Similarity and Congruence

Size: px
Start display at page:

Download "Similarity and Congruence"

Transcription

1 Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244

2

3 SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles re equl. Similr figures hve the sme shpe, ut not necessrily the sme size. In this ook, it is shown how similr nd congruent shpes cn e useful in solving prolems. Try to nswer these questions now, efore working through the chpter. I used to think: The symol for congruent is /. Wht do you think it mens to sy T / TF? If squre with side length 4cm hs een enlrged y scle fctor of 2, then wht is the side length of the lrge squre? If two tringles re the sme except for one ngle, re they congruent? nswer these questions, fter working through the chpter. ut now I think: The symol for congruent is /. Wht do you think it mens to sy T / TF? If squre with side length 4cm hs een enlrged y scle fctor of 2, then wht is the side length of the lrge squre? If two tringles re the sme except for one ngle, re they congruent? Wht do I know now tht I didn t know efore? J 1 SRIS TOPI

4 sics ongruent Tringles (/) ongruent tringles re shpes tht re exctly the sme in every wy (side lengths nd interior ngles re ll equl). If even one side or one ngle re not equl, then the tringles re not congruent. ongruent Tringles These tringles re not congruent 60c c 40c Q 40c R 80c c P 60c c c F N 48c M 22.6 P ngles: + + P + + Q + + R Sides: PQ QR RP ngle is different ll sides nd ngles re equl therefore tringles re congruent. No ngle in T F is equl to + N. ` These tringles re NOT congruent. From ove, nd PQR re congruent. Using the proper nottion, this is written s / PQR. It is importnt to mke sure the ngles mtch when using the / symol. Here is n exmple: Show tht these tringles re congruent 75c 14 M 75c 14 65c 15 40c N 65c 15 40c P In nd MNP: + + M 75c + + N 65c ngles MN NP sides + + P 40c MP ` / MNP Notice the order of the letters when using /. The equl ngles re written in the sme order. qul ngles written in the sme order (correct): qul ngles not written in the sme order (incorrect): T / TMNP T / TNPM T / TPMN T / TPMN T / TPNM T / TMNP 2 J SRIS TOPI

5 sics Similr Figures ( ) Similr figures hve the sme shpe, ut not necessrily the sme size. These shpes re similr. cm 5 cm 9 cm cm J 20 cm K 18 cm 15 cm M 30 cm L Similr figures hve two importnt properties: Their corresponding ngles re equl. Their corresponding sides re in the sme rtio. In the ove similr shpes, the rtio of the corresponding sides is 2 since the sides in the igger shpe re doule the length in the smller shpe. These shpes re similr 3 cm P 8 cm 45c Q 24 cm S R 18 cm 135c Find the length of. Find the size of +. PQRS nd re similr. PQRS nd re similr. ` ` PS PQ ` + + Q ` + 45c ` 9cm c Find the size of + R. d Find the size of RS. PQRS nd re similr. ` + R + ` + R 135c PQRS nd re similr. ` RS RS PQ 8 ` ` RS 6cm J 3 SRIS TOPI

6 Questions sics 1. Show these tringles re congruent, nd then use / symol to stte congruency. T F V 8 U 14 G 67c 67c 67c F c 60c 25c F 95c 25c d P 23c M 5 67c 12 23c 13 K R 67c 5 Q L 4 J SRIS TOPI

7 Questions sics 2. Find the missing vlues in these similr shpes (ll mesurements in cm): G 6 85c 7 1c 95c 9 70c F S 14 P Q c 1c 9 150c 15 T S 0c P 3 50c Q 18 R R PQ QR PT ST RS + P + + P + S + Q QR + c L d F M J 4 K 40c 8 15 H 5 G N M 12 45c L K LM Given LM 2. Find the length of KN. F + M + J 5 SRIS TOPI

8 Knowing More Testing for ongruent Tringles ongruent tringles hve ll 3 corresponding sides equl, nd ll 3 corresponding ngles equl tht is 6 properties. However, there re tests for congruent tringles tht don t require showing ll 6 properties. There re four tests: Side Side Side (SSS) If the corresponding sides of two tringles re equl, then the tringles re congruent (SSS). Q Side ngle Side (SS) If 2 sides nd the included ngle re respectively equl, then the tringles re congruent (SS). M P R L N In nd PQR: PQ QR PR In nd LMN: LM + + L LN T / TPQR (SSS) T / TLMN (SS) Side ngle ngle (S) If 2 corresponding sides nd corresponding ngle re equl, then the tringles re congruent (S). K Right ngle, Hypotenuse, Side (RHS) If two right ngled tringles hve the sme hypotenuse, nd corresponding side, then the tringles re congruent (RHS). N L M M O In nd KLM: KL + + K + + M In nd NOM: NM NO + + M 90c T / TKLM (S) T / TNOM (RHS) 6 J SRIS TOPI

9 Knowing More Here re some exmples: Show tht these tringles re congruent: L I In TIJK 62c + J 180c-93c- 62c (ngle sum of tringle) 25c 93c M K 93c 12 cm 12 cm 25c ` + N + J (oth re 25c ) N J In T LMN nd T IJK : + N + J + M + K 93c MN KJ 12cm ` TLMN / TIKJ (Proved ove) (Given) (Given) (S) In T F nd T GF : F GF (Given) G (Given) G F is common F ` TF / TGF (SSS) Here is n exmple where congruence is used to show something is true. Show tht isects + in the digrm elow In T nd T : (Given) is common c (Given) ` + / + (RHS) ` + + (ongruent tringles, T / T ) ` isecting + J 7 SRIS TOPI

10 Knowing More Similr Tringles ( ) There re two wys to show tht tringles re similr: Show tht their corresponding sides re in proportion. Show tht they hve equl ngles (). If two tringles re similr, the symol is used. Show tht these tringles re similr: In : 78c + 180c - 58c - 78c 44c (ngle sum of tringle) In GHI: 58c +G 180c - 58c - 44c 78c (ngle sum of tringle) G 44c H In nd GHI: + +H (oth re 44c ) 58c + +G (oth re 78c ) I + +I (oth re 58c ) ` GIH () Q 12 R 22 In QRS nd TUV: TU RQ S U UV RS TV QS ` ll sides in proportion TU UV TV c m RQ RS QS T 33 ` QRS TUV 27 V 8 J SRIS TOPI

11 Questions Knowing More 1. xplin wht the following men: / SSS c SS d S e RHS f J 9 SRIS TOPI

12 Questions Knowing More 2. Prove these tringles re congruent: F 3 c P S 75c 75c R M 75c N Q J SRIS TOPI

13 Questions Knowing More 3. In the digrm elow, show tht. 4. Prove tht JKL STU. J T 8 15 L 6 K S 12 U J 11 SRIS TOPI

14 Using Our Knowledge Scle Fctor in Similr Tringles When tringles re similr, their ngles re equl () nd their corresponding sides re in proportion. The rtio tht their sides re in proportion is clled the Scle Fctor. Scle Fctor either enlrges (scles up) or reduces (scles down). G F Scle fctor Scle fctor 2 H I In nd F: In nd GHI: F GI F scle fctor of F to HI scle fctor of GHI to GH ` T T F ` T TGHI If the scle fctor is igger thn 1, the tringle is enlrged. If the scle fctor is etween 0 nd 1 (deciml or frction), the tringle is reduced. Show these tringles re similr nd find their scle fctor of PQR to LMN 6cm N 3cm M 7cm L In LMN nd PQR: PQ LM QR MN cm P RP NL ` LMN PQR (orresponding sides re in proportion) R 21cm PQ QR RP 3 LM MN NL 9cm ` The scle fctor of PQR to LMN is 3. Q 12 J SRIS TOPI

15 Using Our Knowledge Using Similr Tringles If tringles re known to e similr, then the properties of similr tringles cn e used to solve prolems. Find the vlues of x nd y if TUV (ll mesurements in cm) x In nd TUV: To find x: T 5 TV TU ` x (Similr tringles, TUV) ` x 18 # 30 ` x 6cm V y To find y: U UV TU y ` 30 5 ` y 5# 30 ` y 15cm (Similr tringles, TUV) J 13 SRIS TOPI

16 Questions Using Our Knowledge 1. Find the scle fctor in these pirs of similr tringles for oth the smller nd lrger tringles: Given RST UVW. R 35 S U 7 V T 5 W Given J SRIS TOPI

17 Questions Using Our Knowledge 2. nswer these questions out the digrm elow: J 12 N 5 8 K M L Show tht JML JNK. Find the length of KL. c Find the length of ML. J 15 SRIS TOPI

18 Questions Using Our Knowledge 3. nswer these questions out the shpe elow: F 8 6 G 25 J H I Show tht GFH GIJ. Find the length of GI. c Find the length of IJ. d Find the scle fctor of the lrger tringle with respect to the smll tringle. 16 J SRIS TOPI

19 Thinking More Using ongruence nd Similrity in Proofs ongruence nd similrity re used to prove properties of tringles, qudrilterls nd other shpes. Show tht the digonls of prllelogrm isect ech other rw in digonls ( nd ) in the prllelogrm : O Given: To prove: O O nd O O Proof: In O nd O ` + + nd + + ` TO / TO ` O O nd O O (Given) (lternte ngles; ) (lternte ngles; ) (Given) (S) (ongruent tringles; TO / TO ) ` The digonls of prllelogrm isect ech other. Similrity cn lso e used in proofs. In the digrm elow, prove tht if Given: To prove: Proof: In nd ` T / T ` / T ` (Given) (orresponding sides in proportion) (Similr tringles; T T ) (Given) J 17 SRIS TOPI

20 Questions Thinking More 1. nswer these questions out PQRS elow given tht PQ RS nd PR QS: P Q R S Prove PRS / SQP. Prove PQRS is prllelogrm (PQ RS nd PS QR). c Prove tht the opposite ngles of prllelogrm re equl. 18 J SRIS TOPI

21 Questions Thinking More 2. KLM is n iscosceles tringle with KL KM. K M N L KN hs een drwn to isect ML. Show tht KMN / KLN. Show tht +MNK +LNK 90c. c Prove tht +M +L. 3. In F, +F +, if the line G isects +F. Prove F is isosceles. G F J 19 SRIS TOPI

22 Questions Thinking More 4. Prove the following out the Rhomus STUV elow: S T O V VOS / TOU U SOT / UOV c Show tht the digonls isect ech other. d Show tht the digonls isect t 90c. 20 J SRIS TOPI

23 nswers sics: Knowing More: 2. PQ 12cm QR 18cm RS 20cm + P 85c + S 95c + Q 1c 1. e RHS mens Right ngle, Hypotenuse, Side. It is one of the four tests tht cn e used to prove two tringles re congruent. If two right ngle tringles hve equl hypotenuse nd n equl corresponding side, then the tringles re congruent. PT 2cm ST 4cm + 0c + P 1c QR 5cm + 50c f symol mens is similr to. It is used to show two tringles hve the sme shpe (corresponding ngles re equl nd corresponding sides re in proportion). c LM cm 2 2 cm 3 + M 45c + 40c 1. Using Our Knowledge: Scle fctor from RST to UVW is 1 5 Scle fctor from UVW to RST is 5 d KN cm Scle fctor from to is Knowing More: Scle fctor from to is / symol mens is congruent to. It is used to show two tringles re exctly the sme in every wy (corresponding sides equl nd corresponding ngles equl). 2. c KL 6cm ML 12cm SSS mens Side, Side, Side. It is one of the four tests tht cn e used to prove two tringles re congruent. If the corresponding sides of two tringles re equl, then the tringles re congruent. 3. c GI 15cm IJ 20cm d Scle fctor from GFH to GIJ is c SS mens Side, ngle, Side. It is one of the four tests tht cn e used to prove two tringles re congruent. If two sides nd the included ngle of two tringles re equl, then the tringles re congruent. d S mens Side, ngle, ngle. It is one of the four tests tht cn e used to prove two tringles re congruent. If corresponding side nd two corresponding ngles of two tringles re equl, then the tringles re congruent. J 21 SRIS TOPI

24 Notes 22 J SRIS TOPI

25 Notes J 23 SRIS TOPI

26 Notes 24 J SRIS TOPI

27

28 Similrity nd ongruence

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

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

Non Right Angled Triangles

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

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

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

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

GM1 Consolidation Worksheet

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

More information

2 Calculate the size of each angle marked by a letter in these triangles.

2 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 information

Simplifying Algebra. Simplifying Algebra. Curriculum Ready.

Simplifying 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 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

Comparing the Pre-image and Image of a Dilation

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

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

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

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

Trigonometry and Constructive Geometry

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

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

8Similarity ONLINE 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 ONLINE 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 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step

More information

A B= ( ) because from A to B is 3 right, 2 down.

A B= ( ) because from A to B is 3 right, 2 down. 8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.

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

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 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 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

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

Shape and measurement

Shape and measurement C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

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

Individual Contest. English Version. Time limit: 90 minutes. Instructions:

Individual 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 information

Special Numbers, Factors and Multiples

Special 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 information

Trigonometry Revision Sheet Q5 of Paper 2

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.

More information

Angles and triangles. Chapter Introduction Angular measurement Minutes and seconds

Angles and triangles. Chapter Introduction Angular measurement Minutes and seconds hpter 0 ngles nd tringles 0.1 Introduction stright line which crosses two prllel lines is clled trnsversl (see MN in igure 0.1). Trigonometry is suject tht involves the mesurement of sides nd ngles of

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

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths

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

More information

Area and Perimeter. Area and Perimeter. Curriculum Ready.

Area 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 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

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

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

Definition :- A shape has a line of symmetry if, when folded over the line. 1 line of symmetry 2 lines of symmetry

Definition :- A shape has a line of symmetry if, when folded over the line. 1 line of symmetry 2 lines of symmetry Symmetry Lines of Symmetry Definition :- A shpe hs line of symmetry if, when folded over the line the hlves of the shpe mtch up exctly. Some shpes hve more thn one line of symmetry : line of symmetry lines

More information

10. AREAS BETWEEN CURVES

10. 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 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

6.2 The Pythagorean Theorems

6.2 The Pythagorean Theorems PythgorenTheorems20052006.nb 1 6.2 The Pythgoren Theorems One of the best known theorems in geometry (nd ll of mthemtics for tht mtter) is the Pythgoren Theorem. You hve probbly lredy worked with this

More information

Lesson Notes: Week 40-Vectors

Lesson Notes: Week 40-Vectors Lesson Notes: Week 40-Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples

More information

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15 Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll

More information

A Study on the Properties of Rational Triangles

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

More information

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. 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 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

9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1

9.1 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 information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Part I: Study the theorem statement.

Part 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 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

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

Alg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A

Alg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A lg 3 h 7.2, 8 1 7.2 Right Tringle Trig ) Use of clcultor sin 10 = sin x =.4741 c ) rete right tringles π 1) If = nd = 25, find 6 c 2) If = 30, nd = 45, = 1 find nd c 3) If in right, with right ngle t,

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Answers for Lesson 3-1, pp Exercises

Answers for Lesson 3-1, pp Exercises 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

More information

Bridging the gap: GCSE AS Level

Bridging 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 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

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

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

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

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

More information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

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

What s in Chapter 13?

What s in Chapter 13? Are nd volume 13 Wht s in Chpter 13? 13 01 re 13 0 Are of circle 13 03 res of trpeziums, kites nd rhomuses 13 04 surfce re of rectngulr prism 13 05 surfce re of tringulr prism 13 06 surfce re of cylinder

More information

Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS

Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (

More information

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors: Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )

More information

= x x 2 = 25 2

= 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 information

m A 1 1 A ! and AC 6

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:

More information

Introduction to Algebra - Part 2

Introduction 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 information

Chapter 1: Fundamentals

Chapter 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 information

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1 Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more

More information

A study of Pythagoras Theorem

A study of Pythagoras Theorem CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

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

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER 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 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

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

Math 154B Elementary Algebra-2 nd Half Spring 2015

Math 154B Elementary Algebra-2 nd Half Spring 2015 Mth 154B Elementry Alger- nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know

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

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = 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

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up

9.5 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 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

Pythagoras Theorem. Pythagoras

Pythagoras Theorem. Pythagoras 11 Pythgors Theorem Pythgors Theorem. onverse of Pythgors theorem. Pythgoren triplets Proof of Pythgors theorem nd its converse Problems nd riders bsed on Pythgors theorem. This unit fcilittes you in,

More information

Similar Right Triangles

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

More information

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

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

More information

I1.1 Pythagoras' Theorem. I1.2 Further Work With Pythagoras' Theorem. I1.3 Sine, Cosine and Tangent. I1.4 Finding Lengths in Right Angled Triangles

I1.1 Pythagoras' Theorem. I1.2 Further Work With Pythagoras' Theorem. I1.3 Sine, Cosine and Tangent. I1.4 Finding Lengths in Right Angled Triangles UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet STRAND I: Geometry nd Trigonometry I1 Pythgors' Theorem nd Trigonometric Rtios Tet Contents Section I1.1 Pythgors' Theorem I1. Further Work With Pythgors'

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

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

Identifying and Classifying 2-D Shapes

Identifying 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 information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

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

Consolidation Worksheet

Consolidation Worksheet Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

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

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

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

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

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

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

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS 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 information

Plotting Ordered Pairs Using Integers

Plotting 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 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

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