Tangents to Circles. Circle The set of all points in a plane that are equidistant from a given point, called the center of the circle


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1 10.1 Tangents to ircles Goals p Identify segments and lines related to circles. p Use properties of a tangent to a circle. VOULRY ircle 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 of a circle to a point on the circle. segment whose endpoints are the center of the circle and a point on the circle. ongruent circles Two circles that have the same radius iameter The distance across a circle, through its center. chord that passes through the center of the circle. hord segment whose endpoints are points on the circle Secant line that intersects a circle in two points Tangent line that intersects a circle in eactly one point Tangent circles oplanar circles that intersect in one point oncentric oplanar circles that have a common center ommon tangent line or segment that is tangent to two coplanar circles Interior of a circle ll points of the plane that are inside a circle Eterior of a circle ll points of the plane that are outside a circle oint of tangency The point of which a tangent line intersects the circle to which it is tangent opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 10
2 Eample 1 Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of. a. G &* b. &* EG c. ^&( a. G &* is a radius because is the center and G is a point on the circle. E F J H G b. &* EG is a diameter because it contains the center. c. ^&( is a tangent because it intersects the circle at one point. THEOREM 10.1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Q l If l is tangent to Q at, then l Q &*. THEOREM 10. In a plane, 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 l If l Q &* at, then l is tangent to Q. THEOREM 10.3 If two segments from the same eterior point are tangent to a circle, then they are congruent. R S If ^&*(and SR ^**( ST are tangent to, then T SR &* c ST & *. opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 11
3 Eample Finding the Radius of a ircle You are standing at R, 4 feet from a fountain. The distance from you to a point of tangency on the fountain is 8 feet. What is the radius of the fountain? r Q r 8 ft 4 ft R Tangent QR ^**( is perpendicular to radius &* Q at Q, so TQR is a right triangle. So, you can use the ythagorean Theorem. (r 4) r 8 ythagorean Theorem r 8r 16 r 64 Square of binomial 8r 48 Subtract r and 16 from each side. r 6 ivide. nswer The radius of the fountain is 6 feet. Eample 3 Using roperties of Tangents ^&( is tangent to at. ^&( is tangent to at. Find the value of. Use Theorem Substitute. 5 Subtract 4 from each side. 5 Find the square roots of heckpoint omplete the following eercise. 1. ^&( is tangent to at. ^&( is tangent to at. Find the value of or opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 1
4 10. rcs and hords Goals p Use properties of arcs of circles. p Use properties of chords of circles. VOULRY entral angle n angle whose verte is the center of a circle Minor arc art of a circle that measures less than 180 Major arc art of a circle that measures between 180 and 360 Semicircle n arc whose endpoints are the endpoints of a diameter of the circle Measure of a minor arc The measure of its central angle Measure of a major arc The difference between 360 and the measure of its associated minor arc ongruent arcs Two arcs of the same circle or of congruent circles that have the same measure OSTULTE 6: R ITION OSTULTE The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m t ms ms opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 13
5 Eample 1 Finding Measures of rcs Find the measure of each arc. a. Gs b. GH t c. Hs a. mgs mfs mfgs b. mgh t mgs mghs c. mhs 360 mgh t G F T 80 H THEOREM 10.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. s cs if and only if w c w. THEOREM 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Ew cefw, Gs c GFs E G F THEOREM 10.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. JKw is a diameter of the circle. J L M K THEOREM 10.7 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. w cw if and only if EFw c EGw. G E F opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 14
6 Eample Using Theorem 10.5 Find mkms using Theorem mkms mmns Theorem (7 16) Substitute Subtract 5 from each side. 16 dd 16 to each side. 8 ivide. nswer mkms 5 (5 p 8) 40. L J N (7 16) K 5 M Eample 3 Using Theorem 10.7 Find QS if MN 16, RT 16, and NQ 10. N ecause MN w and RTw are congruent chords, they are equidistant from the center. So, Qw cqsw. To find QS, first find N. Qw MN w, so Qw bisects MN w. ecause MN 16, N M R S 10 Q T Then use N to find Q. N 8 and NQ 10, so TNQ is a right triangle. So, Q 6. Finally, use Q to find QS. nswer ecause QSw cqw, QS Q 6. heckpoint omplete the following eercises. 1. Use Theorem 10.5 to. Find HK if G JL 4, find mrsr. and H S R (3 4) W F G V H J T K L 4 5 opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 15
7 10.3 Inscribed ngles Goals p Use inscribed angles to solve problems. p Use properties of inscribed polygons. VOULRY Inscribed angle n angle whose verte is on a circle and whose sides contain chords of the circle Intercepted arc The arc that lies in the interior of an inscribed angle and has endpoints on the angle Inscribed polygon polygon whose vertices all lie on a circle ircumscribed circle circle with an inscribed polygon THEOREM 10.8: MESURE OF N INSRIE NGLE If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. ma 1 mr THEOREM 10.9 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. a ca opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 16
8 Eample 1 Measures of rcs and Inscribed ngles Find the measure of the arc or angle. a. b. X 140 Q R Y Z 80 S a. maxyz 1 mxzs 1 ( 140 ) 70 b. mqr t ma SR ( 80 ) 160 Eample Finding the Measure of an ngle It is given that may 6. What is maz? X ay and az both intercept WX s, so ay c az. nswer So, ma Z ma Y 6. W 6 Y Z heckpoint Find the measure of the arc or angle. 1. amn. RVT t 3. a N M 88 R V 91 S T opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 17
9 THEOREM If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. a is a right angle if and only if &* is a diameter of the circle. THEOREM quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary., E, F, and G lie on some circle,, if and only if ma maf 180 and mae mag 180. E F G Eample 3 Using Theorem Find the value of. &* SV is a diameter. So, at is a right angle and mat S T 15 W V heckpoint omplete the following eercise. 4. In the diagram, WXYZ is inscribed in. Find the values of and y. 6; y 3 Z 18y Y 7 16 X W 18 opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 18
10 10.4 Other ngle Relationships in ircles Goals p Use angles formed by tangents and chords to solve problems in geometry. p Use angles formed by lines that intersect a circle to solve problems. THEOREM 10.1 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. ma1 1 mt, ma 1 m t 1 Eample 1 Finding an ngle Measure In the diagram below, ^&( KL is tangent to the circle. Find maklm. maklm 1 mmjl s K J 4 1 (1 100) L 4 (1 100) nswer maklm ( 4 p 5 ) 100 M heckpoint omplete the following eercise. 1. ^&( QR is tangent to the circle. Find maqrs. (8 3) Q 10 S (5 10) R opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 19
11 THEOREM If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 1 ma1 1 (m t m s ), ma 1 (m s m s ) THEOREM If a tangent and a secant, two tangents, or two secants intersect in the eterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 1 ma1 1 (m s m s ) R Q W 3 Z Y X ma 1 (m QR t m Rs ) ma3 1 (m XYs m WZs ) Eample Measure of an ngle Formed by Two hords Find the value of. F G 1 (m FHs m GJs ) pply Theorem ( ) Substitute. H J 143 Simplify. opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 0
12 Eample 3 Using Theorem Find the value of. a. b. K L N M a. malk 1 (mlmn t mls) pply Theorem Q T 4 R S 58 1 ( 64 ) Substitute Multiply each side by. 180 Solve for. b. 1 (mqt s mrss) pply Theorem ( 68 4 ) Substitute. 1 ( 44 ) Subtract. Multiply. heckpoint Find the value of J 106 M 84 K 86 L H J K F G 7 66 opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 1
13 10.5 Segment Lengths in ircles Goals p Find the lengths of segments of chords. p Find the lengths of segments of tangents and secants. THEOREM If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. THEOREM If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its eternal segment equals the product of the length of the other secant segment and the length of its eternal segment. THEOREM If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its eternal segment equals the square of the length of the tangent segment. E E p E E p E E E p E E p E E ( E ) E p E opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide
14 Eample 1 Finding Segment Lengths hords JN&* and KM&* intersect inside the circle. Find the value of. J LJ p LN LK p LM Theorem p 8 p 5 Substitute Simplify. K 10 L 5 8 N M 4 ivide each side by 10. Eample Finding Segment Lengths Find the value of. 1 E 9 15 p p E Theorem p ( 1 ) 15 p ( 9 15 ) Substitute Simplify. 18 Solve for. Eample 3 Estimating the Radius of a ircle You are standing at a point Q, about 9 feet from a large circular tent. The distance from you to a point of tangency on the tent is about 4 feet. Estimate the radius of the tent. 9 ft S Q r 4 ft (Q) QS p QT Use Theorem p ( r 9 ) Substitute r 81 Simplify r Subtract 81 from each side. 7.5 r ivide each side by 18. nswer So, the radius of the tent is about 7.5 feet. r T opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 3
15 Eample 4 Finding Segment Lengths Use the figure at the right to find the value of. Z V 1 10 W Y ( VW ) WY p WZ Theorem p ( 1 ) Substitute Write in standard form (1)( ) Use Quadratic Formula Simplify. Use the positive solution, because lengths cannot be negative. nswer So, heckpoint omplete the following eercises. 1. hords M *& and NQ *& intersect. Find the value of. inside the circle. Find the H 10 value of. G M 3 N Q K J 7 F 3. Find the value of opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 4
16 10.6 Equations of ircles Goals p Write the equation of a circle. p Use the equation of a circle and its graph to solve problems. VOULRY Standard equation of a circle: circle with radius r and center (h, k) has this standard equation: ( h) (y k) r. Eample 1 Writing a Standard Equation of a ircle Write the standard equation of the circle with center (0, 6) and radius 3.6. ( h) (y k) r Standard equation of a circle ( 0 ) (y 6 ) 3.6 Substitute. (y 6 ) 1.96 Simplify. Eample Writing a Standard Equation of a ircle The point ( 1, 1) is on a circle whose center is ( 3, 4). Write the standard equation of the circle. Find the radius. The radius is the distance from the point ( 1, 1) to the center ( 3, 4). r ( 3 )) ( 1 ( 4 1) r ( ) 3 r 13 Use the istance Formula. Simplify. Simplify. Substitute (h, k) ( 3, 4) and r 13 into the standard equation of a circle. ( ( 3 )) (y 4 ) ( 13 ) Standard equation of a circle ( 3 ) (y 4 ) 13 Simplify. opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 5
17 Eample 3 Graphing a ircle The equation of a circle is ( 3) (y 1) 16. Graph the circle. Rewrite the equation to find the center y 3 and radius: ( 3) (y 1) 16 1 ( 3) [y ( 1 )] 4 The center is ( 3, 1 ) and the radius is 4. To graph the circle, place the point of a compass at ( 3, 1 ), set the radius at 4 units, and swing the compass to draw a full circle (3, 1) heckpoint omplete the following eercises. 1. Write the standard equation of a circle with center ( 5, 3) and radius 5.. ( 5) (y 3) The point (4, 5) is on a circle whose center is (, 3). Write the standard equation of the circle. ( ) (y 3) Graph the equation ( 1) (y 3) 9. y (1, 3) 5 opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 6
18 Eample 4 pplying Graphs of ircles Skydiving skydiving instruction school has a field with multiple landing targets. Each target is circle shaped. coordinate plane is used to arrange the targets in the field, with the corner of the field as the origin. The equation ( 8) (y 4) 9 represents one of the targets. a. Graph the landing target. b. The landing spots by the following skydivers are located as follows: Marika is at (7, 3), le is at (3, 4), Julia is at (10, 7), and aleb is at (9, 6). Which skydivers landed on the target? a. Rewrite the equation to find the center and radius: ( 8) (y 4) 9 ( 8) (y 4) 3 The center is ( 8, 4 ) and the radius is 3. Graph the circle below. 7 y Julia (10, 7) le (3, 4) aleb (9, 6) (8, 4) Marika (7, 3) b. Graph the landing spots of the skydivers. The graph shows that Marika and aleb both landed on the target. opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 7
19 10.7 Locus Goals p raw the locus of points that satisfy a given condition. p raw the locus of points that satisfy two or more conditions. VOULRY Locus locus in a plane is the set of all points in a plane that satisfy a given condition or a set of given conditions. Eample 1 Finding a Locus raw a line m. raw and describe the locus of all points that are 1 centimeter from the line. 1. raw a line m. Locate several points 1 centimeter from m.. Recognize a pattern: the points lie on two lines. 3. raw the lines. nswer The locus of points that are 1 centimeter from m are two lines parallel to m. m opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 8
20 heckpoint omplete the following eercise. 1. escribe the locus of points equidistant from the vertices of an equilateral equiangular triangle. The locus of points is the intersection point of the diagonals. Eample Locus Satisfying Two onditions Lines c and d are in a plane. What is the locus of points in the plane that are equidistant from c and d and within units from the origin? The locus of all points that are equidistant from c and d are the lines 0 and y 0. c y d The locus of all points that are a distance of units from c and d is a circle centered at the origin with a radius of. c y d The intersection of the loci, or locus points, are the line segments from ( 0, ) to ( 0, ) and (, 0 ) to (, 0 ). c y d (0, ) (, 0) (, 0) (0, ) opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 9
21 Eample 3 Locus Satisfying Three onditions oints R(, 4), S(6, 4), and T(7, 4) lie in a plane. What is the locus of points in the plane equidistant from R and S, units from R and 3 units from T? The locus of points equidistant from R and S is the perpendicular bisector of RSw. The locus of points units from R is a circle. The locus of points 3 units from T is a circle. raw the circles and perpendicular bisector in the coordinate plane. 7 y 5 3 R S T nswer The locus is at ( 4, 4 ). heckpoint omplete the following eercises.. oints V and W lie in a plane. What is the locus of points 3 centimeters from V and equidistant from V and W? The locus of points 3 centimeters from V is a circle with center V and radius 3 centimeters. The locus of points equidistant from V and W is a line halfway between V and W. The locus can be 0, 1, or points. 3. Three circles with centers at (, ), (6, ), and (4, 4) each have a radius of units. raw the locus of points and find the point of intersection of the three circles y (4, ) ; opyright Mcougal Littell/Houghton Mifflin ompany ll rights reserved. hapter 10 Geometry Notetaking Guide 30
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