Area. Ⅱ Rectangles. Ⅲ Parallelograms A. Ⅳ Triangles. ABCD=a 2 The area of a square of side a is a 2
|
|
|
- Sherman Hamilton
- 6 years ago
- Views:
Transcription
1 Ⅰ Sques e Letue: iu ng Mtemtis dution oundtion Pesident Wen-Hsien SUN Ⅱ Retngles = Te e of sque of side is Ⅲ Pllelogms = Te e of etngle of sides nd is = Te e of pllelogm is te podut of te lengt of one side nd te oesponding eigt Ⅳ Tingles () Rigt-ngled tingle () ity tingle = =
2 Ⅴ Tpezis e= ( + ) = m Te e of tpezium is te podut of its mid-line nd its eigt. Te mid-line of tpezium is pllel to te se lines nd lf s long s tei sum. Ⅵ Kites Te e of kite is lf te podut of te lengts of te digonls. () Romus () Kites m m x Ⅶ iles e= e= + = x ( ) + x = Te e of ile is te podut of π nd te sque of te dius d O e= π = π d 4 Ⅷ Seto of ile O θ s θ π e= 360 Te e of te seto depends on te entl ngle. Te lengt of te ounding te seto/ iumfeene of te ile = s
3 Lemm. If two tingles ve te sme se nd te sme ltitude, ten tey ve te sme e. Speil se: te medin of tingle uts te tingle into two tingles of equl e Lemm. If two tingles ve te sme ltitude, ten te tio of tei e is equl to te tio of tei ses Lemm 3. If two tingles ve te sme ses, ten te tio of tei e is equl to te tio of tei ltitude In tingle, extend te lengt of side y times to point, extend te lengt of side y 3 times to point, extend te lengt of side y 4 times to point. Wt is te tio of e nd? Teoem of Pytgos ² +²=²
4 Poof Poof Poof 3 Poof 4 Poof 5
5 Poof 6 Poved y 0t pesident of US Jmes. Gfield (876 ) = + = + = ( + ) Poof 7 G = + = + = + Ⅸ Rtio Sques Te tio of te e of two sques =Te sque of te lengt of its sides =²/² =(/) ² Tingles k k If te side lengts of tingle e enlged k times, ten its e is enlged k times. Retngles k If two etngles ve one side lengt in ommon, ten te tio of tei es equls te tio of tei seond sides: / =/
6 In te figue, te e of te tee etngles e 4, 30, 40. Wt is te e of te fout etngle? ? Tee sques wit te sme side lengt e put in lge sque. Te numes sown in te figue e te e we n see. Wt is te totl e of te sded pts? 30 4 line pllel to tingle s side uts te two ote sides in te sme tio. = = = Speil se: =, =, = Ⅹ Simil tingles Two tingles e simil if () te tion etween tee oesponding sides e equl ; () te tee oesponding ngles e equl ; (3) te two oesponding ngles e equl ; (4) te tio etween two sides of one is equl to te tio etween te oesponding sides of te ote nd te enlosed ngles e equl. Te ltitude on te ypotenuse divides igt-ngled tingle into two tingles tt e ot simil to te oiginl tingle nd ene to e ote. Gvity vey medin is divided in te tio : y te ente of gvity, te longe pt djoining te vetex. =,=,=, G=G,G=G, G=G G G
7 If two simil tingles ve lengts of oesponding sides in te tio of k, ten tei es e in te tio of k ''/ =''/ =''/ = k, ~ ''' ''/ = ''/=k '''/ = k² Some fts of tio d si Popety: = d =, 0, 0 d ± n d ± n oesponding ddition: = =, 0, 0 oesponding ddition nd suttion: d f n = = = =, 0, 0, e 0,, m 0 e m ± d + f + + v =, + + e + + m e + + m XI lge lgei sums e multiplied y multiplying evey tem of one sum into evey tem of te ote nd dding tese poduts. ' ' ' ' (+)=+ d d d (+d) (+) = +d++d y x xy x y xy (x+y)² = x²+xy+y² x y x y x x²-y² =(x+y)(x-y) x xy (x-y) y y y (x-y)² =x²-xy+y² y x x-y y x-y
8 =( ) 4 +( )( 3-4 ) +( + )( - 3 ) ommon Side Teoem Let P nd Q ve ommon side, nd ve lines nd PQ meet t P PM point M. Ten = Q QM P P P P M Q G Q M M Q Let te medins nd of inteset t point G, ten G=G : G/ G = / = G/ G = / = G = G = G = /3 / G = / G = 3 Q M Teoem of ev Let point P e ny point not olline wit ny two veties of tingle, nd let te lines P, P nd P inteset te lines, nd t points, nd, espetively. Ten = : P /= P/ P,/= P/ P, /= P/ P, /././ = P/ P. P/ P. P/ P =
9 ommon ngle teoem If two tingles ve equl (o supplementy) of oesponding ngle, ten tei es e in te tio of te podut of two sides tt fom tis ngle. () qul ngles =' / ''' =( / '') ( ''/ ''') =(/'').( /'') ' () Supplementy ngles +'=80 / ''' =( / '') ( ''/ ''') =(/'').(/'') (') ' ' In, =, so = = / =(.)/(.)= / = ' (') ngle iseto teoem Te ngle iseto of tingle divides te opposite side into two segments wose lengts e in te sme tio s te sides of te ngle: /=/ / =./.=/ / =/ /=/ Let e tingle, nd let points, nd e on te sides,, espetively, su tt =, = nd =. Te lines nd inteset t point P, te lines nd inteset t point R, Te lines nd inteset t point Q. ind te tio of es of PQR nd.
10 Q P R Q/ Q=/=/ Q/ Q=/= = Q+ Q+ Q =(++/) Q Q =/7 Similly, P= R=/7 PQR = - P- Q- R =/7 If =, =, nd =, ten te tio of es of PQR nd is (-) /(++)(++)(++) Speil se (ev s Teoem):,, e onuent, o PQR=0 - =0 = / / /= ue Te ue s 6 fes, nd e fe is sque. Te segments wee two fes inteset e edges. ue s edges nd ll edges ve te sme lengt. Te points wee tee edges inteset e veties. ue s 8 veties. In te igt digm, Sufe e of te ue =6 =6 Volume of te ue = = 3 Lengt of segment = Lengt of segment = 3 Retngul uoid Te etngul uoid lso s six fes. fe is etngul o sque. etngul uoid lso s edges nd 8 veties. It is lso pism, wit volume = (se e) (eigt). In te igt digm, Sufe e of te uoid = ( + + ) Volume of te uoid = Lengt of segment = + Lengt of segment = + +
11 xmple: Te figue to te igt is wooden etngul uoid. n nt is t point, nd it wnts to wl to point long te sufe of te uoid. Wt is te sotest pt it n tke? (Te dimensions of te uoid e lengt 5m, widt 4m, eigt 3m.) Solution: Te nt wnts to wl fom point to point. Te uoid is solid, so te nt must wl on te sufe nd not inside te uoid. Tee e infinitely mny pts on te uoid s sufe fom to, ut we n lssify tem into only tee types: (I) pt toug fe, pssing toug te edge nd eing toug fe. Rotte te fe out 90, tus mking te fes nd opln. Sine te sotest pt etween two points is te line joining tem, te sotest pt fom to is te segment. y te Pytgoen Teoem, = = 9 3 ( + ) + = (5 + 4) = 8+ 9 = m (II) pt toug fe, edge, nd fe to point. fte simil ottion, we get y te Pytgoen Teoem = ( + ) + = (3 + 5) + 4 = = = m (III) pt toug fe, edge, nd fe to point. Rotte nd pply te Pytgoen Teoem similly: = = ( + ) = 5 + (3 + 4) + = = m Tee e infinitely mny pts fom to. Te tee types ove e ve fixed minimum nd te tid type s te sotest pt. nswe: Te sotest pt is fo te nt to wl toug fe, edge, nd fe to. Te sotest pt s lengt ppoximtely 8.60m. Polem: n nt wls on te sufe of etngul uoid. Te distne etween two points on te sufe is te lengt of te sotest pt te nt n tke etween te two points. om te point of view of te nt, e te two points futest fom e 5m 3m 4m
12 ote lwys two digonlly opposite veties of te uoid (tese two veties e symmeti wit espet to te ente of te uoid)? Solution: Unde tis definition of distne, te nswe is negtive. Suppose points nd e opposite veties of uoid. Let point e point on te 4 4 fe tt ontins, su tt te distne fom to te two edges of lengt 4 tt ontin e e. Tee e two wys to unfold te uoid. Let s see if te distne etween nd is te longest distne on te uoid. In te left digm, = + 4 = 60, = + 3 = 30 ;in te igt digm, = = 8, = = 30. y definition, te distne etween two points on te uoid is te distne te nt must wl to get fom one point to te ote. So te distne fom to is 8, ut te distne fom to is 30. lely, is fte fom tn is. ylinde () iul ylinde Te solid of evolution of etngle out one of its sides is iul ylinde. Volume of ylinde =te e of te se eigt =π Sufe e of ylinde=te e of te top se + te e of te ottom se + te e of te side-fes =π + π + π = π ( + ) () Rigt pism Tke two onguent polygons in pllel plnes nd onnet te oesponding points su tt e line is pependiul to te se plne. Te esulting solid is igt pism. Volume of igt pism =se e eigt= Sufe e of igt pism =Top se e+lowe se e+ltel sufe e =+se peimete
13 () Olique pism Tke two onguent polygons in pllel plnes nd onnet te oesponding points su tt te lines e not pependiul to te se plne. Te esulting solid is n olique pism. Te distne etween te two plnes is te eigt, wile te distne etween two oesponding points is te slnt eigt. Volume of n olique pism =se e eigt Sufe e of n olique pism =Top se e + Lowe se e + Ltel sufe e one () Rigt one Te solid of evolution of igt tingle out one of its legs is igt one. In te igt digm, is te xis, is te geneto (o te slnt eigt), is te one s pex. Volume of one= se e eigt 3 = 3 π Sufe e of one=se e + ltel e=π + π g g g π lttened out, te ltel e is iul seto wit e π g = π g () Rigt pymid If te segment onneting te ente of te pymid s se nd te pex is pependiul to te se, te pymid is lled igt pymid. s in te digm, is te eigt of te pymid, is te side-edge, nd te ltitude of tingle, ee, is te slnt eigt. Volume of igt pymid = se e eigt 3 Sufe e of igt pymid =se e + ltel sufe e Ltel sufe e of igt pymid is peimete slnt eigt
14 Regul Tetedon (Regul tingul pymid) Te solid omposed of fou equiltel tingle fes is te egul tetedon. s in te pitue, if =, we ve: Slnt eigt M= 3, Te sufe e is Heigt O= 3 4 = 3, O = 3 Te volume of te egul tetedon is Spee 3 = 3 = = Te solid of evolution of ile out its dimete is spee. 4 3 Te e of te spee is 3 π. Te sufe e of te spee is 4π. Questions:. In pentgon, te digonl inteset te digonls nd t points nd G espetively. If :=5:4, G:G=:, :G:G =::3 ind te tio of es of nd. (983 Russi Mt Olympid ). In qudiltel, te tio of es of, nd is 3:4:,points M nd N on segments nd espetively su tt M:= N:, nd points, M nd N e olline. Pove tt points M nd N e midpoint of nd espetively. (983 in Mt Olympid) 3. In exgon, te digonls nd e points M nd N divide in tio M:=N:=. If points, M nd N e olline, find. (98 IMO) 4. ind te e of te sded potions. 3 O M π [(4 -π ) 4] (4 -π ) 4
GEOMETRY Properties of lines
www.sscexmtuto.com GEOMETRY Popeties of lines Intesecting Lines nd ngles If two lines intesect t point, ten opposite ngles e clled veticl ngles nd tey ve te sme mesue. Pependicul Lines n ngle tt mesues
Properties and Formulas
Popeties nd Fomuls Cpte 1 Ode of Opetions 1. Pefom ny opetion(s) inside gouping symols. 2. Simplify powes. 3. Multiply nd divide in ode fom left to igt. 4. Add nd sutt in ode fom left to igt. Identity
The Area of a Triangle
The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest
Reference. Reference. Properties of Equality. Properties of Segment and Angle Congruence. Other Properties. Triangle Inequalities
Refeene opeties opeties of qulity ddition opety of qulity If =, ten + = +. Multiplition opety of qulity If =, ten =, 0. Reflexive opety of qulity = Tnsitive opety of qulity If = nd =, ten =. Suttion opety
Baltimore County ARML Team Formula Sheet, v2.1 (08 Apr 2008) By Raymond Cheong. Difference of squares Difference of cubes Sum of cubes.
Bltimoe Couty ARML Tem Fomul Seet, v. (08 Ap 008) By Rymo Ceog POLYNOMIALS Ftoig Diffeee of sques Diffeee of ues Sum of ues Ay itege O iteges ( )( ) 3 3 ( )( ) 3 3 ( )( ) ( )(... ) ( )(... ) Biomil expsio
10.3 The Quadratic Formula
. Te Qudti Fomul We mentioned in te lst setion tt ompleting te sque n e used to solve ny qudti eqution. So we n use it to solve 0. We poeed s follows 0 0 Te lst line of tis we ll te qudti fomul. Te Qudti
Section 2.1 Special Right Triangles
Se..1 Speil Rigt Tringles 49 Te --90 Tringle Setion.1 Speil Rigt Tringles Te --90 tringle (or just 0-60-90) is so nme euse of its ngle mesures. Te lengts of te sies, toug, ve very speifi pttern to tem
Pythagorean Theorem and Trigonometry
Ptgoren Teorem nd Trigonometr Te Ptgoren Teorem is nient, well-known, nd importnt. It s lrge numer of different proofs, inluding one disovered merin President Jmes. Grfield. Te we site ttp://www.ut-te-knot.org/ptgors/inde.stml
Shape and measurement
PTR 10 Spe nd mesuement PTR ONTNTS 10 Pytgos teoem 10 Pytgos teoem in tee dimensions 10 Peimete nd e 10 Totl sufce e (TS) 10 Volume 10F pcity 10G Simil figues 10 Simil tingles 10I Symmety IGIT O doc-9608
D Properties and Measurement
APPENDIX D. Review of Alge, Geomet, nd Tigonomet A D Popetie nd Meuement D. Review of Alge, Geomet, nd Tigonomet Alge Popetie of Logitm Geomet Plne Anltic Geomet Solid Anltic Geomet Tigonomet Li of Function
Appendix D: Formulas, Properties and Measurements
Appendi D: Fomul, Popetie nd Meuement Review of Alge, Geomet, nd Tigonomet Unit of Meuement D. REVIEW OF ALGEBRA, GEOMETRY, AND TRIGONOMETRY Alge Popetie of Logitm Geomet Plne Anltic Geomet Solid Anltic
Mathematical Reflections, Issue 5, INEQUALITIES ON RATIOS OF RADII OF TANGENT CIRCLES. Y.N. Aliyev
themtil efletions, Issue 5, 015 INEQULITIES ON TIOS OF DII OF TNGENT ILES YN liev stt Some inequlities involving tios of dii of intenll tngent iles whih inteset the given line in fied points e studied
Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R Pen Towe Rod No Conttos Ae Bistupu Jmshedpu 8 Tel (67)89 www.penlsses.om IIT JEE themtis Ppe II PART III ATHEATICS SECTION I (Totl ks : ) (Single Coet Answe Type) This setion ontins 8 multiple hoie questions.
Week 8. Topic 2 Properties of Logarithms
Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e
A 2 ab bc ca. Surface areas of basic solids Cube of side a. Sphere of radius r. Cuboid. Torus, with a circular cross section of radius r
Sufce e f ic lid Cue f ide R See f diu 6 Cuid c c Elliticl cectin c Cylinde, wit diu nd eigt Tu, wit cicul c ectin f diu R R Futum, ( tuncted ymid) f e eimete, t e eimete nd lnt eigt. nd e te eective e
April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
Chapter Seven Notes N P U1C7
Chpte Seven Notes N P UC7 Nme Peiod Setion 7.: Angles nd Thei Mesue In fling, hitetue, nd multitude of othe fields, ngles e used. An ngle is two diffeent s tht hve the sme initil (o stting) point. The
Trigonometry. cosθ. sinθ tanθ. Mathletics Instant Workbooks. Copyright
Student Book - Series K- sinθ tnθ osθ Mtletis Instnt Workooks Copyrigt Student Book - Series K Contents Topis Topi - Nming te sides of rigt-ngled tringle Topi 2 - Te trigonometri rtios Topi 3 - Using lultor
m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
MAT 1275: Introduction to Mathematical Analysis
1 MT 1275: Intrdutin t Mtemtil nlysis Dr Rzenlyum Slving Olique Tringles Lw f Sines Olique tringles tringles tt re nt neessry rigt tringles We re ging t slve tem It mens t find its si elements sides nd
CHAPTER 7 Applications of Integration
CHAPTER 7 Applitions of Integtion Setion 7. Ae of Region Between Two Cuves.......... Setion 7. Volume: The Disk Method................. Setion 7. Volume: The Shell Method................ Setion 7. A Length
EXERCISE - 01 CHECK YOUR GRASP
SLUTIN F TRINGLE EXERISE - 0 HEK YUR GRSP 4 4R sin 4R sin 4R sin sin sin sin 4R (sin sin sin ) sin sin 6R os sin R sin sin sin R 8R 4R 5 p p p 6 p p p (s ) ( + + s) os tn os 8 + + s s pplying hlf ngle
1.3 Using Formulas to Solve Problems
Section 1.3 Uing Fomul to Solve Polem 73 1.3 Uing Fomul to Solve Polem OBJECTIVES 1 Solve fo Vile in Fomul 2 Ue Fomul to Solve Polem Peping fo Fomul Befoe getting tted, tke ti edine quiz. If you get polem
PLEASE DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO THEN ENSURE THAT YOU HAVE THE CORRECT EXAM PAPER
OLLSCOIL NA ÉIREANN, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA OLLSCOILE, CORCAIGH UNIVERSITY COLLEGE, CORK 4/5 Autumn Suppement 5 MS Integ Ccuus nd Diffeenti Equtions Pof. P.J. Rippon
INTRODUCTION AND MATHEMATICAL CONCEPTS
Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips
1. Viscosities: μ = ρν. 2. Newton s viscosity law: 3. Infinitesimal surface force df. 4. Moment about the point o, dm
3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess Motition Gien elocit field o ppoimted elocit field, we wnt to be ble to estimte
Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as
Dte: 6.1 Lw of Sines Syllus Ojetie: 3.5 Te student will sole pplition prolems inoling tringles (Lw of Sines). Deriing te Lw of Sines: Consider te two tringles. C C In te ute tringle, sin In te otuse tringle,
HYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3..
HYPERBOLA AIEEE Sllus. Stndrd eqution nd definitions. Conjugte Hperol. Prmetric eqution of te Hperol. Position of point P(, ) wit respect to Hperol 5. Line nd Hperol 6. Eqution of te Tngent Totl No. of
A Cornucopia of Pythagorean triangles
A onucopi of Pytgoen tingles onstntine Zelto Deptment of temtics 0 ckey Hll 9 Univesity Plce Univesity of Pittsbug Pittsbug PA 60 USA Also: onstntine Zelto PO Bo 80 Pittsbug PA 0 USA e-mil ddesses: ) onstntine_zelto@yoocom
Non Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit
PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS
PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides
Electric Field F E. q Q R Q. ˆ 4 r r - - Electric field intensity depends on the medium! origin
1 1 Electic Field + + q F Q R oigin E 0 0 F E ˆ E 4 4 R q Q R Q - - Electic field intensity depends on the medium! Electic Flux Density We intoduce new vecto field D independent of medium. D E So, electic
STATICS. CENTROIDS OF MASSES, AREAS, LENGTHS, AND VOLUMES The following formulas are for discrete masses, areas, lengths, and volumes: r c
STTS FORE foe is veto qutit. t is defied we its () mgitude, () oit of litio, d () dietio e kow. Te veto fom of foe is F F i F j RESULTNT (TWO DMENSONS) Te esultt, F, of foes wit omoets F,i d F,i s te mgitude
INTRODUCTION AND MATHEMATICAL CONCEPTS
INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,
SSC [PRE+MAINS] Mock Test 131 [Answer with Solution]
![SSC [PRE+MAINS] Mock Test 131 [Answer with Solution]](/thumbs/93/112515487.jpg "SSC [PRE+MAINS] Mock Test 131 [Answer with Solution]")
SS [PRE+MINS] Mock Test [nswe with Solution]. () Put 0 in the given epession we get, LHS 0 0. () Given. () Putting nd b in b + bc + c 0 we get, + c 0 c /, b, c / o,, b, c. () bc b c c b 0. b b b b nd hee,
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
Chapter 1 Functions and Graphs
Capte Functions and Gaps Section.... 6 7. 6 8 8 6. 6 6 8 8.... 6.. 6. n n n n n n n 6 n 6 n n 7. 8 7 7..8..8 8.. 8. a b ± ± 6 c ± 6 ± 8 8 o 8 6. 8y 8y 7 8y y 8y y 8 o y y. 7 7 o 7 7 Capte : Functions and
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
π,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4
MATHEMATICS IV MARKS. If + + 6 + c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce
Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:
. Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo
Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:
Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,
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
4.3 The Sine Law and the Cosine Law
4.3 Te Sine Lw nd te osine Lw Te ee Tower is te tllest prt of nd s rliment uildings. ronze mst, wi flies te ndin flg, stnds on top of te ee Tower. From point 25 m from te foot of te tower, te ngle of elevtion
Invert and multiply. Fractions express a ratio of two quantities. For example, the fraction
Appendi E: Mnipuling Fions Te ules fo mnipuling fions involve lgei epessions e el e sme s e ules fo mnipuling fions involve numes Te fundmenl ules fo omining nd mnipuling fions e lised elow Te uses of
Inspiration and formalism
Inspirtion n formlism Answers Skills hek P(, ) Q(, ) PQ + ( ) PQ A(, ) (, ) grient ( ) + Eerise A opposite sies of regulr hegon re equl n prllel A ED i FC n ED ii AD, DA, E, E n FC No, sies of pentgon
Similar Right Triangles
Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems
MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER
MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER I One M Queston Fnd the unt veto n the deton of Let ˆ ˆ 9 Let & If Ae the vetos & equl? But vetos e not equl sne the oespondng omponents e dstnt e detons
CELESTIAL MECHANICS. Advisor: Dr. Steve Surace Assistant: Margaret Senese
CELESTIAL MECHANICS Ei Ce, Tyle Enst, Minqi Jing, Steve Kuei, Dniel Levine, Piy Mte, Jeemy Silve, Antony Svs, Stefn Tinte, Dmity Vgne, Stepnie Wng ABSTACT Adviso: D. Steve Sue Assistnt: Mget Senese Te
m A 1 1 A ! and AC 6
REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:
ALGEBRA. ( ) is a point on the line ( ) + ( ) = + ( ) + + ) + ( Distance Formula The distance d between two points x, y
ALGEBRA Popeties of Asoute Vue Fo e umes : 0, 0 + + Tige Iequity Popeties of Itege Epoets Ris Assume tt m e positive iteges, tt e oegtive, tt eomitos e ozeo. See Appeies B D fo gps fute isussio. + ( )
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.
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.](/thumbs/78/78412113.jpg "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
10 Statistical Distributions Solutions
Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques
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)
Trigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides
Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the
FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3
![FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3](/thumbs/78/77696958.jpg "FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3")
FORMULAE Algeba 1. (a + b) = a + b + ab = (a b) + 4ab. (a b) = a + b ab = (a + b) 4ab 3. a b = (a b) (a + b) 4. a + b = (a + b) ab = (a b) + ab 5. (a + b) + (a b) = (a + b ) 6. (a + b) (a b) = 4ab 7. (a
1 Review: Volumes of Solids (Stewart )
Lecture : Some Bic Appliction of Te Integrl (Stewrt 6.,6.,.,.) ul Krin eview: Volume of Solid (Stewrt 6.-6.) ecll: we d provided two metod for determining te volume of olid of revolution. Te rt w by dic
PROPERTIES OF TRIANGLES
PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties.
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
Physical Security Countermeasures. This entire sheet. I m going to put a heptadecagon into game.
Phsicl Secuit Countemesues This entie sheet Telmo, AHI I m going to put heptdecgon into gme. Cssie Hung Mechnicl lockpicking is mechnicked geometic constuctions with compss nd stightedge. Ech lock will
SSC Mains Mock Test 226 [Answer with Solution]
![SSC Mains Mock Test 226 [Answer with Solution]](/thumbs/88/115543390.jpg "SSC Mains Mock Test 226 [Answer with Solution]")
SS Mins Mock Test [nswe with Solution]. () Requied weight +.. () The sum of ges of the two olde plyes 0 + yes vege ge incesed months So, totl ge incesed months Sum of the ges of two new plyes yes + months
sec set D (sp 2014): PaPer 1
sec set D (sp 4): PPe Question (5 mks) Question () (i) w + i w ( ) + ( ) + 4 w Im( z) tnθ tn α, wee α is te elted ngle in te fist qudnt. p α tn 6 o p p θ p p + + p + w cos np isin np in genel pol fom.
Chapter 5 Worked Solutions to the Problems
Mtemtis for Queenslnd, Yer Mtemtis, Grpis lultor ppro 00. Kiddy olger, Rex oggs, Rond Frger, Jon elwrd pter 5 Worked Solutions to te Problems Hints. Strt by writing formul for te re of tringle. Note tt
GM1 Consolidation Worksheet
Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up
Ellipses. The second type of conic is called an ellipse.
Ellipses The seond type of oni is lled n ellipse. Definition of Ellipse An ellipse is the set of ll points (, y) in plne, the sum of whose distnes from two distint fied points (foi) is onstnt. (, y) d
Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
Mathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
No. Diagram Given Condition Conclusion Abbreviation a and b are adjacent angles on a straight a b 180. a, b and c are angles at a point
Pge 46 REVITION USE IN EUTIVE GEOMETR. Properties of Plne Geometry No. igrm Given ondition onlusion revition nd re djent 1 ngles on stright 180 dj. s on st. line line 2, nd re ngles t point 360 s t pt.
1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
Mathematics 10 Page 1 of 5 Properties of Triangle s and Quadrilaterals. Isosceles Triangle. - 2 sides and 2 corresponding.
Mthemtis 10 Pge 1 of 5 Properties of s Pthgoren Theorem 2 2 2 used to find the length of sides of right tringle Tpe of s nd Some s Theorems ngles s Slene Isoseles Equilterl ute - ll ngles re less thn 90
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
Trigonometry and Constructive Geometry
Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties
Physics Courseware Electromagnetism
Pysics Cousewae lectomagnetism lectic field Poblem.- a) Find te electic field at point P poduced by te wie sown in te figue. Conside tat te wie as a unifom linea cage distibution of λ.5µ C / m b) Find
This immediately suggests an inverse-square law for a "piece" of current along the line.
Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line
Use of Trigonometric Functions
Unit 03 Use of Trigonometric Functions 1. Introduction Lerning Ojectives of tis UNIT 1. Lern ow te trigonometric functions re relted to te rtios of sides of rigt ngle tringle. 2. Be le to determine te
Static Surface Forces. Forces on Plane Areas: Horizontal surfaces. Forces on Plane Areas. Hydrostatic Forces on Plane Surfaces
Hdrostti ores on Plne Surfes Stti Surfe ores ores on lne res ores on urved surfes Buont fore Stbilit of floting nd submerged bodies ores on Plne res Two tes of roblems Horizontl surfes (ressure is ) onstnt
Optimization. x = 22 corresponds to local maximum by second derivative test
Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible
MATHEMATICS PAPER & SOLUTION
MATHEMATICS PAPER & SOLUTION Code: SS--Mtemtis Time : Hours M.M. 8 GENERAL INSTRUCTIONS TO THE EXAMINEES:. Cndidte must write first is / er Roll No. on te question pper ompulsorily.. All te questions re
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
Exercise sheet 6: Solutions
Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd
Similar Right Triangles
9.3 EX EENIL KNOWLEGE N KILL G.8. G.8. imilar igt riangles Essential Question How are altitudes and geometric means of rigt triangles related? Writing a onjecture Work wit a partner. a. Use dnamic geometr
STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0
STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed
Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013
Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo
THREE DIMENSIONAL GEOMETRY
MD THREE DIMENSIONAL GEOMETRY CA CB C Coordintes of point in spe There re infinite numer of points in spe We wnt to identif eh nd ever point of spe with the help of three mutull perpendiulr oordintes es
A Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
Course Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week.
Couse Updtes http://www.phys.hwii.edu/~vne/phys7-sp1/physics7.html Remindes: 1) Assignment #8 vilble ) Chpte 8 this week Lectue 3 iot-svt s Lw (Continued) θ d θ P R R θ R d θ d Mgnetic Fields fom long
General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface
Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept
Physics 217 Practice Final Exam: Solutions
Physis 17 Ptie Finl Em: Solutions Fll This ws the Physis 17 finl em in Fll 199 Twenty-thee students took the em The vege soe ws 11 out of 15 (731%), nd the stndd devition 9 The high nd low soes wee 145
Comparing the Pre-image and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity
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
VECTOR ALGEBRA. Syllabus :
MV VECTOR ALGEBRA Syllus : Vetors nd Slrs, ddition of vetors, omponent of vetor, omponents of vetor in two dimensions nd three dimensionl spe, slr nd vetor produts, slr nd vetor triple produt. Einstein
Tutorial on Strehl ratio, wavefront power series expansion, Zernike polynomials expansion in small aberrated optical systems By Sheng Yuan
Tutoial on Stel atio, wavefont powe seies expansion, Zenike polynomials expansion in small abeated optical systems By Seng Yuan. Stel Ratio Te wave abeation function, (x,y, is defined as te distance, in
Chapter 6 Area and Volume
Capte 6 Aea and Volume Execise 6. Q. (i) Aea of paallelogam ( ax)( x) Aea of ectangle ax ( x + ax)( x) x x ( + a) a x a Faction x ( + a) + a (ii) Aea of paallelogam Aea of ectangle 5 ax (( + 5 x)( x ax)
LESSON 11: TRIANGLE FORMULAE
. THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.
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
1. QUESTION BANK ( ) Class - XII Subject - MATHEMATICS (ONE MARK QUESTIONS)
. QUESTION BANK (-) Clss - XII Sujet - MATHEMATICS (ONE MARK QUESTIONS) os sin. If A = find < < sin os when A A' = I.. If B is skew smmeti mti wite whethe the mti (ABA') is smmeti o skew smmeti.. If A
Chapter 2. Numerical Integration also called quadrature. 2.2 Trapezoidal Rule. 2.1 A basic principle Extending the Trapezoidal Rule DRAWINGS
S Cpter Numericl Integrtion lso clled qudrture Te gol of numericl integrtion is to pproximte numericlly. f(x)dx Tis is useful for difficult integrls like sin(x) ; sin(x ); x + x 4 Or worse still for multiple-dimensionl
Calculating Tank Wetted Area Saving time, increasing accuracy
Clulting Tnk Wetted Are ving time, inresing ur B n Jones, P.., P.E. C lulting wetted re in rtillfilled orizontl or vertil lindril or ellitil tnk n e omlited, deending on fluid eigt nd te se of te eds (ends)