Pythagoras Theorem PYTHAGORAS THEOREM.
|
|
- Luke Warren
- 5 years ago
- Views:
Transcription
1 Pthgors Theorem PYTHAGORAS THEOREM
2
3 How oes it work? Solutions Pthgors Theorem Pge 3 questions Right-ngle tringles D E x z Hotenuse is sie: F Hotenuse is sie: DF Q k j l Hotenuse is sie: k P R Hotenuse is sie: PQ Nme the hotenuse for eh of these l rwn tringles. M L N Hotenuse is sie: Hotenuse is sie: MN Mthletis Pssort 3P Lerning I SERIES TOPIC
4 How oes it work? Solutions Pthgors Theorem Pge 5 questions Squres n right-ngle tringles 3 units Are 5 units # 5 units 5 units 3 Are units # units 44 units 5 units Are 3 3 units # 3 units 69 units Are + Are 5 units + 44 units 69 units units Are 3 Are 6 units # 6 units 36 units 3 Are 8 units # 8 units 64 units 0 units 6 units Are 3 0 units # 0 units 00 units 8 units Are + Are 36 units + 64 units 00 units Are 3 I SERIES TOPIC Mthletis Pssort 3P Lerning
5 How oes it work? Solutions Pthgors Theorem Pge questions Pthgors Theorem for right-ngle tringles Right-ngle Not right-ngle Right-ngle Not right-ngle Right-ngle Not right-ngle Right-ngle Not right-ngle e f Right-ngle Not right-ngle Right-ngle Not right-ngle Mthletis Pssort 3P Lerning I SERIES TOPIC 3
6 How oes it work? Solutions Pthgors Theorem Pge 8 questions Pthgors Theorem for right-ngle tringles A J K I 4.5 H J B M K 9 A 5 0 C ! 56 L 48 N ! H 5 G The right-ngle tringles re: ΔAJK, ΔHIJ, ΔGHK 4 I SERIES TOPIC Mthletis Pssort 3P Lerning
7 How oes it work? Solutions Pthgors Theorem Pge 8 questions Pthgors Theorem for right-ngle tringles 3 Ern n wesome ssort with this one! Nme ll the right-ngle tringles in this imge n mrk where the right-ngles re with the orret smol. R 65 S The right-ngle tringles re: 5 T 380 ΔPUV ΔQRU P 5 Q 5 6 U ΔRSU ΔSTU V Pge 9 questions Pthgors Theorem for right-ngle tringles Assuming the sle of the ge is the sme s the originl rint, the mesurements shoul e s follows: NOTE: if not the sme sle, the sme reltionshi etween our mesurements shoul work mm mm mm mm mm 56 4 mm mm 5 36 mm mm mm 96 mm mm mm 8 40 mm mm mm mm mm Mthletis Pssort 3P Lerning I SERIES TOPIC 5
8 Where oes it work? Solutions Pthgors Theorem Pge questions Clulting the length of the hotenuse g g g g 89 0 g Pge questions Clulting the length of the hotenuse 3 h h h h 3. in ext squre root form n + 35 n n n 389 in ext squre root form 4 ( 0 units) + ( 9 units) ( 59. units) + ( 34. units) 00units + 8units units units 8units units 8 units units units units units to eiml les. 68. units to eiml les 6 I SERIES TOPIC Mthletis Pssort 3P Lerning
9 Where oes it work? Solutions Pthgors Theorem Pge 3 questions Clulting the length of the hotenuse 5 Stge Stge Stge m 55m m m to eiml les The totl length of the 3 stge flight th m m Pge 5 questions Clulting the length of short sie j 0-56 (8. units) -(8 units) j units -34 units j units j units j 4 9. units Mthletis Pssort 3P Lerning I SERIES TOPIC
10 Where oes it work? Solutions Pthgors Theorem Pge 6 questions Clulting the length of short sie 3 - w w w in ext squre root form w 65 in ext squre root form x x x x x to eiml oint x to eiml oint 8 I SERIES TOPIC Mthletis Pssort 3P Lerning
11 Where oes it work? Solutions Pthgors Theorem Pge questions Comintion of hotenuse n short sie lultions The seil nme given right-ngle tringle whih is extl one hlf of n equilterl tringle: H E M I E Q tringle I M 4. E Q e E 6 6. h H 30 g 60 6 Mthletis Pssort 3P Lerning I SERIES TOPIC 9
12 Where oes it work? Solutions Pthgors Theorem Pge 9 questions Alitions of Pthgors Theorem x (3 m) -( m) x 69m -44 m 3 m x m x 5m x 5 m x 5m 4 m ut (4 m) + (34 m) 34 m ut ut ut 64 m + 56 m 90 m ut 90 m ut m ut. 54 to nerest whole m 3. km Strt (i) (. km) + (3.9 km).89km + 5.km 3.9 km 8.km 8. km km Finish. 4.5km to eiml oints (ii) To voi the swm, Mil wlke 3.9 km +. km 5.6 km Mil wlke further 5.6 km km..35 km 0 I SERIES TOPIC Mthletis Pssort 3P Lerning
13 Where oes it work? Solutions Pthgors Theorem Pge 0 questions Alitions of Pthgors Theorem 4.6 m m (i) se se ( m) -(.6 m) 89m -6.6 m se 8.4 m se se se 84. m 6.8 m (ii) Are (se # height) ' Are Are Are ( 6. 8m 6. m) 43.68m '. 84 m # ' 5 C AB (8 m) + (3.3 m) AB 34m m B 3.3 m 54.4 m AB AB AB m m 8.3m A 8 m Digrm not rwn to sle. BC (54.4 m) + (3.3 m) BC m m BC 90.5m BC m BC 54.5m Distne roun wll 9m Distne AC AB + BC 8.3m+ 54.5m.8 m Shortest th Mthletis Pssort 3P Lerning I SERIES TOPIC
14 Where oes it work? Solutions Pthgors Theorem Pge 0 questions Alitions of Pthgors Theorem 6 m (i) ( m- 3 m) + (0 m) (35 m) + (0 m) 0 m 5 m m 565 m 3 m 565 m 5 m (ii) Perimeter of the trezium m+ 0 m+ 3m+ 5 m 554 m Pge questions Alitions of Pthgors Theorem Y WY YZ -WZ WX XZ -WZ XY WY -WX WY 3969 WX 900 X 65 WY 63 WX W 6 I SERIES TOPIC Mthletis Pssort 3P Lerning
15 Where oes it work? Solutions Pthgors Theorem Pge questions Alitions of Pthgors Theorem 8 Clulte the length of the le suort BD on the rne iture elow if CD 9.5 m, AB 6 m n BC 8.5 m 8.5 m C B 9.5 m 6 m D A AC BC -AB AD AC -DC BD AD + AB m AC BD 00 AC. 5 m BD 0 m Mthletis Pssort 3P Lerning I SERIES TOPIC 3
16 Wht else n ou o? Solutions Pthgors Theorem Pge 3 questions Pthgoren tris ", 6, 0, 0, 4, 6 sst! Note tht the re written in orer of size. ", ", 35, 3, " 9, 40, 4, 40 Show whether these sets of ositive integers form Pthgoren tri or not. ", 4, 5, " 4, 48, 50, ", 34, 36, + 4 5? ? ? ? ? ? 300! 96 Yes No Yes No Yes No " 5, 36, 39, e " 6, 60, 63, f ", 30, 3, ? ? ? 3856! 3969? ? ? 044! 96 Yes No Yes No Yes No 4 I SERIES TOPIC Mthletis Pssort 3P Lerning
17 Wht else n ou o? Solutions Pthgors Theorem Pge 5 questions Euli s formul for Pthgoren tris q - q q + q Tri # # { 3, 4, 5 } 3-8 # 3 # { 6, 8, 0 } 5 - # 5 # { 0,, 9 } # # { 3, 84, 85 } - 3 # # { 66,, 30 } - 8 # # {, 56, 65 } (i) Fin Pthgoren tri in whih n - q is equl to 33 - q 33 - q q q 6 q q 4 q # # q Pthgoren tri is { 33, 56, 65 } (ii) Fin Pthgoren tri in whih q 5 n + q is equl to 6 - q q # 6 # q 6-5 Pthgoren tri is {, 60, 6 } Mthletis Pssort 3P Lerning I SERIES TOPIC 5
18 Wht else n ou o? Solutions Pthgors Theorem Pge 6 questions Pthgoren tris 3 Fin grou of three integers tht inlues the numer 4 n forms Pthgoren tri. { - q, q, + q } q 4 # # q 4 # q n q ( q) + q q - 48 Pthgoren tri is: { 4, 48, 50 } 4 { - q, q, + q } hint: Pthgoren tris n e me using ositive integers onl. Forml exlntion: smll integer other smll integer lrgest integer From hint, Pthgoren tris re me using ositive integers. ie. ositive whole numers onl. One of the smller integers is foun using - q If the vlue of ws smller thn the vlue of q, then the nswer woul e negtive. So this oul not e use euse onl ositive whole numers re llowe. Showing using hosen vlues n q : When n q, - q - 3 (this is ositive integer n is llowe) If we sw these roun, so n q - q (this is negtive integer n is not llowe) This will lws hen if the vlue of is smller thn the vlue of q when using Euli s formul. Negtive numers re not llowe euse eh integer reresents the length of the sie of right-ngle tringle. So sie length of -3 oes not mke sense. 6 I SERIES TOPIC Mthletis Pssort 3P Lerning
19 Wht else n ou o? Solutions Pthgors Theorem Pge 8 questions Wheel of Theoorus n so on Mthletis Pssort 3P Lerning I SERIES TOPIC
20 Jigsw Wht else Puzzle n ou o? Solutions Pthgors Theorem Pge 3 questions Squres n right-ngle tringles: Jigsw Puzzle I SERIES TOPIC Mthletis Pssort 3P Lerning
21 Pthgors Theorem Notes Mthletis Pssort 3P Lerning I SERIES TOPIC 9
22 Pthgors Theorem Notes 0 I SERIES TOPIC Mthletis Pssort 3P Lerning
23
24 PYTHAGORAS THEOREM APPLICATIONS OF PYTHAGORAS THEOREM.../.../0... APPLICATIONS OF TRIANGLES RIGHT-ANGLED TRIANGLES RIGHT-ANGLED.../.../0... EUCLID S FORMULA FOR PYTHAGOREAN TRAIDS * ", - q, q, + q.../.../0... SQUARES AND RIGHT- ANGLED TRIANGLES SQUARES AND RIGHT- ANGLED TRIANGLES.../.../0... * AWESOME *.../.../0... * AWESOME *
Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.
Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45 www.mthletis.om Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived
More informationArea and Perimeter. Area and Perimeter. Solutions. Curriculum Ready.
Are n Perimeter Are n Perimeter Solutions Curriulum Rey www.mthletis.om How oes it work? Solutions Are n Perimeter Pge questions Are using unit squres Are = whole squres Are = 6 whole squres = units =
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationPerimeter and Area. Mathletics Instant Workbooks. Copyright
Perimeter nd Are Student Book - Series J- L B Mthletis Instnt Workooks Copyright Student Book - Series J Contents Topis Topi - Plne shpes Topi 2 - Perimeter of regulr shpes Topi 3 - Perimeter of irregulr
More informationSection 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 informationNon 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
More informationSurds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.
More informationMCH T 111 Handout Triangle Review Page 1 of 3
Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:
More informationGM1 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
More information1 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
More informationTrigonometry 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.
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More information3 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 informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give this
More informationSection 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
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More information9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up
9.5 Strt Thinking In Lesson 9.4, we discussed the tngent rtio which involves the two legs of right tringle. In this lesson, we will discuss the sine nd cosine rtios, which re trigonometric rtios for cute
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationAlgebra & Functions (Maths ) opposite side
Instructor: Dr. R.A.G. Seel Trigonometr Algebr & Functions (Mths 0 0) 0th Prctice Assignment hpotenuse hpotenuse side opposite side sin = opposite hpotenuse tn = opposite. Find sin, cos nd tn in 9 sin
More information1 Find the volume of each solid, correct to one decimal place where necessary. 12 cm 14 m. 25 mm. p c 5 ffiffiffi
1 Find the volume of eh solid, orret to one deiml le where neessry. 8 m 6 m m 14 m 65 m 2 2 m d 7.6 mm 2 m 4 m 4 m 7 m 25 mm Stge 5.3 See Chter 1 See Chter 7 See Chter 9 See Chter 9 See Chter 13 2 Simlify
More informationDirected Numbers. Directed Numbers. Curriculum Ready.
Direte Numers Curriulum Rey www.mthletis.om Numers ome in ll sizes n forms. They n e positive or negtive, whole numers, frtions or eimls n rtionl or irrtionl. Before you strt, investigte these terms n
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationH SERIES. Algebra Basics. Algebra Basics. Solutions. Curriculum Ready.
Alger Bsis H SERIES Alger Bsis Curriulum Rey www.mthletis.om Copyright 009 P Lerning. All rights reserve. First eition printe 009 in Austrli. A tlogue reor for this ook is ville from P Lerning Lt. ISBN
More informationSIMPLE NONLINEAR GRAPHS
S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle
More informationPYTHAGORAS 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
More information青藜苑教育 The digrm shows the position of ferry siling between Folkestone nd lis. The ferry is t X. X 4km The pos
青藜苑教育 www.thetopedu.com 010-6895997 1301951457 Revision Topic 9: Pythgors Theorem Pythgors Theorem Pythgors Theorem llows you to work out the length of sides in right-ngled tringle. c The side opposite
More informationCARLETON UNIVERSITY. 1.0 Problems and Most Solutions, Sect B, 2005
RLETON UNIVERSIT eprtment of Eletronis ELE 2607 Swithing iruits erury 28, 05; 0 pm.0 Prolems n Most Solutions, Set, 2005 Jn. 2, #8 n #0; Simplify, Prove Prolem. #8 Simplify + + + Reue to four letters (literls).
More informationUNIT 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 informationIndividual Contest. English Version. Time limit: 90 minutes. Instructions:
Elementry Mthemtics Interntionl Contest Instructions: Individul Contest Time limit: 90 minutes Do not turn to the first pge until you re told to do so. Write down your nme, your contestnt numer nd your
More informationIndividual Group. Individual Events I1 If 4 a = 25 b 1 1. = 10, find the value of.
Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 00-0 33 3 7 7 5 Individul 6 7 7 3.5 75 9 9 0 36 00-0 Grou 60 36 3 0 5 6 7 7 0 9 3 0 Individul Events I If = 5 = 0, find the vlue
More information1Measurement and geometry. Pythagoras theorem
1Mesurement n geometry Pythgors theorem rhiteture, engineering, surveying n mny other fiels. ws known to numer of nient ivilistions, inluing the Egytins who use the rule to onstrut their yrmis s fr k s
More information9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1
9.1 Dy 1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 1, 2017 Geometry 9.1 The Pythgoren Theorem 1 9.1 Dy 2 Wrm Up Use the Pythgoren
More informationGeometry 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 informationMinnesota 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= x x 2 = 25 2
9.1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 7, 2016 Geometry 9.1 The Pythgoren Theorem 1 Geometry 9.1 The Pythgoren Theorem 9.1
More informationComparing 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
More informationPythagoras 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
More information3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h
PAKTURK 8 th Ntionl Interschool Mths Olmpid,.9. Q: Evlute 6.9. 6 6 6... 8 8...... Q: Evlute bc bc. b. c bc.9.9b.9.9bc Q: Find the vlue of h in the eqution h 7 9 7.. bc. bc bc. b. c bc bc bc bc......9 h
More informationWhat else can you do?
Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationUNIT-4. File downloaded from For the things of this world cannot be made known without a knowledge of mathematics.
File downloded from http://jsuniltutoril.weely.com UNIT- For the things of this world cnnot e mde known without knowledge of mthemtics.. Solve y fctoriztion. - + ( ) = 0 Ans: + ( - ) = 0. [( + ) + ( -
More informationT 1 T 2 T 3 T 4 They may be illustrated by triangular patterns of numbers (hence their name) as shown:
TOPIC 3: VISUAL EXPLANATIONS (PROOFS) (Pge references to Proof re to Bndll, P R et l, Proof in Mthemtics, KMEP, 2002). 3. The tringulr numbers form the sequence, 3, 6, 0,, 2,... T T 2 T 3 T 4 The m be
More informationMATH 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 informationSolutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
More informationTrigonometry. 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
More informationSpecial Numbers, Factors and Multiples
Specil s, nd Student Book - Series H- + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book - Series H Contents Topics Topic - Odd, even, prime nd composite numers Topic - Divisiility tests
More information2 Calculate the size of each angle marked by a letter in these triangles.
Cmridge Essentils Mthemtics Support 8 GM1.1 GM1.1 1 Clculte the size of ech ngle mrked y letter. c 2 Clculte the size of ech ngle mrked y letter in these tringles. c d 3 Clculte the size of ech ngle mrked
More informationPlotting Ordered Pairs Using Integers
SAMPLE Plotting Ordered Pirs Using Integers Ple two elsti nds on geoord to form oordinte xes shown on the right to help you solve these prolems.. Wht letter of the lphet does eh set of pirs nme?. (, )
More informationArea and Perimeter. Area and Perimeter. Curriculum Ready.
Are nd Perimeter Curriculum Redy www.mthletics.com This ooklet shows how to clculte the re nd perimeter of common plne shpes. Footll fields use rectngles, circles, qudrnts nd minor segments with specific
More information332:221 Principles of Electrical Engineering I Fall Hourly Exam 2 November 6, 2006
2:221 Principles of Electricl Engineering I Fll 2006 Nme of the student nd ID numer: Hourly Exm 2 Novemer 6, 2006 This is closed-ook closed-notes exm. Do ll your work on these sheets. If more spce is required,
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationICSE 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 informationS56 (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 informationSimilar 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
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationLesson 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 informationm 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:
More informationIdentifying and Classifying 2-D Shapes
Ientifying n Clssifying -D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte
More informationProportions: 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 informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationLesson 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 informationLESSON 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.
More informationPart I: Study the theorem statement.
Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for
More informationSimplifying Algebra. Simplifying Algebra. Solutions. Curriculum Ready.
Siplifing Alger Siplifing Alger Curriulu Re www.thletis.o How oes it work? Siplifing Alger Pge questions Multipling n iviing 6 6 6 0 0 e f p r q p r q 6 6pqr p r q p q r g g g g g g g g g h n n n n n
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationR(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 informationSTRAND I: Geometry and Trigonometry. UNIT 32 Angles, Circles and Tangents: Student Text Contents. Section Compass Bearings
ME Jmi: STR I UIT 32 ngles, irles n Tngents: Stuent Tet ontents STR I: Geometry n Trigonometry Unit 32 ngles, irles n Tngents Stuent Tet ontents Setion 32.1 ompss erings 32.2 ngles n irles 1 32.3 ngles
More information10 If 3, a, b, c, 23 are in A.S., then a + b + c = 15 Find the perimeter of the sector in the figure. A. 1:3. A. 2.25cm B. 3cm
HK MTHS Pper II P. If f ( x ) = 0 x, then f ( y ) = 6 0 y 0 + y 0 y 0 8 y 0 y If s = ind the gretest vlue of x + y if ( x, y ) is point lying in the region O (including the boundry). n [ + (n )d ], then
More informationTrigonometry. Trigonometry. Solutions. Curriculum Ready ACMMG: 223, 224, 245.
Trgonometry Trgonometry Solutons Currulum Redy CMMG:, 4, 4 www.mthlets.om Trgonometry Solutons Bss Pge questons. Identfy f the followng trngles re rght ngled or not. Trngles,, d, e re rght ngled ndted
More informationLecture 6: Coding theory
Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those
More informationPolynomials. Polynomials. Curriculum Ready ACMNA:
Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression
More information50 AMC Lectures Problem Book 2 (36) Substitution Method
0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem
More informationMath Lesson 4-5 The Law of Cosines
Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude
More informationAnswers 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 informationLesson 55 - Inverse of Matrices & Determinants
// () Review Lesson - nverse of Mtries & Determinnts Mth Honors - Sntowski - t this stge of stuying mtries, we know how to, subtrt n multiply mtries i.e. if Then evlute: () + B (b) - () B () B (e) B n
More informationIntroduction to Algebra - Part 2
Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite
More informationK 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c:
Qudrti Equtions The Null Ftor Lw Let's sy there re two numers nd. If # = then = or = (or oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More informationAlg. 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 informationPrecalculus 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 informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationUSA 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 informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More informationH SERIES. Area and Perimeter. Curriculum Ready ACMMG: 109, 159, 196,
Are n Perimeter Curriulum Rey ACMMG: 0, 5, 6, 6 www.mthletis.om Copyright 00 3P Lerning. All rights reserve. First eition printe 00 in Austrli. A tlogue reor for this ook is ville from 3P Lerning Lt. ISBN
More informationMomentum and Energy Review
Momentum n Energy Review Nme: Dte: 1. A 0.0600-kilogrm ll trveling t 60.0 meters per seon hits onrete wll. Wht spee must 0.0100-kilogrm ullet hve in orer to hit the wll with the sme mgnitue of momentum
More informationMid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours
Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into
More information1. 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 informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationMEP 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 informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
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
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 informationHS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem
HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More information