Answers for Lesson 3-1, pp Exercises

Size: px
Start display at page:

Download "Answers for Lesson 3-1, pp Exercises"

Transcription

1 Answers for Lesson -, pp. Eercises * ) PQ * ) PS * ) PS * ) PS * ) SR * ) QR * ) QR * ) QR. nd with trnsversl ; lt. int. '. nd with trnsversl ; lt. int. '. nd with trnsversl ; sme-side int. '. nd with trnsversl ; corr. ' 5. nd : corr. ' 6. nd : sme-side int. ' nd : lt. int. ' nd : corr. ' 5 nd 6: corr. ' 5 nd 6: corr. ' 7. nd : corr. ' 8. lt. int. ' nd : sme-side int. ' 5 nd 6: lt. int. ' 9.. Sme-Side Int. Angles Thm.. Sme-Side Int. Angles Thm. 5. Congruent Supplements Thm. 0.. (Given). (Alt. Int. ' Thm.). c d (Given). (Corr. ' Post.) 5. (Trns. Prop.) * ) SQ * ) SQ * ) PQ * ) SR. m 75 ecuse corr. ' of lines re ; m 05 ecuse sme-side int. ' of lines re suppl.. m 0 ecuse corr. ' of lines re ; m 60 ecuse sme-side int. ' of lines re suppl.. m 00 ecuse sme-side int. ' of lines re suppl.; m 70 ecuse lt. int. ' of lines hve mesure. Geometry Chpter 5

2 Answers for Lesson -, pp. Eercises (cont.). 70; 70, ; 65, ; 00, m m m 6 m 8 m 9 m m m 5 5; m m m 5 m 7 m 0 m m 8 8. You must find the mesure of one.all ' tht re vert., corr., or lt. int. to tht will hve tht mesure. All other ' will e the suppl. of tht mesure. 9. two 0. four. two. four.. 76, y 7, v, w , y 5 6. Trns mens cross or over. A trnsversl cuts cross other lines. 7. Answers my vry. Smple: E illustrtes corr. ' ( nd, nd ) nd sme-side int. ' ( nd, nd ); I illustrtes lt. int. ' ( nd, nd ) nd sme-side int. ' ( nd, nd ). 8.. (Given). re suppl. (Sme Side Int. ' Thm.). re suppl. (Sme Side Int. ' Thm.). (Given) 5. re suppl. (Sust.) 6. ( Suppl. Thm.) Geometry Chpter 5

3 Answers for Lesson -, pp. Eercises (cont.) 9. Since, ecuse they re corr. '. Also nd re supplementry y the Add. Post. So y Sust., nd re supplementry sme-side int. '.. lt. int. '. He knew tht lt. int. ' of lines re.. The ' leled re corr. ' nd should e. If you solve 60 60, you get 0. This would e impossile since 60 nd 60 would equl 0.. Never; the two plnes do not intersect.. Sometimes; if they re. 5. Sometimes; they my e skew. 6. Sometimes; they my e. Geometry Chpter 5

4 Answers for Lesson -, pp. 7 9 Eercises * BE ) * CG ). ; Conv. of Corr. ' Post.. CA HR; Conv. of Corr. ' Post.. JO LM; if two lines nd trnsversl form sme-side int. ' tht re suppl., then the lines re.. PQ ST; Conv. of Alt. Int. ' Thm The corr. ' re, so the lines re y the Conv. of Corr. ' Post. 0. ; if two lines nd trnsversl form sme-side int. ' tht re suppl., then the lines re.. ; if two lines nd trnsversl form sme-side int. ' tht re suppl., then the lines re.. ; if two lines nd trnsversl form sme side et. ' tht re suppl., then the two lines re.. none. ; Conv. of Corr. ' Post. 5. none 6. ; Conv. of Alt. Int. ' Thm. 7., m; Conv. of Corr. ' Post. 8. ; if two lines nd trnsversl form lt. et. ' tht re congruent, then the two lines re. 9. ; Conv. of Corr. ' Post. 0. none., m; Conv. of Alt. Int. ' Thm. Geometry Chpter 5

5 Answers for Lesson -, pp. 7 9 Eercises (cont.)... c. d. e. Conv. of Corr. '. The corr. ' he drws re C 7. m m 80 (Given). m m 80 ( Add. Post.). m m m m (Sustitution). m m (Sut. Prop. of )., n (Conv. of Corres. ' Post.) 8. 0; m m ; m m ; m m 0..5; m m 0. The corr. ' re, nd the ors re y the Conv. of Corr. ' Post.. Answers my vry. Smple: 9; j k y Conv. of the Alt. Int. ' Thm.. Answers my vry. Smple: 9; j k y Conv. of the Alt. Int.' Thm. nd, m y Conv. of Sme-Side Int. ' Thm. 5. Answers my vry. Smple: ;, m y Conv. of the Alt. Int. ' Thm. nd j k y Conv. of Corr. ' Post. 6. Answers my vry. Smple: nd re suppl.; j k y the Conv. of Corr. ' Post. 7. Vert. ' Thm. nd Conv. of Corr. ' Post. Geometry Chpter 55

6 Answers for Lesson -, pp. 7 9 Eercises (cont.) 8.., n. Given. 8. Corres. ' Post.. 8. Given.. Trns. Prop. of 5. j k 5. Conv. of Corres. ' Post. 9.. j k. Given. m 9 m 80. Sme-Side Int. ' Thm.. m 8 m Given 0.. m 9 m. Trns. Prop. of m 8 m 9 5. m m 8 5. Sut. Prop. of 6., n 6. Conv. of Corres. ' Post. nd re supp. Given m Conv. of Corr. Post. m m 80 Supp. Thm. Add. Post. nd re supp. Def. of suppl.. PL NA nd PN LA y Conv. of Sme-Side Int. ' Thm.. PL NA y Conv. of Sme-Side Int. ' Thm. Geometry Chpter 56

7 Answers for Lesson -, pp. 7 9 Eercises (cont.). none. PN LA y Conv. of Sme-Side Int. ' Thm. 5.. Answers my vry. Smple: e c A O B d X Z. Given: with trnsversl e, c isects AOB, d isects AXZ. c. Prove: c d d. To prove tht c d, show tht. if AOB OXZ. AOB OXZ y the Corr. ' Post. e.. (Given). AOB AXZ (Corr. ' Post.). m AOB m AXZ (Def. of '). m AOB m m ; m AXZ m m ( Add. Post.) 5. c isects AOB; d isects AXZ. (Given) 6. m m ; m m (Def. of isector) 7. m m m m (Trns. Prop. of ) 8. m m m m (Sust.) 9. m m (Add. Prop.) 0. m m (Div. Prop.). c d (Conv. of Corr. ' Post.) Geometry Chpter 57

8 Answers for Lesson -, pp. Eercises... Slt D is perp. to slts B nd C. Eplntions my vry. Smple: Slt D is perp. to slt A. Slts A, B, nd C re prllel, so y Theorem -, slt D is lso perp. to B nd C.. is perp. to nd c is perp. to, so is prllel to c ecuse two lines perp. to the sme line re prllel. d is prllel to c nd is prllel to c, so y Thm. -9, is prllel to d... corr. ',c., (ny order) d. Converse of Corr.. Answers my vry. Smples re given.. The rungs re prllel to ech other ecuse they re ll perpendiculr to the sme side. 5. All of the rungs re perpendiculr to one side. The side is perp. to the top rung, nd ecuse ll of the rungs re prllel to ech other, the side is perp. to ll of the rungs. 6. The rungs re perpendiculr to oth sides. The rungs re perp. to one of two prllel lines, so they re perp. to oth lines. 7. The rungs re prllel to ech other ecuse they re ll perpendiculr to one side. The sides re prllel ecuse they re oth perpendiculr to one rung. 8. The sides re prllel ecuse they re oth perpendiculr to one rung. 9. All of the rungs re prllel. All of the rungs re prllel to one rung, so they re ll prllel to ech other. Geometry Chpter 58

9 Answers for Lesson -, pp. Eercises 0. The rungs re prllel ecuse they re ll perpendiculr to one side.. In the digrm, # mens the mrked is rt.. c mens tht the corres. formed y nd c is rt., so # c.. Thm. -0. Answers my vry. Smple: In the digrm, AB # BH nd AB # BD, ut BH 6 BD. They intersect.. d 5. # d 6. # d 7. # d 8. d 9. d 0. d. # d. Refleive: ; flse; ny line intersects itself. Symmetric: If, then ; true; nd re coplnr nd do not intersect. Trnsitive: In generl, if, nd c, then c; true; however, when, nd, it does not follow tht.. Refleive: # ; flse; # lines re two lines tht intersect to form right '. Symmetric: If #, then # ; true; nd intersect to form right '. Trnsitive: If #, nd # c, then # c; flse; in plne, two lines # to the sme line re. Geometry Chpter 59

10 Answers for Lesson -, pp Eercises ; y 0; z ; y right, sclene 8. cute, equingulr, equilterl 9. otuse, isosceles 0.. Not possile; right will lwys hve one longest side opp. the right , 6, 8. nd for 5 nd for 6 nd for 8 c. They re vert. '. Geometry Chpter 60

11 Answers for Lesson -, pp Eercises (cont.) m 9; m 88. 7, y. 6, 8. 7; 55, 5, 90; right. 7; 7, 65, 78; cute 5. 8, y 6, z 90; ABD: 6, 90, 5; right; BCD: 90, 5, 8; right; ABC: 7, 5, 5; cute 6. 67, 58, c 5, d, e 90; FGH: 58, 67, 55; cute; FEH: 5,, ; otuse; EFG: 67,, 90; right 7. 60; Yes, n equilterl is isosc. ecuse if three sides of re, then t lest two sides re. No, the third side of n isosc. does not need to e to the other two. 9. eight Right isosceles 0. A. 0 nd , 60, 80. cute Otuse sclene Acute isosceles Right sclene Geometry Chpter 6

12 Answers for Lesson -, pp Eercises (cont.). Check students work. Answers my vry. Smple: The two et. ' formed t verte A re vert. ' nd thus hve the sme mesure.. By the definition of right ngle, m C 90. By the Tringle Angle-Sum Theorem, m A m B m C 80. Sutrcting 90 from ech side gives m A m B 90, so A nd B re complementry y the definition of comp. ngles. 5. m m 80 y the Add. Postulte. m m m 80 y the -Sum Theorem. m m m m m y the Trns. Property of Equlity. m m m y the Sutr. Property of Equlity. 6. ; since the third is 68, the lrgest et. is Check students work , 6, 7 c. cute 9... There re no such tringles c. isosceles tringle.. Answers my vry. Smple: The mesure of the et. is to the sum of the mesures of the two remote int. '. Since these ' re, the ' formed y the isector of the et. re to ech of them. Therefore, the isector is to the included side of the remote int. ' y the Conv. of the Alt Int. ' Thm. Geometry Chpter 6

13 Answers for Lesson -5, pp. 6 6 Eercises. yes. No; it hs no sides.. No; it is not plne figure.. No; two sides intersect etween endpoints. 5. MWBFX; sides: MW, WB, BF, FX, XM; ': M, W, B, F, X 6. KCLP; sides: KC, CL, LP, PK; ': K, C, L, P 7. HEPTAGN; sides: HE, EP, PT, TA, AG, GN, NH; ': H, E, P, T, A, G, N 8. pentgon; conve 9. decgon; concve 0. pentgon; concve , , 60,., 9. 08; 7. 50; 0 0, 0. 60; ; , 5, C 7. octgon; m 5; m 5 90 Geometry Chpter 6

14 Answers for Lesson -5, pp. 6 6 Eercises (cont.) 8. If you solve 0, you get n 7.. This numer is not n integer ; ; 5. ; 0. 6; 0. 50;. 80 ; n 80. (n )80 c. 80n 80(n ) 60 d. Polygon Et. -Sum Thm. 7. y 0; z 70; qud. 8. w 7, 59, y 9, z ; 9. 6, 7, 08, ; qud Answers my vry. Smples re given Yes; the sum of the mesures of ' t the int. point is 60. The sum of the mesures of ll the > is 80n. 80n 60 (n ) Answers my vry. Smple: The figure is conve equilterl qud. The sum of its ' is 80 or octgon (n )80 n Geometry Chpter 6

15 Answers for Lesson -5, pp. 6 6 Eercises (cont.) 57.. (0, 6), (0, 7), (60, 7), (80, 75.5), (00, 76.), (0, 77), (0, 77.), (60, 77.75), (80, 78), (00, 78.). 8 Interior Angle Mesure of Regulr Polygons c. It is very close to 80. d. No, two sides cnnot e colliner [80(n )] n Interior Angle Mesure n n 60 n 5. As n gets lrger, the size of the ngles get closer to 80. The more sides it hs, the closer the polygon is to circle Answers my vry. Smples re given. 60 n Not possile; opp. sides would overlp Not possile; opp. nd dj. sides would overlp. Geometry Chpter 65

16 Answers for Lesson -6, pp Eercises. y. y. 7 5 y O O O. y 5. y 6. O O 5 y 0 O 0 7. y 8. y 9. O 8 0. y. y. y O. y. y y O 5 y O y 6 y O 6 y O O O Geometry Chpter 66

17 Answers for Lesson -6, pp Eercises (cont.) 5. y 6. y y O O y 7. y ( ) 8. y ( ) 9. y 5 ( ) 0. y 6 ( ). y ( 6). y ( 0) or y 8. Equtions my vry from the pt. chosen. Smples re given.. y 5 ( 0). y ( 6) 5. y 6 ( ) 6. y ( ) 7. y 0 ( ) 8. y 0 ( 8) 9.. y y.... y.. y y. y 5. O 5 O 6 O 6 y 6 6. y y O O Geometry Chpter 67

18 Answers for Lesson -6, pp Eercises (cont.) the cost per min c..95 d. the initil chrge for cll 9. No; line with no slope is verticl line. 0 slope is horizontl line. 0.. m 0; it is horizontl line.. y 0.. Undefined; it is verticl line.. 0. Answers my vry. Smples re given.. The eq. is in stndrd form; chnge to slope-intercept form, ecuse it is esy to grph the eq. from tht form.. The eq. is in slope-int. form; use slope-int. form, ecuse the eq. is lredy in tht form.. The eq. is in point-slope form; use point-slope form, ecuse the eq. is lredy in tht form. 5. y 6. O The slopes re the sme, nd the y-intercepts re different. 7. Check students work. O y The slopes re ll different, nd the y-intercepts re the sme. Geometry Chpter 68

19 Answers for Lesson -6, pp Eercises (cont.) 8. O y (6, ) y 50. y 5. y (, ) O y 0 y 0 O (0, 0) y 6 y 5 O (, 5) , 0.08; 0 ; it is possile only if the rmp zigzgs. 5. The y-intercepts re the sme, nd the lines hve the sme steepness. One line rises from left to right while the other flls from left to right. 5. Answers my vry. Smple: 5, y 6 ( 5), y 55. (, 0), (0, ); m 0 y 0 ( ), y or y y 0 ( 0) or y 5 0. y 5 ( ) or y 0 c. The s. vlue of the slopes is the sme, ut one slope is pos. nd the other is neg. One y-int. is t (0, 0) nd the other is t (0, 0). 57. Yes; the slope of AB the slope of BC. 58. No; the slope of DE the slope of EF. 59. Yes; the slope of GH the slope of HI. 5 5 Geometry Chpter 69

20 Answers for Lesson -6, pp Eercises (cont.) 60. Yes; the slope of JK the slope of KL. 6. y ( ); y 8 6. y 5 ( 5); y 5 6. y 6 ( ); y Geometry Chpter 70

21 Answers for Lesson -7, pp Eercises. Yes; oth slopes.. No; the slope of, nd the slope of,.. No; the slope of,, nd the slope of,.. Yes; oth slopes. 5. Yes; oth slopes Yes; the lines oth hve slope of ut different y-intercepts. 7. Yes; the lines oth hve slope of ut different y-intercepts. 8. Yes; the lines oth hve slope of ut different y-intercepts. 9. No; one slope 7 nd the other slope No; one slope nd the other slope.. Yes; the lines oth hve slope of 5 ut different y-intercepts.. y ( 0) or y. y 0 ( 6) or y ( 6). y ( ) 5. y ( 6) 6. Yes; the slope of,, nd the slope of, ;. 7. Yes; the slope of,, nd the slope of, ;. 8. No; the slope of,, nd the slope of, 5; Yes; the slope of,, nd the slope of, ;. Geometry Chpter 7

22 Answers for Lesson -7, pp Eercises (cont.) 0. Answers my vry. Smples re given. 0. y 6 ( 6). y ( ). y ( ). y. y 5. Yes; ( ). 6. Yes; one is verticl nd the other is horizontl. 7. No; Q. 8. A 7 7 R 9. slope of AB slope of CD ; AB CD slope of BC slope of AD ; BC AD 0. slope of AB slope of CD ; AB CD slope of BC slope of AD ; BC AD. slope of AB ; slope of CD ; AB slope of BC ; slope of AD ; BC. slope of AB slope of CD 0; AB CD slope of BC nd slope of AD ; BC. Answers my vry. Smple: y 5, y 5. No; two lines with the sme y-intercept re the sme line. 5. RS nd VU re horizontl with slope 0; RS VU; slope of RW slope of UT ; RW UT; slope of WV slope of ST ; WV ST 6. No; ecuse no pirs of slopes hve product of. 7. The lines will hve the sme slope CD 6 AD 6 AD 5 Geometry Chpter 7

23 Answers for Lesson -7, pp Eercises (cont.) 8. When lines re #, the product of their slopes is. So, two lines # to the sme line must hve the sme slope. 9.. y 0 ( 5). ecuse you re given point nd cn quickly find the slope 0.. #. neither. #. # 5. AC: d "(7 9) ( ) "0 BD: d "( ) (7 5) "0 AC BD 6. slope of AC 5; slope of BD 5; since 5 5, AC ' BD; midpoint AC (8, 6); midpoint BD (8, 6); since the midpoints re the sme, the digonls isect ech other. 7.. Answers my vry. Smple: y O c. The other possile loctions for S re (, ) nd (8, 7). 8. y 5 ( ) 6 R Q P S (6, ) 6 Geometry Chpter 7

24 Answers for Lesson -8, pp Eercises.. J A B J A,,.. J * AB ) B A AB B A B J 5 7. Constructions my vry. Smples using the following segments re shown:, AB, AB , A ' P * AB ) B Geometry Chpter 7

25 Answers for Lesson -8, pp Eercises (cont.) A P, ' B * AB ) P, ' R * RS ) S.. P S, ', '.. R < ' 5. Construct lt. int. ; then drw the line. 6.. P R R * RS ) * RS ) S P m A * RS ) S B C Geometry Chpter 75

26 Answers for Lesson -8, pp Eercises (cont.) 7. Constructions my vry. Smples re given c c m The sides re nd. c. Check students work... c c Geometry Chpter 76

27 Answers for Lesson -8, pp Eercises (cont.). c. c d. The sides of the smller re hlf the length of the sides of the lrger tht they re to. e. Check students work. 5. D 6.. Check students work. c. p m; in plne, two lines # to third line re Answers my vry. Smples re given D G D D G G The qud. is rectngle. The qud. is squre Geometry Chpter 77

28 Answers for Lesson -8, pp Eercises (cont.).... c c c c 5. Not possile; if c, then c or c. The smller sides would meet t the midpoint of the longer side, forming segment. 6. Not possile; the smller sides would meet t the midpoint of the longer side, forming segment. Geometry Chpter 78

GEOMETRICAL PROPERTIES OF ANGLES AND CIRCLES, ANGLES PROPERTIES OF TRIANGLES, QUADRILATERALS AND POLYGONS:

GEOMETRICAL PROPERTIES OF ANGLES AND CIRCLES, ANGLES PROPERTIES OF TRIANGLES, QUADRILATERALS AND POLYGONS: GEOMETRICL PROPERTIES OF NGLES ND CIRCLES, NGLES PROPERTIES OF TRINGLES, QUDRILTERLS ND POLYGONS: 1.1 TYPES OF NGLES: CUTE NGLE RIGHT NGLE OTUSE NGLE STRIGHT NGLE REFLEX NGLE 40 0 4 0 90 0 156 0 180 0

More information

Use the diagram to identify each angle pair as a linear pair, vertical angles, or neither.

Use the diagram to identify each angle pair as a linear pair, vertical angles, or neither. inl xm Review hpter 1 6 & hpter 9 Nme Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points

More information

Geometry AP Book 8, Part 2: Unit 3

Geometry AP Book 8, Part 2: Unit 3 Geometry ook 8, rt 2: Unit 3 IMRTNT NTE: For mny questions in this unit, there re multiple correct nswers, e.g. line segment cn e written s, RST is the sme s TSR, etc. Where pproprite, techers should e

More information

Unit 5 Review. For each problem (1-4) a and b are segment lengths; x and y are angle measures.

Unit 5 Review. For each problem (1-4) a and b are segment lengths; x and y are angle measures. For ech problem (1-4) nd b re segment lengths; x nd y re ngle mesures. 1. Figure is Prllelogrm 2. Figure is Squre 21 36 16 b 104 3 2 6 b = 21 b = 16 x = 104 y = 40 = 3 2 b = 6 x = 45 y = 90 3. Figure is

More information

2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B

2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B Review Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points in Plne M. [],, [],, [],, [],,

More information

Answers to Exercises. c 2 2ab b 2 2ab a 2 c 2 a 2 b 2

Answers to Exercises. c 2 2ab b 2 2ab a 2 c 2 a 2 b 2 Answers to Eercises CHAPTER 9 CHAPTER LESSON 9. CHAPTER 9 CHAPTER. c 9. cm. cm. b 5. cm. d 0 cm 5. s cm. c 8.5 cm 7. b cm 8.. cm 9. 0 cm 0. s.5 cm. r cm. 7 ft. 5 m.. cm 5.,, 5. 8 m 7. The re of the lrge

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

Name Class Date. Line AB is parallel to line CD. skew. ABDC } plane EFHG. In Exercises 4 7, use the diagram to name each of the following.

Name Class Date. Line AB is parallel to line CD. skew. ABDC } plane EFHG. In Exercises 4 7, use the diagram to name each of the following. Reteching Lines nd Angles Not ll lines nd plnes intersect. prllel plnes. prllel. } shows tht lines or plnes re prllel: < > < > A } ens Line A is prllel to line. skew. A } plne EFHG A plne FH } plne AEG

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

Lesson-5 ELLIPSE 2 1 = 0

Lesson-5 ELLIPSE 2 1 = 0 Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).

More information

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

More information

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS www.mthletics.com.u Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the

More information

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs

More information

P 1 (x 1, y 1 ) is given by,.

P 1 (x 1, y 1 ) is given by,. MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

More information

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)

More information

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the

More information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year 1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

More information

+ R 2 where R 1. MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark)

+ R 2 where R 1. MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark) 2. C h p t e r t G l n c e is the set of ll points in plne which re t constnt distnce from fixed point clled centre nd constnt distnce is known s rdius of circle. A tngent t ny point of circle is perpendiculr

More information

1. ~, rectangle, rhombus, square 2. parallelogram 3. trapezoid 4. ~, rhombus 5. kite 6. trapezoid, isosc. trapezoid 7. rhombus 8.

1. ~, rectangle, rhombus, square 2. parallelogram 3. trapezoid 4. ~, rhombus 5. kite 6. trapezoid, isosc. trapezoid 7. rhombus 8. Answers for Lesson 6-1, pp. 308 311 Exercises 1. ~, rectangle, rhomus, square. parallelogram 3. trapezoid 4. ~, rhomus 5. kite 6. trapezoid, isosc. trapezoid 7. rhomus 8. parallelogram 9. rhomus 10. rectangle

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q., 3., 31 8 p 98 q 8 53 17 79 3 50. r, s, 5.,. t t 85 s 100 r 30 100 7 31 7. s 8. m 9. m s 7 35 m c c 10. Find

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

UNCORRECTED. 9Geometry in the plane and proof

UNCORRECTED. 9Geometry in the plane and proof 9Geometry in the plne nd proof Ojectives To consider necessry nd sufficient conditions for two lines to e prllel. To determine the ngle sum of polygon. To define congruence of two figures. To determine

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,

More information

MATHEMATICS AND STATISTICS 1.6

MATHEMATICS AND STATISTICS 1.6 MTHMTIS N STTISTIS 1.6 pply geometri resoning in solving prolems ternlly ssessed 4 redits S 91031 inding unknown ngles When finding the size of unknown ngles in figure, t lest two steps of resoning will

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.

More information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting

More information

( β ) touches the x-axis if = 1

( β ) touches the x-axis if = 1 Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without

More information

4. Statements Reasons

4. Statements Reasons Chpter 9 Answers Prentie-Hll In. Alterntive Ativity 9-. Chek students work.. Opposite sides re prllel. 3. Opposite sides re ongruent. 4. Opposite ngles re ongruent. 5. Digonls iset eh other. 6. Students

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider

More information

JEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse

More information

Set 1 Paper 2. 1 Pearson Education Asia Limited 2017

Set 1 Paper 2. 1 Pearson Education Asia Limited 2017 . A. A. C. B. C 6. A 7. A 8. B 9. C. D. A. B. A. B. C 6. D 7. C 8. B 9. C. D. C. A. B. A. A 6. A 7. A 8. D 9. B. C. B. D. D. D. D 6. D 7. B 8. C 9. C. D. B. B. A. D. C Section A. A (68 ) [ ( ) n ( n 6n

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12. Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k

More information

Triangles The following examples explore aspects of triangles:

Triangles The following examples explore aspects of triangles: Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., 3., 31 82 p 98 q 28 53 17 79 23 50 4. r, s, 5., 6. t t s r 100 85 100 30 4 7 31 7. s 8. m 9. m s 76 35

More information

Similarity and Congruence

Similarity and Congruence Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles

More information

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

More information

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1 8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.

More information

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

1 (=0.5) I3 a 7 I4 a 15 I5 a (=0.5) c 4 N 10 1 (=0.5) N 6 A 52 S 2

1 (=0.5) I3 a 7 I4 a 15 I5 a (=0.5) c 4 N 10 1 (=0.5) N 6 A 52 S 2 Answers: (98-84 HKMO Finl Events) Creted by Mr. Frncis Hung Lst updted: December 05 Individul Events SI 900 I 0 I (=0.5) I 7 I4 5 I5 80 b 7 b b 5 b 6 b 8 b 4 c c 4 c 0 x (=0.5) c 4 N 0 d 9 d 5 d 5 y d

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles. 3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)

More information

Chapter 12. Lesson Geometry Worked-Out Solution Key. Prerequisite Skills (p. 790) A 5 } perimeter Guided Practice (pp.

Chapter 12. Lesson Geometry Worked-Out Solution Key. Prerequisite Skills (p. 790) A 5 } perimeter Guided Practice (pp. Chpter 1 Prerequisite Skills (p. 790) 1. The re of regulr polygon is given by the formul A 5 1 p P, where is the pothem nd P is the perimeter.. Two polygons re similr if their corresponding ngles re congruent

More information

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r= 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

More information

MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK. 10th Grade Unit 5 MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

More information

Mathematics Extension 1

Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

More information

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A. 378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

Prove Lines are Parallel. p Use angle relationships to prove that lines are parallel. VOCABULARY. Paragraph proof

Prove Lines are Parallel. p Use angle relationships to prove that lines are parallel. VOCABULARY. Paragraph proof Your Notes Prove Lines re Prllel Gol p Use ngle reltionships to prove tht lines re prllel VOCABULARY Prgrph proof POSTULATE 6 CORRESPONDING ANGLES If two lines re cut trnsversl so the corresponding ngles

More information

SOLUTION OF TRIANGLES

SOLUTION OF TRIANGLES SOLUTION OF TIANGLES DPP by VK Sir B.TEH., IIT DELHI VK lsses, -9-40, Indr Vihr, Kot. Mob. No. 989060 . If cos A + cosb + cos = then the sides of the AB re in A.P. G.P H.P. none. If in tringle sin A :

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

Parallel Projection Theorem (Midpoint Connector Theorem):

Parallel Projection Theorem (Midpoint Connector Theorem): rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length one-hlf the third side. onversely, If line isects

More information

MDPT Practice Test 1 (Math Analysis)

MDPT Practice Test 1 (Math Analysis) MDPT Prctice Test (Mth Anlysis). Wht is the rdin mesure of n ngle whose degree mesure is 7? ) 5 π π 5 c) π 5 d) 5 5. In the figure to the right, AB is the dimeter of the circle with center O. If the length

More information

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims

More information

Answers: ( HKMO Heat Events) Created by: Mr. Francis Hung Last updated: 15 December 2017

Answers: ( HKMO Heat Events) Created by: Mr. Francis Hung Last updated: 15 December 2017 Answers: (0- HKMO Het Events) reted y: Mr. Frncis Hung Lst updted: 5 Decemer 07 - Individul - Group Individul Events 6 80 0 4 5 5 0 6 4 7 8 5 9 9 0 9 609 4 808 5 0 6 6 7 6 8 0 9 67 0 0 I Simplify 94 0.

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,

More information

Essential Question What conjectures can you make about perpendicular lines?

Essential Question What conjectures can you make about perpendicular lines? 3. roofs with erpendiculr Lines Essentil Question Wht conjectures cn ou ke out perpendiculr lines? Writing onjectures Work with prtner. Fold piece of pper in hlf twice. Lel points on the two creses, s

More information

First Semester Review Calculus BC

First Semester Review Calculus BC First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

More information

CHAPTER 6 Introduction to Vectors

CHAPTER 6 Introduction to Vectors CHAPTER 6 Introduction to Vectors Review of Prerequisite Skills, p. 73 "3 ".. e. "3. "3 d. f.. Find BC using the Pthgoren theorem, AC AB BC. BC AC AB 6 64 BC 8 Net, use the rtio tn A opposite tn A BC djcent.

More information

ICSE Board Class IX Mathematics Paper 4 Solution

ICSE Board Class IX Mathematics Paper 4 Solution ICSE Bord Clss IX Mthemtics Pper Solution SECTION A (0 Mrks) Q.. () Consider x y 6 5 5 x y 6 5 5 0 6 0 6 x y 6 50 8 5 6 7 6 x y 6 7 6 x y 6 x 7,y (b) Dimensions of the brick: Length (l) = 0 cm, bredth

More information

If C = 60 and = P, find the value of P. c 2 = a 2 + b 2 2abcos 60 = a 2 + b 2 ab a 2 + b 2 = c 2 + ab. c a

If C = 60 and = P, find the value of P. c 2 = a 2 + b 2 2abcos 60 = a 2 + b 2 ab a 2 + b 2 = c 2 + ab. c a Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Individul Events I P I P I P I P 5 7 0 0 S S S S Group Events G G G G 80 00 0 c 8 c c c d d 6 d 5 d 85 Individul Event I.,

More information

PARABOLA EXERCISE 3(B)

PARABOLA EXERCISE 3(B) PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution

More information

Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: A ratio is the quotient of two numbers. For example, 2 3 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

More information

Individual Events I3 a 10 I4. d 90 angle 57 d Group Events. d 220 Probability

Individual Events I3 a 10 I4. d 90 angle 57 d Group Events. d 220 Probability Answers: (98-8 HKMO Finl Events) Creted by: Mr. Frncis Hung Lst updted: 8 Jnury 08 I 800 I Individul Events I 0 I4 no. of routes 6 I5 + + b b 0 b b c *8 missing c 0 c c See the remrk 600 d d 90 ngle 57

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

Chapter 6. Worked-Out Solutions. Chapter 6 Maintaining Mathematical Proficiency (p. 299)

Chapter 6. Worked-Out Solutions. Chapter 6 Maintaining Mathematical Proficiency (p. 299) hapter 6 hapter 6 Maintaining Mathematical Proficiency (p. 99) 1. Slope perpendicular to y = 1 x 5 is. y = x + b 1 = + b 1 = 9 + b 10 = b n equation of the line is y = x + 10.. Slope perpendicular to y

More information

Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38

Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38 Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score

More information

On the diagram below the displacement is represented by the directed line segment OA.

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

HKDSE2018 Mathematics (Compulsory Part) Paper 2 Solution 1. B 4 (2 ) = (2 ) 2. D. α + β. x x. α β 3. C. h h k k ( 4 ) 6( 2 )

HKDSE2018 Mathematics (Compulsory Part) Paper 2 Solution 1. B 4 (2 ) = (2 ) 2. D. α + β. x x. α β 3. C. h h k k ( 4 ) 6( 2 ) HKDSE08 Mthemtics (Compulsory Prt) Pper Solution. B n+ 8 n+ 4 ( ) ( ) n+ n+ 6n+ 6n+ (6n+ ) (6n+ ). D α β x x α x β ( x) α x β β x α x + β x β ( α + β ) x β β x α + β. C 6 4 h h k k ( 4 ) 6( ) h k h + k

More information

MTH 4-16a Trigonometry

MTH 4-16a Trigonometry MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

More information

Minnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017

Minnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017 Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll.

More information

126 Holt McDougal Geometry

126 Holt McDougal Geometry test prep 51. m Q = m S 3x + 5 = 5x - 5 30 = x x = 15 5. J 53. 6.4 P = + + + = + + + = (5 + 8.) = 6.4 challenge and extend 54. Let given pts. be (0, 5), (4, 0), (8, 5), and possible 4th pts. be X, Y, Z.

More information

S56 (5.3) Vectors.notebook January 29, 2016

S56 (5.3) Vectors.notebook January 29, 2016 Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution

More information

UNIT 31 Angles and Symmetry: Data Sheets

UNIT 31 Angles and Symmetry: Data Sheets UNIT 31 Angles nd Symmetry Dt Sheets Dt Sheets 31.1 Line nd Rottionl Symmetry 31.2 Angle Properties 31.3 Angles in Tringles 31.4 Angles nd Prllel Lines: Results 31.5 Angles nd Prllel Lines: Exmple 31.6

More information

Stage 11 Prompt Sheet

Stage 11 Prompt Sheet Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions

More information

Set 6 Paper 2. Set 6 Paper 2. 1 Pearson Education Asia Limited 2017

Set 6 Paper 2. Set 6 Paper 2. 1 Pearson Education Asia Limited 2017 Set 6 Pper Set 6 Pper. C. C. A. D. B 6. D 7. D 8. A 9. D 0. A. B. B. A. B. B 6. B 7. D 8. C 9. D 0. D. A. A. B. B. C 6. C 7. A 8. B 9. A 0. A. C. D. B. B. B 6. A 7. D 8. A 9. C 0. C. C. D. C. C. D Section

More information

Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Valley High School Calculus Summertime Fun Packet Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

More information

Coimisiún na Scrúduithe Stáit State Examinations Commission

Coimisiún na Scrúduithe Stáit State Examinations Commission M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions

More information

JEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS

JEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODE-D. Let, b nd c be three non-zero vectors such tht no two of them re colliner nd, b c b c. If is the ngle

More information

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically. Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it

More information

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.

Precalculus Due Tuesday/Wednesday, Sept. 12/13th  Mr. Zawolo with questions. Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

Topics Covered AP Calculus AB

Topics Covered AP Calculus AB Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

More information

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1. Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

More information

π = tanc 1 + tan x ...(i)

π = tanc 1 + tan x ...(i) Solutions to RSPL/ π. Let, I log ( tn ) d Using f () d f ( ) d π π I log( tnc d m log( cot ) d...(ii) On dding (i) nd (ii), we get +,. Given f() + ), For continuit t lim " lim f () " ( ) \ Continuous t.

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( ) UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

More information

Summary Information and Formulae MTH109 College Algebra

Summary Information and Formulae MTH109 College Algebra Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,

More information