at BigIdeasMath.comTell how many common tangents the circles have and draw them. State whetherthe tangents are external tangents or internal tangents.3.4.Section 10.1hs geo pe 1001.indd 5315.Lines and Segments That Intersect Circles5311/19/15 2:31 PM
Using Properties of TangentsTheoremsTheorem 10.1 Tangent Line to Circle TheoremIn a plane, a line is tangent to a circle if and only ifthe line is perpendicular to a radius of the circle atits endpoint on the circle.PQmLine m is tangent to Qif and only if m QP.Proof Ex. 47, p. 536Theorem 10.2 External Tangent Congruence TheoremTangent segments from a common external pointare congruent.RSPTIf SR and ST are tangentsegments, then SR ST.Proof Ex. 46, p. 536Verifying a Tangent to a Circle— tangent to P?Is ST35S37T12PSOLUTIONUse the Converse of the Pythagorean Theorem (Theorem 9.2). Because 122 352 372,— PT—. So, ST— is perpendicular to a radius of P at its PTS is a right triangle and STendpoint on P.— is tangent to P.By the Tangent Line to Circle Theorem, STFinding the Radius of a CircleIn the diagram, point B is a point of tangency. Findthe radius r of C.A50 ftCrr80 ftBSOLUTION— BC—, so ABC isYou know from the Tangent Line to Circle Theorem that ABa right triangle. You can use the Pythagorean Theorem (Theorem 9.1).AC 2 BC 2 AB2(r 50)2 r 2 802r 2 100r 2500 r 2 6400100r 3900r 39Pythagorean TheoremSubstitute.Multiply.Subtract r 2 and 2500 from each side.Divide each side by 100.The radius is 39 feet.532Chapter 10hs geo pe 1001.indd 532Circles1/19/15 2:31 PM
Constructing a Tangent to a CircleGiven C and point A, construct a line tangentto C that passes through A. Use a compassand straightedge.CASOLUTIONStep 1Step 2CMStep 3ACMACBFind a midpoint—. Construct the bisectorDraw ACof the segment and label themidpoint M.MABDraw a circleConstruct M with radius MA.Label one of the points where M intersects C as point B.Construct a tangent lineDraw ⃖ ⃗AB. It is a tangentto C that passes through A.Using Properties of Tangents— is tangent to C at S, and RT— is tangent to C at T. Find the value of x.RSS28RC3x 4TSOLUTIONRS RTExternal Tangent Congruence Theorem28 3x 4Substitute.8 xSolve for x.The value of x is 8.Monitoring Progress—6. Is DE tangent to C?3CD42 ESection 10.1hs geo pe 1001.indd 533Help in English and Spanish at BigIdeasMath.com—8. Points M and N are7. ST is tangent to Q.Find the radius of Q.QrSpoints of tangency.Find the value(s) of x.Mr24x2P18TN9Lines and Segments That Intersect Circles5331/19/15 2:31 PM
10.1 ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. WRITING How are chords and secants alike? How are they different?2. WRITING Explain how you can determine from the context whether the words radius anddiameter are referring to segments or lengths.3. COMPLETE THE SENTENCE Coplanar circles that have a common center are called .4. WHICH ONE DOESN’T BELONG? Which segment does not belong with the other three?Explain your reasoning.chordradiustangentdiameterMonitoring Progress and Modeling with MathematicsIn Exercises 5–10, use the diagram. (See Example 1.)5. Name the circle.6. Name two radii.B8. Name a diameter.J9. Name a secant.— is tangent to C.In Exercises 19–22, tell whether ABExplain your reasoning. (See Example 3.)CDHG18.KA7. Name two rr14AChapter 10hs geo pe 1001.indd 5347Br61624C9ArBIn Exercises 15–18, tell whether the common tangent isinternal or external.5348B23.r16.CIn Exercises 23–26, point B is a point of tangency. Findthe radius r of C. (See Example 4.)C15.1216A13.18CA21.A1595310. Name a tangent and a point of tangency.In Exercises 11–14, copy the diagram. Tell how manycommon tangents the circles have and draw them.(See Example 2.)BrA26.B30rCCrA18Circles1/19/15 2:31 PM
CONSTRUCTION In Exercises 27 and 28, construct Cwith the given radius and point A outside of C. Thenconstruct a line tangent to C that passes through A.27. r 2 in.37. USING STRUCTURE Each side of quadrilateralTVWX is tangent to Y. Find the perimeter ofthe quadrilateral.28. r 4.5 cm1.2 TIn Exercises 29–32, points B and D are points oftangency. Find the value(s) of x. (See Example 5.)29.B2x 730.B4.5VX3x 103.3YA8.3AC5x 831.32.2x2 4A—C2x 5BA3x2 2x 7D33. ERROR ANALYSIS Describe and correct the error in— is tangent to Z.determining whether XY 60Z11Y61X—38. LOGIC In C, radii CA and CB are perpendicular.DC22WDDB3.17x 6CBecause 112 602 612, XYZ is a— is tangent to Z.right triangle. So, XY⃖ ⃗ are tangent to C.⃖ ⃗BD and AD—, CB—, BD⃖ ⃗, and ⃖ ⃗a. Sketch C, CAAD.b. What type of quadrilateral is CADB? Explainyour reasoning.39. MAKING AN ARGUMENT Two bike paths are tangentto an approximately circular pond. Your class isbuilding a nature trail that begins at the intersection Bof the bike paths and runs between the bike paths andover a bridge through the center P of the pond. Yourclassmate uses the Converse of the Angle BisectorTheorem (Theorem 6.4) to conclude that the trail mustbisect the angle formed by the bike paths. Is yourclassmate correct? Explain your reasoning.EP34. ERROR ANALYSIS Describe and correct the error infinding the radius of T. BMU39TS40. MODELING WITH MATHEMATICS A bicycle chain36V392 362 152So, the radius is 15.— is a common tangent ofis pulled tightly so that MNthe gears. Find the distance between the centers ofthe gears.17.6 in.35. ABSTRACT REASONING For a point outside of acircle, how many lines exist tangent to the circle thatpass through the point? How many such lines existfor a point on the circle? inside the circle? Explainyour reasoning.1.8 in.MLN4.3 in.P41. WRITING Explain why the diameter of a circle is thelongest chord of the circle.36. CRITICAL THINKING When will two lines tangent tothe same circle not intersect? Justify your answer.Section 10.1hs geo pe 1001.indd 535Lines and Segments That Intersect Circles5351/19/15 2:31 PM
42. HOW DO YOU SEE IT? In the figure, ⃗PA is tangent46. PROVING A THEOREM Prove the External Tangentto the dime, ⃗PC is tangent to the quarter, and ⃗PB is acommon internal tangent. How do you know that— PB— PC—?PACongruence Theorem (Theorem 10.2).RPPST— and ST— are tangent to P.Given SR— ST—Prove SRCAB47. PROVING A THEOREM Use the diagram to prove eachpart of the biconditional in the Tangent Line to CircleTheorem (Theorem 10.1).—43. PROOF In the diagram, RS is a common internalAC RCtangent to A and B. Prove that — —.BC SCQRAmBCa. Prove indirectly that if a line is tangent to a circle,then it is perpendicular to a radius. (Hint: If you—, thenassume line m is not perpendicular to QPthe perpendicular segment from point Q to line mmust intersect line m at some other point R.)Given Line m is tangent to Q at point P.—Prove m QPS44. THOUGHT PROVOKING A polygon is circumscribedabout a circle when every side of the polygon istangent to the circle. In the diagram, quadrilateralABCD is circumscribed about Q. Is it always truethat AB CD AD BC? Justify your answer.YDXBWAb. Prove indirectly that if a line is perpendicular toa radius at its endpoint, then the line is tangent tothe circle.—Given m QPProve Line m is tangent to Q.CQZ48. REASONING In the diagram, AB AC 12, BC 8,and all three segments are tangent to P. What is theradius of P? Justify your answer.45. MATHEMATICAL CONNECTIONS Find the values of xBand y. Justify your answer.P4y 1 RQ2x 5x 8Tx 6PDPEFCSMaintaining Mathematical ProficiencyAReviewing what you learned in previous grades and lessonsFind the indicated measure. (Section 1.2 and Section 1.5)49. m JKM50. ABJ15 L10AB7C28 K536Chapter 10hs geo pe 1001.indd 536MCircles1/19/15 2:31 PM
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 .
Types of Lines Horizontal Vertical Oblique Horizontal Vertical Oblique Lines Segments Lines Segments Parallel Lines Intersecting Perpendicular Intersecting ObliqueParallel Segments Angles † Angles form when lines, line segments, or rays intersect. † A vertex is where 2 sides of an angle meet. vertex side angle † Angles can be right, acute .
Skew Lines Skew lines are lines that are and do not . In this diagram, planes R and W are parallel. DEand FGare lines. Perpendicular lines are not skew lines, because they're in the same . Parallel lines are skew lines,
1. Lines that do not intersect are parallel lines. 2. Skew lines are coplanar. 3. Transversal is a line that intersects two or more lines. 4. Perpendicular lines are intersecting lines. 5. If two lines are parallel to a third line, then the two lines are parallel. You have just tried describing parallel and perpendicular lines. In
Given a set Sof n line segments in three-dimensional space, finding all the lines that si-multaneously intersect at least of line segments in Sis a fundamental problem that arises in . cases are: A line segment may degenerate to a point, several segments may intersect, be coplanar, parallel, concurrent, lie on the same supporting line, or .
2CURVATURA 3-Dimensional System Components The CURVATURA 3-D System is a “pre-engineered” collection of 16 curved “vault” main tee segments and 16 curved main “valley” tee segments and 4 straight main tee segments. Custom curved segments are also available. Combine the segments to create an infinite number of undulating waves and sweeping curves.
Figure 1: A clockwise component and its C-polygon We define notation for this paper. A polygonal chain is a concatenation of line segments. The endpoints of the segments are called vertices} and the segments themselves are edges, If the segments intersect only at the endpoints of adjacent segments, then the
All segments skew to MN _ 10. All segments parallel to IK _ 11. All segments skew HJ _ Name the pars of the pyramid shown at the right. 12. A pairs of parallel segments 13. A pairs of skew segments 14. All panes parallel to plane EDC 15. All planes that interest to form the line BC Draw and Label the following to illustrate each .
segments. OR A polygon is a closed curve (figure) formed by the line segments such that: (i) No two line segments intersect except at their end points. (ii) No two line segments with a common end point are coincident. The smallest possible polygon is made up of three sides called as Triangle. A polygon made up of four line segments is called as .
