Use Properties of Tangents

Transcription

1 6.1 Georgia Performance Standard(s) MM2G3a, MM2G3d Your Notes Use Properties of Tangents Goal p Use properties of a tangent to a circle. VOULRY ircle enter Radius hord iameter Secant Tangent Example 1 Identify special segments and lines Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of (. a. } E c. ###$ E F E Solution a. } is a because is the center and is a point on the circle. E is a because it is a line that intersects the circle in two points. c. ###$ E is a ray because it is contained in a line that intersects the circle in exactly one point. 192 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

2 Your Notes Example 2 Find lengths in circles in a coordinate plane Use the diagram to find the given lengths. a. Radius of ( b. iameter of ( c. Radius of ( d. iameter of ( Solution a. The radius of ( is units. b. The diameter of ( is units. c. The radius of ( is units. d. The diameter of ( is units. y 1 1 x heckpoint omplete the following exercises. 1. In Example 1, tell whether } is best described as a radius, chord, diameter, secant, or tangent. Explain. 2. Use the diagram to find (a) the radius of ( and (b) the diameter of (. y 1 1 x opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 193

3 Your Notes Example 3 raw common tangents Tell how many common tangents the circles have and draw them. a. b. c. Solution a. common b. common c. common tangent(s) tangent(s) tangent(s) heckpoint Tell how many common tangents the circles have and draw them THEOREM 6.1 In a plane, a line is tangent to a circle if and only if the line is to a radius of the circle at its endpoint on the circle. O P 194 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

4 Your Notes Example 4 Verify a tangent to a circle In the diagram, } RS is a radius of (R. Is } ST tangent to (R? Solution Use the onverse of the Pythagorean Theorem. ecause , nrst is a and } RS. So, is perpendicular to a radius of (R at its endpoint on (R. y, } ST is to (R. T S R 10 heckpoint } RS is a radius of (R. Is } ST tangent to (R? 5. R 5 S 8 12 T 6. S T R 7 Example 5 In the diagram, is a point of tangency. Find the radius r of (. 77 r Solution You know from Theorem 6.1 that } }, so n is a. You can use the Pythagorean Theorem Pythagorean Theorem (r 1 49) 2 5 r Substitute. r 2 1 r 1 5 r 2 1 Multiply. r 5 r 5 The radius of ( is. Find the radius of a circle 49 Subtract from each side. r ivide each side by. opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 195

5 Your Notes heckpoint omplete the following exercise. 7. In the diagram, K is a point of K r tangency. Find the radius r of (L. L 56 r 32 J THEOREM 6.2 Tangent segments from a common external point are. P R S T Example 6 Use properties of tangents } QR is tangent to ( at R and } QS is tangent to ( at S. Find the value of x. Solution QR 5 QS 5 Substitute. 5 x Solve for x. S R 32 3x 1 5 Tangent segments from a common external point are. Q TRINGLE SIMILRITY POSTULTES N THEOREMS ngle-ngle () Similarity Postulate: If two angles of one triangle are to two angles of another, then the two triangles are. Theorem 6.3 Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are, then the triangles are. Theorem 6.4 Side-ngle-Side (SS) Similarity Theorem: If an angle of one triangle is to an angle of a second triangle and the lengths of the sides including these angles are, then the triangles are. 196 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

6 Your Notes Example 7 Use tangents with similar triangles In the diagram, both circles are centered at. } E is tangent to the inner circle at and } is tangent to the outer circle at. Use similar triangles to show that } 5 } E. E Solution } } E and } } ù E efinition of ll right? are ù. Similarity Postulate orr. side lengths of,ns are proportional. heckpoint omplete the following exercises. 8. } RS is tangent to ( at S and } RT is tangent to ( at T. Find the value(s) of x. R 49 x 2 S T 9. In the diagram, } ST is a common internal tangent to (M and (P. Use similar triangles to show that MN } PN 5 } SN TN. M S N T P Homework opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 197

7 Name ate LESSON 6.1 Practice Match the notation with the term that best describes it. 1.. enter FH. hord } }. iameter. Radius H F G E 5. E. Point of tangency 6. } F. ommon external tangent E G. ommon internal tangent H. Secant Use the diagram at the right. 9. What are the diameter and radius of (? y 10. What are the diameter and radius of (? 1 1 x 11. escribe the intersection of the two circles. 12. escribe all the common tangents of the two circles. Use (P to draw the part of the circle described or to answer the question. 13. raw a diameter }. 14. raw tangent 15. raw chord }. 16. raw a secant through point. 17. What is the name of a radius of the circle? P 198 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

8 Name ate LESSON 6.1 Practice continued Tell how many common tangents the circles have and draw them raw two circles with the given number of common tangents In the diagram, } is a radius of (. etermine whether } is tangent to (. Explain your reasoning In the diagram, } is tangent to ( at point. Find the radius r of ( r r 28. r r r r 15 9 opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 199

9 Name ate LESSON 6.1 Practice continued } JK is tangent to (L at K and } JM is tangent to (L at M. Find the value of x. 30. K x 31. 3x K 32. 5x 2 4 J K L M 22 J J x 1 12 M L 2x 1 2 M L 33. Softball On a softball field, home plate is 38 feet from the pitching circle. Home plate is about 45.3 feet from a point of tangency on the circle. r r 45.3 ft 38 ft a. How far is it from home plate to a point of tangency on the other side of the pitching circle? b. What is the radius of the pitching circle? 34. In the diagram, } QT is tangent to both circles and } RT is tangent to both circles. Use similar triangles to show that } PS ST 5 } QR RT. R P S T 200 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

10 6.2 Find rc Measures Georgia Performance Standard(s) MM2G3b, MM2G3d Your Notes Goal p Use angle measures to find arc measures. VOULRY entral angle Semicircle rc Minor arc Major arc Measure of a minor arc Measure of a major arc ongruent circles ongruent arcs MESURING RS The measure of a minor arc is the measure of its central angle. The expression m is read as the measure of arc. The measure of the entire circle is 508. The measure of a major arc is the difference between and the measure of the related minor arc. m The measure of a semicircle is. m opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 201

11 Your Notes Example 1 Find measures of arcs Find the measure of each arc of (, where } F is a diameter. a. E b. FE c. EF E F 1178 a. E is a arc, so me 5 m 5. b. FE is a arc, so mfe c. } F is a diameter, so EF is a, and mef 5. heckpoint omplete the following exercise. 1. Find mrts in the diagram at the right. T P R 458 S R ITION POSTULTE The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m 5 m 1 m Example 2 Find measures of arcs Money You join a new bank and divide your money several ways, as shown in the circle graph at the right. Find the indicated arc measures. a. m b. m hecking 558 Savings 1058 Money Market 1408 onds 608 Solution a. m 5 m 1 m b. m m Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

12 Your Notes Example 3 Identify congruent arcs Tell whether the given arcs are congruent. Explain why or why not. a. and E b. and c. FG and HJ G E 808 F H J Solution a. E because they are in and m me. b. and have the same, but they are because they are arcs of circles that are. c. FG HJ because they are in and mfg mhj. heckpoint omplete the following exercises. 2. In Example 2, find (a) m and (b) m. Homework 3. In the diagram at the right, is PQ > SR? Explain why or why not. P R S opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 203

13 Name ate LESSON 6.2 Practice Name the arc shown in bold S P R } and } FE are diameters of (. etermine whether the given arc is a minor arc, major arc, or semicircle. 4. E 5. E F 6. FE 7. F E 8. F 9. E F In (O, } MQ and } NR are diameters. Find the indicated measure. 12. m MN 13. m NQ 14. m NQR 15. m MRP N M O R 16. m QR 17. m MR P 18. m QMR 20. m PRN 19. m PQ 21. m MQN 204 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

14 Name ate LESSON 6.2 Practice continued Find the indicated arc measure. 22. m 23. m 24. m Use the information given about a central angle of a circle to find the measure of its corresponding arc. 25. The central angle is a right angle. 26. The central angle is a diameter. 27. The central angle is complementary to a 308 angle. 28. The central angle is supplementary to a 588 angle. opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 205

15 Name ate LESSON 6.2 Practice continued Tell whether ù. Explain Keeping Time In the clock face shown at the right, the positions of the numbers determine congruent arcs along the circle. 33. What is the measure of the arc between any two consecutive numbers? n arc is traced out by the end of the second hand as it moves from the 12 to the 4. Is this a minor arc or a major arc? Starting at the 2, what number does the end of the second hand reach as it completes a semicircle? 36. When the second hand moves from the 8 to the 3, what is the measure of the arc? 37. When the second hand moves from halfway between the 1 and the 2 to } 4 of the way 5 from the 1 to the 2, what is the measure of the arc? 38. The second hand moves from the 3 to the 7. What is the measure of the corresponding major arc? 206 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

16 6.3 pply Properties of hords Georgia Performance Standard(s) MM2G3a, MM2G3d Your Notes Goal p Use relationships of arcs and chords in a circle. THEOREM 6.5 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are > if and congruent. only if >. Example 1 Use congruent chords to find an arc measure In the diagram, ( > (, } > } EF, and mef Find m Solution ecause } and } EF are congruent in congruent, the corresponding minor arcs and EF are. So, m 5 mef 5. F E THEOREM 6.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. If } QS is a perpendicular bisector of } TR, then diameter of the circle. S T P R is a THEOREM 6.7 If a diameter of a circle is perpendicular E to a chord, then the diameter bisects the chord and its arc. If } EG is a diameter and } EG } F, then } H > } HF and >. F H G opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 207

17 Your Notes Example 2 Use perpendicular bisectors Journalism journalist is writing a a story about three sculptures, arranged as shown at the right. Where should the journalist place a camera so that it is the same distance from each sculpture? Solution Step 1 Label the sculptures,, and. raw segments } and }. Step 2 raw the of } and }. y, these bisectors are diameters of the circle containing,, and. Step 3 Find the point where these bisectors. This is the center of the circle containing,, and, and so it is from each point. amera heckpoint omplete the following exercises. 1. If mtv , find mrs. S T 6 6 R V 2. Find the measures of, E, 4x8 and E. (80 2 x)8 E 208 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany. 6.indd 127 4/16/07 9:01:40 M

18 Your Notes THEOREM 6.8 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. F E G } > } if and only if 5. Example 3 Use Theorem 6.8 In the diagram of (F, Find EF. Solution hords } and } are congruent, so by Theorem 6.8 they are from F. Therefore, EF 5. EF 5 Use Theorem x 5 Substitute. x 5 Solve for x. So, EF 5 3x 5 3( ) G 7x 2 8 F 3x E 12 heckpoint omplete the following exercise. 3. In the diagram in Example 3, suppose 5 27 and EF 5 GF 5 7. Find. opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 209

19 Your Notes Example 4 Use chords with triangle similarity 8 In (S, SP 5 5, MP 5 8, ST 5 SU, } N QN } MP, and NRQ is a right angle. Show that npts, nnrq. M 1. etermine the side lengths of npts. iameter } QN is perpendicular to } } MP, so by QN bisects } MP. Therefore, PT 5 1 } } 2 ( ) 5. } SP has a given length of. ecause } QN is perpendicular to } MP, PTS is a and TS 5 Ï }} ( ) 2 2 ( ) 2 5 Ï }} 2 5. The side lengths of npts are SP 5, PT 5, and TS etermine the side lengths of nnrq. The radius } SP has a length of, so the diameter QN 5 2( ) 5 } 2( ) 5. y NR ù } MP, so NR 5 MP 5. ecause NRQ is a, RQ 5 Ï }} ( ) 2 2 ( ) 2 5 Ï }} 2 5. The side lengths of nnrq are QN 5, NR 5, and RQ Find the ratios of corresponding sides. } PT NR 5 5, TS } RQ 5 5, and SP } QN 5 5. ecause the side lengths are proportional, npts, nnrq by the. R U T S 5 P heckpoint omplete the following exercise. Homework 4. In Example 4, suppose in (S, QN 5 26, NR 5 24, ST 5 SU, } QN } MP, and NRQ is a right angle. Show that npts, nnrq. 210 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

20 Name ate LESSON 6.3 Practice Find the chord length FG 3. KM E 6 G F K J 3 5 M L Find the arc length. 4. m XY 5. m TU 6. m X W Y Z 1058 S R 858 T 1008 U Tell whether } QS is a diameter of the circle. If not, explain why R P S R P 19 S R 20 T 9 9 S opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 211

21 Name ate LESSON 6.3 Practice continued Tell whether the measures are equal. 10. LK and KN 11. m ST and m RS 12. m and m L M K P N S T R U E Find the given measure m JK 15. m MN 5 H 558 L J K M N E G L 618 P Tell whether the lengths are equal. 16. and EF 17. JK and LM 18. TQ and UQ F E K 9 J 8 M L P R T U N S 212 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

22 Name ate LESSON 6.3 Practice continued Find the given measure. 19. Q 20. HJ 21. m TU G E 7 F J H 4 4 K L R S T U Find the value of x x 1 1 8x 2 5 E 23. 4x 2 1 3x 1 6 Q x8 Y X (35x 2 16)8 Z 25. oins You notice that the word LIERTY on the heads side of a quarter makes about the same arc as the words QURTER OLLR on the tails side. Explain how you can use a straight ruler to check whether this is true. 26. In (, } FE } G, FE is a right angle, } ù } H, E 5 16, and FE Show that n, nfe. H 16 G F E 20 opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 213

23 6.4 Use Inscribed ngles and Polygons Georgia Performance Standard(s) MM2G3b, MM2G3d Your Notes Goal p Use inscribed angles of circles. VOULRY Inscribed angle Intercepted arc Inscribed polygon ircumscribed circle THEOREM 6.9: MESURE OF N INSRIE NGLE THEOREM The measure of an inscribed angle is one half the measure of its m 5 } 1 intercepted arc. 2 Example 1 Use inscribed angles Find the indicated measure in (P. a. m S b. mrq Solution R 608 T 378 P Q S a. m S 5 1 } } 2 ( ) 5 b. mqs 5 2m 5 2 p 5. ecause RQS is a semicircle, mrq Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

24 Your Notes heckpoint Find the indicated measure. 1. m GHJ 2. m H F 408 G 1008 J THEOREM 6.10 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. > THEOREM 6.11 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. m if and only if is a diameter of the circle. THEOREM 6.12 quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary., E, F, and G lie on ( if and only if m 1 m F 5 m E 1 m G 5. E F G opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 215

25 Your Notes Example 2 Use a circumscribed circle Security security camera that rotates 908 is placed in a location where the only thing viewed when rotating is the wall. You want to change the camera s location. Where else can it be placed so that the wall is viewed in the same way? Solution From Theorem 6.11, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a of the circle. So, draw the circle that has the width of the wall as a. The wall fits perfectly with your camera s 908 rotation from any point on the in front of the wall. heckpoint omplete the following exercises. 3. Find m RTS. R Q 318 S T 4. right triangle is inscribed in a circle. The radius of the circle is 5.6 centimeters. What is the length of the hypotenuse of the right triangle? 216 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

26 Your Notes Example 3 Use Theorem 6.12 Find the value of each variable. a. Q b. K 888 3y8 P 1008 x8 S y8 R J 8x8 3y8 M 4x8 L Solution a. PQRS is inscribed in a circle, so its opposite angles are. m P 1 m R 5 m Q 1 m S 5 1 y8 5 1 x8 5 y 5 x 5 b. JKLM is inscribed in a circle, so its opposite angles are. 1 m L 5 1 m M 5 1 4x y x 5 y 5 heckpoint omplete the following exercise. 5. Find the values of a and b. 5a8 4b8 968 Homework opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 217

27 Name ate LESSON 6.4 Practice Find the indicated measure. 1. m 2. m 3. m m 5. m 6. m m 8. m 9. m Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

28 Name ate LESSON 6.4 Practice continued Find the indicated measure in (M. 10. m PNO 1188 N Q M 11. m QNP P 688 O 12. m PQ 13. m QO 14. m NMO 15. m NOP 16. m QMP 17. m OQN opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 219

29 Name ate LESSON 6.4 Practice continued Name two pairs of congruent angles. 18. E 19. X Y F W Z ecide whether a circle can be circumscribed about the quadrilateral Find the values of the variables. 23. x8 y8 24. M x8 N y8 25. x8 E y O F P 220 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

30 Name ate LESSON 6.4 Practice continued Find the values of the variables J x E F y8 R U x8 S y8 408 T 1148 M 528 L y8 808 K 1088 H x G Find m and m (2x 1 8)8 (3x 2 24)8 32. Stained Glass You are making the stained glass ornament shown at the right. The kite is symmetric, so >, } is a diameter of the circle, and m What are the measures of,, and? opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 221

31 6.5 pply Other ngle Relationships in ircles Georgia Performance Standard(s) MM2G3b, MM2G3d Your Notes Goal p Find the measures of angles inside or outside a circle. THEOREM 6.13 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 m 1 5 } 1 2 intercepted arc. m 2 5 } Example 1 Find angle and arc measures Line m is tangent to the circle. Find the indicated measure. a. m 1 b. mef m E 1 m F Solution a. m 1 5 (1328) 5 b. mef 5 (1108) 5 heckpoint Find the indicated measure. 1. m Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

32 Your Notes THEOREM 6.14: NGLES INSIE THE IRLE THEOREM If two chords intersect inside 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. 2 m } 2 (m 1 m ) m } 2 (m 1 m ) 1 Example 2 Find the value of x. Solution Find an angle measure inside a circle The chords } FH and } GJ intersect inside the circle. G F 1128 x J H x8 5 1 } 2 (m 1 m ) Use Theorem x8 5 1 } 2 ( 1 ) Substitute. x 5 Simplify. heckpoint Find the value of x. 2. F x8 J G H opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 223

33 Your Notes THEOREM 6.15: NGLES OUTSIE THE IRLE THEOREM If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 1 2 P Q m } 2 (m 2 m ) m } 2 (mpqr 2 mpr ) R X W 3 m 3 5 } 1 2 (mxy 2 mwz ) Z Y Example 3 Find the value of x. Find an angle measure outside a circle J Solution The tangent ###$ GF and the secant ###$ GJ intersect outside the circle. H G x8 728 F 1708 m FGH 5 1 } 2 (m 2 m ) Use Theorem x85 1 } 2 ( 2 ) Substitute. x 5 Simplify. heckpoint Find the value of y. Homework 3. G 268 H J 308 F y8 K 224 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

34 Name ate LESSON 6.5 Practice Find the measure of each numbered angle or arc opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 225

35 Name ate LESSON 6.5 Practice continued Find the measure of each numbered angle or arc Find the value of x x x x (3x 2 1) (5x 1 33) 226 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

36 Name ate LESSON 6.5 Practice continued Find the value of x x 21. x 45 (10x 2 1) 22. Satellites satellite is taking pictures of Earth from 4000 miles above its surface. What is the measure of RT, which corresponds to the portion of Earth that can be photographed from the satellite? 8000 mi R U T S 4000 mi Not drawn to scale 23. Theater play is being presented on a circular stage. The two main characters are at positions and at the back of the stage. Use the diagram to answer the 808 following questions. a. What angle of view between the main characters does an actor at position at center stage have? 388 E b. What angle of view between the main characters does the orchestra conductor at point have? c. What angle of view between the main characters does an audience member at point E have? opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 227

37 6.6 Find Segment Lengths in ircles Georgia Performance Standard(s) MM2G3a, MM2G3d Your Notes Goal p Find segment lengths in circles. VOULRY Segments of a chord Secant segment External segment THEOREM 6.16: SEGMENTS OF HORS 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 E lengths of the segments of the other chord. E p 5 E p Example 1 Find lengths using Theorem 6.16 Find ML and JK. NK p NJ 5 p x p (x 1 5) 5 ( ) p ( ) x 2 1 5x 5 x 5 Find ML and JK by substitution. L x 1 3 x N K x 1 5 M J x 1 1 ML 5 ( ) 1 ( ) JK 5 1 ( ) Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

38 Your Notes THEOREM 6.17: SEGMENTS OF SENTS THEOREM If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its E external segment. p E 5 E p Example 2 Find lengths using Theorem 6.17 Find the value of x. Solution T P 4 Q 6 R 5 x S RQ p RP 5 RS p RT Use Theorem. p ( 1 ) 5 p (x 1 ) Substitute. 5 x 1 Simplify. 5 x Solve for x. 1. heckpoint Find the value of x. 5 x x 6 7 opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 229

39 Your Notes THEOREM 6.18: SEGMENTS OF SENTS N TNGENTS THEOREM If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the E lengths of the secant segment and its external segment equals the square of the length of the tangent segment. E 2 5 p Example 3 Find lengths using Theorem 6.18 Find RS.. 14 R Q x S 12 Solution RQ 2 5 RS p T Use Theorem. 2 5 x p ( ) Substitute. 5 x 2 1 x Simplify. 0 5 x 2 1 x 2 Write in standard form. 6 Ï }}} x 5 2 4( )( ) Use quadratic }}} 2( ) formula. x 5 Simplify. Lengths cannot be, so use the solution. So, x 5 <, and RS <. Homework heckpoint omplete the following exercise. 3. Find JK. L 6 M 8 K x J 230 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

40 Name ate LESSON 6.6 Practice Fill in the blanks. Then find the value of x. 1. x p 5 5 p 2. 6 p 5 3 p 3. x p 5 8 p 9 3 x x 8 x p 5 5 p 5. 3 p 5 4 p 6. 3 p 5 5 p x x 3 5 x x p 8. x p 9. x p 4 12 x x 2 x opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 231

41 Name ate LESSON 6.6 Practice continued Find the value of x x x x x x x x x x 232 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

42 Name ate LESSON 6.6 Practice continued Find the value of x x 20. 3x x 21. x x x x 3 5x x oorway n arch over a doorway is an arc that is 160 centimeters wide and 50 centimeters high. You are curious about the size of the entire circle containing the arch. y drawing and labeling a circle passing through the arc as shown in the diagram, you can use the following steps to find the radius of the circle. a. Find. 160 cm O E 50 cm b. Find and. c. Use,, and to find E. d. Find E. e. Find the length of the radius, EO. opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 233

43 6.7 ircumference and rc Length Georgia Performance Standard(s) MM2G3c Your Notes Goal p Find arc lengths and other measures of circles. VOULRY ircumference rc length THEOREM 6.19: IRUMFERENE OF IRLE The circumference of a circle is r 5 or 5, where d d is the diameter of the circle and r is the radius of the circle. 5 5 Example 1 Use the formula for circumference Find the indicated measure. a. ircumference of a circle b. Radius of a circle with with radius 11 meters circumference 18 yards Solution a. 5 2πr b. 5 2πr 5 2 pπp 5 2πr 5 π 5 r < m yd < r heckpoint omplete the following exercise. 1. Find the circumference of a circle with diameter 23 inches. 234 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

44 Your Notes R LENGTH OROLLRY 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 rc length of }} 2πr 5 m }, or 3608 rc length of 5 m } 3608 p 2πr P r Example 2 Find and use arc lengths Find the indicated measure. a. rc length of b. mrs 12.3 ft S P m R 38 ft a. rc length of 5 p 2π( ) < meters b. rc length of RS }} 2πr 5 m RS } Write equation π1 2 5 m RS } 3608 Substitute. p 2π1 2 5 mrs Multiply each side by. < mrs Use a calculator. heckpoint Find the indicated measure. Homework 2. rc length of 3. ircumference of (Z 4 ft X P 978 Z cm Y opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 235

45 Name ate LESSON 6.7 Practice Use the diagram to find the indicated measure. 1. Find the circumference. 2. Find the circumference. 3. Find the radius. Find the indicated measure. 4. The exact radius of a circle with circumference 36 meters 5. The exact diameter of a circle with circumference 29 feet 6. The exact circumference of a circle with diameter 26 inches 7. The exact circumference of a circle with radius 15 centimeters Find the length of Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

46 Name ate LESSON 6.7 Practice continued In ( shown below, >. Find the indicated measure. 11. m 12. m 13. Length of 14. Length of 15. m 16. Length of Find the indicated measure. 17. Length of 18. ircumference of (Q 19. Radius of (Q opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 237

47 Name ate LESSON 6.7 Practice continued Find the perimeter of the region In the table below, refers to the arc of a circle. omplete the table. Radius m Length of Scooter Wheel The wheel on a manual scooter has a diameter of 4 } 1 inches, as shown. 2 a. Find the circumference of the wheel. b. How many feet does the wheel travel when it makes 150 revolutions? Round to the nearest foot in. 24. irthday ake birthday cake is sliced into 8 equal pieces. The arc length of one piece of cake is 6.28 inches, as shown. Find the diameter of the cake. 238 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

48 6.8 reas of ircles and Sectors Georgia Performance Standard(s) MM2G3c, MM2G3d Your Notes Goal p Find the areas of circles and sectors. VOULRY Sector of a circle THEOREM 6.20: RE OF IRLE The area of a circle is π times the square of the radius. 5 r Example 1 Find the indicated measure. a. rea 4.2 m Use the formula for area of a circle b. iameter in. 2 Solution a. 5 πr 2 Write formula for area of a circle. 5 π( ) 2 Substitute for r. 5 π Simplify. < m 2 Use a calculator. b. 5 πr 2 Write formula for area of a circle. 5 πr 2 Substitute for. 5 r 2 ivide each side by. in. < r Find the positive square root of each side. The radius is about inches, so the diameter is about inches. opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 239

49 Your Notes THEOREM 6.21: RE OF SETOR The ratio of the area of a sector of a circle to the area of the whole circle (πr 2 ) is equal to the ratio of the measure of the intercepted arc to rea of sector P , or P r rea of sector P p Example 2 Find areas of sectors Find the areas of the sectors formed by RQS. P R S Solution Step 1 Find the measures of the minor and major arcs. ecause m RQS 5, mrs 5 and mrps Step 2 Find the areas of the small and large sectors. rea of small sector 5 m RS } pπr pπp 2 < rea of large sector 5 m RPS } 3608 pπr 2 5 pπp < The areas of the small and large sectors are about square units and square units, respectively. 240 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

50 Your Notes Example 3 Use the rea of a Sector Theorem Use the diagram to find the area of (. Solution rea of sector 5 m } p rea of ( m p rea of ( The area of ( is 5 rea of ( square meters. heckpoint omplete the following exercises. 1. Find the radius of the circle ft 2 2. Find the areas of the sectors formed by RQS. P R ft S Homework 3. Find the area of (G. F yd G H opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 241

51 Name ate LESSON 6.8 Practice Find the exact area of the circle. Then find the area of the circle to the nearest hundredth Find the indicated measure. 4. The area of a circle is 58 square inches. Find the radius. 5. The area of a circle is 37 square meters. Find the radius. 6. The area of a circle is 106 square centimeters. Find the diameter. 7. The area of a circle is 249 square feet. Find the diameter. Find the areas of the sectors formed by Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

52 Name ate LESSON 6.8 Practice continued Use the diagram to find the indicated measure. 11. Find the area of (S. 12. Find the area of (S. 13. Find the radius of (S. The area of (Z is square centimeters. The area of sector XZY is 28 square centimeters. Find the indicated measure. 14. Radius of (Z 15. ircumference of (Z 16. m XY 17. Length of XY 18. Perimeter of shaded region 19. Perimeter of unshaded region opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 243

53 Name ate LESSON 6.8 Practice continued Find the area of the shaded region Hockey face-off circle from a hockey rink is shown at the right. The diameter of the circle is 30 inches. Find the area of the face-off circle. 24. Pizza pizza is cut into 8 congruent pieces, as shown. The diameter of the pizza is 16 inches. Find the area of one piece of pizza. 25. lock wall clock has an area of square inches. Find the diameter of the clock. Then find the area of the sector formed when the time is 3:00, as shown. 244 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

54 6.9 Surface rea and Volume of Spheres Georgia Performance Standard(s) MM2G4a, MM2G4b Your Notes Goal p Find surface areas and volumes of spheres. VOULRY Sphere enter of a sphere Radius of a sphere hord of a sphere iameter of a sphere Great circle Hemisphere THEOREM 6.22: SURFE RE OF SPHERE The surface area S of a sphere is S 5, where r is the radius of the sphere. r opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 245

55 Your Notes Example 1 Find the surface area of a sphere Find the surface area of the sphere. 7 cm Solution S 5 4πr 2 Formula for surface area of a sphere 5 4π( ) 2 Substitute for r. 5 π Simplify. < Use a calculator. The surface area of the sphere is about square centimeters. Example 2 Find the circumference of a sphere The circumference of a sphere is 12π feet. Find the surface area of the sphere. Solution 5 2πr Formula for circumference 5 2πr Substitute for. 5 r ivide each side by. S 5 4πr 2 Formula for surface area 5 4π( ) 2 Substitute for r. ø Use a calculator. The surface area of the sphere is about feet. square 246 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

56 Your Notes heckpoint omplete the following exercises. 1. The diameter of a sphere is 1 } Ï } π meter. Find the surface area of the sphere. 2. The circumference of a sphere is 2π inches. Find the surface area of the sphere. THEOREM 6.23: VOLUME OF SPHERE The volume V of a sphere is V 5, where r is the radius r of the sphere. Example 3 Find the volume of a sphere Find the volume of the sphere. 3 mm Solution V 5 4 } 3 πr 3 Formula for volume of a sphere 5 4 } 3 π( ) 3 Substitute for r. < Use a calculator. The volume of the sphere is about millimeters. cubic opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 247

57 Your Notes Example 4 etermine the effects of a change in radius sphere has a radius of 6 meters. The radius of a second sphere is 3 meters, which is one half the radius of the first sphere. How do the surface area and volume of the second sphere compare to the surface area and volume of the first sphere? Solution First sphere: Substitute for r and simplify. S 5 4πr 2 5 4π( ) 2 5 π m 2 V 5 } 4 3 πr 3 5 } 4 3 π( ) 3 5 π m 3 Second sphere: Substitute for r and simplify. S 5 4πr 2 5 4π( ) 2 5 π m 2 V 5 } 4 3 πr 3 5 } 4 3 π( ) 3 5 π m 3 The surface area of the second sphere is, or the surface area of the first sphere. The volume of the second sphere is, or the volume of the first sphere. heckpoint omplete the following exercises. 3. The radius of a sphere is 2.4 centimeters. Find the volume of the sphere. Round your answer to two decimal places. Homework 4. The radius of a sphere is 4 inches. Explain how the surface area and volume change if the radius is doubled. 248 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

58 Name ate LESSON 6.9 Practice 1. Name the center of the sphere. R 2. Name a chord of the sphere. P T S 7 m 3. Name a radius of the sphere. 4. Name a diameter of the sphere. 5. Find the circumference of the great circle that contains P and S. Write your answer in terms of π. Find the surface area of the sphere. Round your answer to two decimal places cm 19 ft 22 in m 15 yd 30 cm opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 249

59 Name ate LESSON 6.9 Practice continued In Exercises 12 15, use the diagram. The center of the sphere is and its circumference is 17π feet. 12. What is half of the sphere called? 13. Find the radius of the sphere. 14. Find the diameter of the sphere. 15. Find the surface area of half of the sphere. Find the radius of a sphere with the given surface area S. 16. S 5 324π cm S 5 4π ft S π m The circumference of a sphere is 338π meters. What is the surface area of the sphere? Round your answer to two decimal places. Find the volume of the sphere. Round your answer to two decimal places m 14 in. 6 ft 250 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

60 Name ate LESSON 6.9 Practice continued Find the volume of the sphere. Round your answer to two decimal places cm 21 yd 28 cm Find the radius of a sphere with the given volume V. 26. V π yd V 5 36π in V mm sphere is inscribed in a cube of volume 8 cubic meters. What are the surface area and volume of the sphere? Round your answers to two decimal places. In Exercises 30 32, use the following information. each all beach ball has a surface area of about square feet. 30. Find the radius of the beach ball. 31. Find the circumference of a great circle of the beach ball. Round your answer to two decimal places. S ft Find the volume of the beach ball. Round your answer to two decimal places. 33. Planets The mean radius of Earth is approximately Earth 6378 kilometers. The mean radius of Mars is approximately 3397 kilometers, or about } 1 2 the mean radius of Earth. How does the surface area of Mars compare to the surface area of Earth? Mars 6378 km 3397 km opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 251

61 Words to Review Give an example of the vocabulary word. ircle enter, radius, diameter hord Secant Tangent entral angle rc, minor arc, major arc Semicircle Measure of a minor arc Measure of a major arc 252 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.

62 ongruent circles ongruent arcs Inscribed angle Intercepted arc Inscribed polygon ircumscribed circle Segments of a chord Secant segment External segment ircumference opyright Mcougal Littell/Houghton Mifflin ompany. Georgia Notetaking Guide, Mathematics 2 253

63 rc length Sector of a circle Sphere, center, radius, chord, diameter Great circle Hemisphere 254 Georgia Notetaking Guide, Mathematics 2 opyright Mcougal Littell/Houghton Mifflin ompany.