I.G.C.S.E. Similarity. You can access the solutions from the end of each question

Size: px
Start display at page:

Download "I.G.C.S.E. Similarity. You can access the solutions from the end of each question"

Transcription

1 I.G.C.S.E. Similarity Inde: Please click on the question number you want Question 1 Question Question Question 4 Question 5 Question 6 Question 7 Question 8 You can access the solutions from the end of each question

2 Question 1 1. Find the sides marked with letters. C 4 cm F b B E cm a 9 cm 10 cm D Click here to read the solution to this question Click here to return to the inde

3 Solution to question 1 Consider! BC and! DEF. C 4 cm F b B E cm a 9 cm 10 cm D We have ˆ D ˆ Bˆ Eˆ Cˆ Fˆ Hence B BC C DE EF FD a 4 10 b 9 a a a cm b 1 cm b 9 b Click here to read the question again Click here to return to the inde

4 Question Find the sides marked with letters. D c C cm 7 cm 8 cm d F B 5 cm E Click here to read the solution to this question Click here to return to the inde

5 Solution to question Consider! BC and! DEF. D c C cm 7 cm 8 cm d F B 5 cm We have ˆ D ˆ Bˆ Eˆ Cˆ Fˆ Hence B BC C DE EF FD 8 6 d 5 7 E c 1 1 5c 1 c 4 cm d b 1 c m d 5 Click here to read the question again Click here to return to the inde

6 Question Find the sides marked with letters. D E 6cm 4 cm 4 cm y B cm C Click here to read the solution to this question Click here to return to the inde

7 Solution to question D E 6 cm 4 cm 4 cm y B cm C! BE and! CD are similar D E 4 cm 4 cm 4+ cm 6 cm y B y + cm C We have ˆ ˆ Bˆ Cˆ Eˆ Dˆ (same angle in both triangles) (corresponding angles) (corresponding angles) Hence B BE E C CD D y 4 4 y cm ( ) 4 y 4( y + ) 6y 4y + 1 6y 1 y y 6 cm 6 y + Click here to read the question again Click here to return to the inde

8 Question 4 Find the sides marked with letters. Z v 5 cm 1 cm W w X 1 cm Y Click here to read the solution to this question Click here to return to the inde

9 Solution to question 4 Z v 5 cm 1 cm W w X 1 cm Y! WYZ and! ZYX are similar Z Z v 1 cm 5 cm 1 cm W w + 1 Y X 1 cm Y We have Wˆ Zˆ Yˆ Yˆ (same angle in both triangles) Zˆ Xˆ (both right-angles) Hence WY YZ ZW ZY YX XZ w v v 65 1v v 5 cm 1 1 w ( w + 1) 169 1w w w cm 1 1 Click here to read the question again Click here to return to the inde

10 Question 5 Find the missing area in each of the following similar triangles. a. 9cm cm 4 cm b. 7 cm 8 cm 18 cm Click here to read the solution to this question Click here to return to the inde

11 Solution to question 5 a. rea ratio cm cm 4 cm Length ratio Length ratio : 4 rea ratio :4 9 : 16 4 rea of the larger triangle 16 area of the smaller triangle m 16c 9 b. rea ratio cm 8 cm 18 cm Length ratio 7 8 Length ratio 7 : 8 rea ratio 7 :8 49 : rea of the smaller triangle area of the larger triangle cm 8 64 Click here to read the question again Click here to return to the inde

12 Question 6 Two spheres have radii of cm and 5 cm respectively. If the volume of the smaller sphere is 7cm, find the volume of the larger sphere. Click here to read the solution to this question Click here to return to the inde

13 Solution to question 6 Volume ratio 15 7 Volume 7cm Length ratio : 5 rea ratio :5 7 : 15 Volume of the larger sphere V c m 7 Length ratio 5 15 volume of the smaller triangle 7 Click here to read the question again Click here to return to the inde

14 Question 7 Two similar jugs have volumes of 54cm and 104 cm respectively. If the height of the larger jug is 64 cm, find the height of the smaller jug. Click here to read the solution to this question Click here to return to the inde

15 Solution to question 7 Volume ratio Volume 54cm Volume 104cm 7 cm Length ratio 8 Volume ratio 54 : : 51 Length ratio 7 : 51 : 8 Height of the smaller jug of the height of the larger jug m 4c 8 Click here to read the question again Click here to return to the inde

16 Question 8 The surface areas of two similar model ships are 4m and 5m respectively. a. If the length of the larger model is 75 m, find the length of the smaller model. b. If the volume of the smaller model is model. m, find the volume of the larger Click here to read the solution to this question Click here to return to the inde

17 Solution to question 8 Volume ratio 15 8 Volume m rea rea ratio 5 4cm 4 rea 5cm Length 75 m rea ratio 4 : Length ratio 5 a. Length ratio 4: 5 :5 Length of the smaller model of the larger model m 5 b. Volume ratio :5 8:15 15 Volume of the larger model of the smaller model m 50 8 Click here to read the question again Click here to return to the inde

DIPOLES III. q const. The voltage produced by such a charge distribution is given by. r r'

DIPOLES III. q const. The voltage produced by such a charge distribution is given by. r r' DIPOLES III We now consider a particularly important charge configuration a dipole. This consists of two equal but opposite charges separated by a small distance. We define the dipole moment as p lim q

More information

Notes 19 Gradient and Laplacian

Notes 19 Gradient and Laplacian ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can

More information

I.G.C.S.E. Algebra 02. You can access the solutions from the end of each question

I.G.C.S.E. Algebra 02. You can access the solutions from the end of each question I.G.C.S.E. Algebra 0 Index: Please click on the question number you want Question 1 Question Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 You can access the

More information

Homework 1/Solutions. Graded Exercises

Homework 1/Solutions. Graded Exercises MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both

More information

I pledge that I have neither given nor received help with this assessment.

I pledge that I have neither given nor received help with this assessment. CORE MATHEMATICS PII Page 1 of 4 HILTON COLLEGE TRIAL EXAMINATION AUGUST 016 Time: 3 hours CORE MATHEMATICS PAPER 150 marks PLEASE READ THE FOLLOWING GENERAL INSTRUCTIONS CAREFULLY. 1. This question paper

More information

MATHEMATICS: PAPER II MARKING GUIDELINES

MATHEMATICS: PAPER II MARKING GUIDELINES NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 05 MATHEMATICS: PAPER II MARKING GUIDELINES Time: 3 hours 50 marks These marking guidelines are prepared for use by examiners and sub-examiners, all of

More information

E(3;2) (4 3) (5 2) r r. 10 ( x 4) ( y 5) 10. y D A(4;5) C(10;3) B(2;-1) SECTION A QUESTION 1 In the diagram below:

E(3;2) (4 3) (5 2) r r. 10 ( x 4) ( y 5) 10. y D A(4;5) C(10;3) B(2;-1) SECTION A QUESTION 1 In the diagram below: SECTION A QUESTION In the diagram below: DC CB A is the centre of the circle. E is the midpoint of AB. The equation of line BA is: y 7 DF is a tangent to the circle at F. y D F A(4;5) E B(;-) C(0;) (a)

More information

MATHEMATICS: PAPER II MARKING GUIDELINES

MATHEMATICS: PAPER II MARKING GUIDELINES NATIONAL SENIOR CERTIFICATE EXAMINATION SUPPLEMENTARY EXAMINATION MARCH 08 MATHEMATICS: PAPER II MARKING GUIDELINES Time: 3 hours 50 marks These marking guidelines are prepared for use by examiners and

More information

GCSE Mathematics (Calculator Paper)

GCSE Mathematics (Calculator Paper) Centre Number Surname Other Names Candidate Number For Examiner s Use Examiner s Initials Candidate Signature GCSE Mathematics (Calculator Paper) Practice Paper Style Questions Topic: Trigonometry (Higher

More information

MATHEMATICS: PAPER II MARKING GUIDELINES

MATHEMATICS: PAPER II MARKING GUIDELINES GRADE 10 IEB STANDARDISATION PROJECT NOVEMBER 01 MATHEMATICS: PAPER II MARKING GUIDELINES Time: hours 100 marks These marking guidelines are prepared for use by examiners and sub-examiners, all of whom

More information

Today in Physics 217: electric potential

Today in Physics 217: electric potential Today in Physics 17: electric potential Finish Friday s discussion of the field from a uniformly-charged sphere, and the gravitational analogue of Gauss Law. Electric potential Example: a field and its

More information

= (, ) V λ (1) λ λ ( + + ) P = [ ( ), (1)] ( ) ( ) = ( ) ( ) ( 0 ) ( 0 ) = ( 0 ) ( 0 ) 0 ( 0 ) ( ( 0 )) ( ( 0 )) = ( ( 0 )) ( ( 0 )) ( + ( 0 )) ( + ( 0 )) = ( + ( 0 )) ( ( 0 )) P V V V V V P V P V V V

More information

I.G.C.S.E. Trigonometry 01. You can access the solutions from the end of each question

I.G.C.S.E. Trigonometry 01. You can access the solutions from the end of each question I.G..S.E. Trigonometry 01 Index: Please click on the question number you want Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 You can access the solutions from the end of each

More information

CSSTP. Given CSSTP. Statements Reasons. Given CSSTP. Mult. Prop. = Div. Prop. = Sym. Prop. = or 1 Mult. Prop. = Div. Prop. =

CSSTP. Given CSSTP. Statements Reasons. Given CSSTP. Mult. Prop. = Div. Prop. = Sym. Prop. = or 1 Mult. Prop. = Div. Prop. = : If the triangles are similar (~), then all of the sides must be congruent proportional (create equal scale fractions). Example: A~ F Before you start your proof, it is important to plan! Setup the three

More information

IB MYP Unit 6 Review

IB MYP Unit 6 Review Name: Date: 1. Two triangles are congruent if 1. A. corresponding angles are congruent B. corresponding sides and corresponding angles are congruent C. the angles in each triangle have a sum of 180 D.

More information

1 h 9 e $ s i n t h e o r y, a p p l i c a t i a n

1 h 9 e $ s i n t h e o r y, a p p l i c a t i a n T : 99 9 \ E \ : \ 4 7 8 \ \ \ \ - \ \ T \ \ \ : \ 99 9 T : 99-9 9 E : 4 7 8 / T V 9 \ E \ \ : 4 \ 7 8 / T \ V \ 9 T - w - - V w w - T w w \ T \ \ \ w \ w \ - \ w \ \ w \ \ \ T \ w \ w \ w \ w \ \ w \

More information

Here are some solutions to the sample problems assigned for Chapter 4. Solution: Consider a function of 3 (independent) variables. treating as real.

Here are some solutions to the sample problems assigned for Chapter 4. Solution: Consider a function of 3 (independent) variables. treating as real. Lecture 11 Appendix B: Some sample problems from Boas Here are some solutions to the sample problems assigned for Chapter 4. 4.1: 3 Solution: Consider a function of 3 (independent) variables,, ln z u v

More information

I.G.C.S.E. Volume & Surface Area. You can access the solutions from the end of each question

I.G.C.S.E. Volume & Surface Area. You can access the solutions from the end of each question I.G.C.S.E. Volume & Surface Area Index: Please click on the question number you want Question 1 Question Question Question 4 Question 5 Question 6 Question 7 Question 8 You can access the solutions from

More information

5200: Similarity of figures. Define: Lemma: proof:

5200: Similarity of figures. Define: Lemma: proof: 5200: Similarity of figures. We understand pretty well figures with the same shape and size. Next we study figures with the same shape but different sizes, called similar figures. The most important ones

More information

MATHEMATICS: PAPER II MARKING GUIDELINES

MATHEMATICS: PAPER II MARKING GUIDELINES NATIONAL SENIOR CERTIFICATE EXAMINATION EXAMINATION 05 MATHEMATICS: PAPER II MARKING GUIDELINES Time: 3 hours 50 marks These marking guidelines are prepared for use by examiners and sub-examiners, all

More information

Questions. Exercise (1)

Questions. Exercise (1) Questions Exercise (1) (1) hoose the correct answer: 1) The acute angle supplements. angle. a) acute b) obtuse c) right d) reflex 2) The right angle complements angle whose measure is. a) 0 b) 45 c) 90

More information

CHAPTER 3 BOOLEAN ALGEBRA

CHAPTER 3 BOOLEAN ALGEBRA CHAPTER 3 BOOLEAN ALGEBRA (continued) This chapter in the book includes: Objectives Study Guide 3.1 Multiplying Out and Factoring Expressions 3.2 Exclusive-OR and Equivalence Operations 3.3 The Consensus

More information

Omm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics

Omm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector

More information

ST MARY S DSG, KLOOF GRADE: 12 SEPTEMBER 2016 MATHEMATICS: PAPER II. 1. This question paper consists of 27 typed pages. There are also 2 blank pages.

ST MARY S DSG, KLOOF GRADE: 12 SEPTEMBER 2016 MATHEMATICS: PAPER II. 1. This question paper consists of 27 typed pages. There are also 2 blank pages. ST MARY S DSG, KLOOF GRADE: 12 SEPTEMBER 2016 MATHEMATICS: PAPER II Examiner: S Drew TIME: 3 HOURS Moderators: J van Rooyen J Kinsey TOTAL: 150 MARKS INSTRUCTIONS: 1. This question paper consists of 27

More information

Electromagnetic Properties of Materials Part 2

Electromagnetic Properties of Materials Part 2 ECE 5322 21 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) 747 6958 rcrumpf@utep.edu Lecture #3 Electromagnetic Properties of Materials Part 2 Nonlinear

More information

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort - 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

Algebra 1. Predicting Patterns & Examining Experiments. Unit 5: Changing on a Plane Section 4: Try Without Angles

Algebra 1. Predicting Patterns & Examining Experiments. Unit 5: Changing on a Plane Section 4: Try Without Angles Section 4 Examines triangles in the coordinate plane, we will mention slope, but not angles (we will visit angles in Unit 6). Students will need to know the definition of collinear, isosceles, and congruent...

More information

APPLICATIONS OF GAUSS S LAW

APPLICATIONS OF GAUSS S LAW APPLICATIONS OF GAUSS S LAW Although Gauss s Law is always correct it is generally only useful in cases with strong symmetries. The basic problem is that it gives the integral of E rather than E itself.

More information

I.G.C.S.E. Trigonometry. You can access the solutions from the end of each question

I.G.C.S.E. Trigonometry. You can access the solutions from the end of each question I.G..S.E. Trigonometry Index: Please click on the question number you want Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 You can access the solutions from the end of each question Question

More information

Example 1. Show that the shaded triangle is a (3, 4, 5) triangle.

Example 1. Show that the shaded triangle is a (3, 4, 5) triangle. Example 1. Show that the shaded triangle is a (3, 4, 5) triangle. Solution to Example 1. Show that the shaded triangle C is a (3, 4, 5)-triangle. E D t C 4 T t 4 4 Solution. Suppose each side of the square

More information

Matrices. Background mathematics review

Matrices. Background mathematics review Matrices Background mathematics review David Miller Matrices Matrix notation Background mathematics review David Miller Matrix notation A matrix is, first of all, a rectangular array of numbers An M N

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

Chapter 6 Summary 6.1. Using the Hypotenuse-Leg (HL) Congruence Theorem. Example

Chapter 6 Summary 6.1. Using the Hypotenuse-Leg (HL) Congruence Theorem. Example Chapter Summary Key Terms corresponding parts of congruent triangles are congruent (CPCTC) (.2) vertex angle of an isosceles triangle (.3) inverse (.4) contrapositive (.4) direct proof (.4) indirect proof

More information

1.5 geo concepts perimeter, segments, and angles 2017 ink.notebook. August 31, Page 34. Page 33 Page Two-Step Geometry Concepts

1.5 geo concepts perimeter, segments, and angles 2017 ink.notebook. August 31, Page 34. Page 33 Page Two-Step Geometry Concepts 1.5 geo concepts perimeter, segments, and angles 2017 ink.notebook Page 34 Page 33 Page 32 1.4 Two-Step Geometry Concepts Lesson Objectives Standards 1.4 Using Algebra for Geometry Concepts Rules Page

More information

5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Construct each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and

More information

UNIT 1: SIMILARITY, CONGRUENCE, AND PROOFS. 1) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of 1. 2 centered at ( 4, 1).

UNIT 1: SIMILARITY, CONGRUENCE, AND PROOFS. 1) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of 1. 2 centered at ( 4, 1). 1) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of 1. 2 centered at ( 4, 1). The dilation is Which statement is true? A. B. C. D. AB B' C' A' B' BC AB BC A' B' B' C' AB BC A' B' D'

More information

5-5 The Triangle Inequality

5-5 The Triangle Inequality Is it possible to form a triangle with the given side lengths? If not, explain why not. 1. 5 cm, 7 cm, 10 cm Yes; 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5 3. 6 m, 14 m, 10 m Yes; 6 + 14 > 10, 6 + 10 > 14,

More information

LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6.2 POLYNOMIAL OPERATIONS I LESSON 6. POLYNOMIAL OPERATIONS I LESSON 6. POLYNOMIALS OPERATIONS I 63 OVERVIEW Here's what you'll learn in this lesson: Adding and Subtracting a. Definition of polynomial, term, and coefficient b. Evaluating

More information

(Chapter 10) (Practical Geometry) (Class VII) Question 1: Exercise 10.1 Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only. Answer 1: To

More information

Chapter 2 One-Dimensional Kinematics

Chapter 2 One-Dimensional Kinematics Review: Chapter 2 One-Dimensional Kinematics Description of motion in one dimension Copyright 2010 Pearson Education, Inc. Review: Motion with Constant Acceleration Free fall: constant acceleration g =

More information

CHAPTER * 2-2.* Pearson Education Limited Problem Solutions, Global Edition Chapter 2. Verification of DeMorgan s Theorem

CHAPTER * 2-2.* Pearson Education Limited Problem Solutions, Global Edition Chapter 2. Verification of DeMorgan s Theorem HPTER 2 2-.* a) XYZ = X + Y + Z Verification of DeMorgan s Theorem Pearson Education Limited 206. X Y Z XYZ XYZ X + Y + Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b) X + YZ = ( X + Y) ( X + Z) The Second

More information

Unit 1. GSE Analytic Geometry EOC Review Name: Units 1 3. Date: Pd:

Unit 1. GSE Analytic Geometry EOC Review Name: Units 1 3. Date: Pd: GSE Analytic Geometry EOC Review Name: Units 1 Date: Pd: Unit 1 1 1. Figure A B C D F is a dilation of figure ABCDF by a scale factor of. The dilation is centered at ( 4, 1). 2 Which statement is true?

More information

1. Town A is 48 km from town B and 32 km from town C as shown in the diagram. A 48km

1. Town A is 48 km from town B and 32 km from town C as shown in the diagram. A 48km 1. Town is 48 km from town and 32 km from town as shown in the diagram. 32km 48km Given that town is 56 km from town, find the size of angle (Total 4 marks) Â to the nearest degree. 2. The diagram shows

More information

(e) (i) Prove that C(x) = C( x) for all x. (2)

(e) (i) Prove that C(x) = C( x) for all x. (2) Revision - chapters and 3 part two. (a) Sketch the graph of f (x) = sin 3x + sin 6x, 0 x. Write down the exact period of the function f. (Total 3 marks). (a) Sketch the graph of the function C ( x) cos

More information

EULER EQUATIONS. We start by considering how time derivatives are effected by rotation. Consider a vector defined in the two systems by

EULER EQUATIONS. We start by considering how time derivatives are effected by rotation. Consider a vector defined in the two systems by EULER EQUATIONS We now consider another approach to rigid body probles based on looking at the change needed in Newton s Laws if an accelerated coordinate syste is used. We start by considering how tie

More information

Identity. "At least one dog has fleas" is translated by an existential quantifier"

Identity. At least one dog has fleas is translated by an existential quantifier Identity Quantifiers are so-called because they say how many. So far, we've only used the quantifiers to give the crudest possible answers to the question "How many dogs have fleas?": "All," "None," "Some,"

More information

I.G.C.S.E. Area. You can access the solutions from the end of each question

I.G.C.S.E. Area. You can access the solutions from the end of each question I.G.C.S.E. Area Index: Please click on the question number you want Question Question Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 You can access the solutions from the

More information

1.4 geo concepts perimeter, segments, and angles ink.notebook. August 31, Page 42 Page Two-Step Geometry Concepts

1.4 geo concepts perimeter, segments, and angles ink.notebook. August 31, Page 42 Page Two-Step Geometry Concepts 1.4 geo concepts perimeter, segments, and angles ink.notebook Page 42 Page 41 1.4 Two-Step Geometry Concepts Lesson Objectives Standards Standards 1.4 Using Algebra for Geometry Concepts Rules Press the

More information

LESSON 7.1 FACTORING POLYNOMIALS I

LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of

More information

UNIT 3 BOOLEAN ALGEBRA (CONT D)

UNIT 3 BOOLEAN ALGEBRA (CONT D) UNIT 3 BOOLEAN ALGEBRA (CONT D) Spring 2011 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic

More information

GEOMETRY UNIT 1 WORKBOOK. CHAPTER 2 Reasoning and Proof

GEOMETRY UNIT 1 WORKBOOK. CHAPTER 2 Reasoning and Proof GEOMETRY UNIT 1 WORKBOOK CHAPTER 2 Reasoning and Proof 1 2 Notes 5 : Using postulates and diagrams, make valid conclusions about points, lines, and planes. I) Reminder: Rules that are accepted without

More information

Quine-McCluskey (Tabular) Minimization

Quine-McCluskey (Tabular) Minimization Quine-McCluskey (Tabular) Minimization Two step process utilizing tabular listings to: Identify prime implicants (implicant tables) Identify minimal PI set (cover tables) All work is done in tabular form

More information

Symmetry. PlayMath, Summer, 2018

Symmetry. PlayMath, Summer, 2018 This essay is about the use of symmetry in solving algebra and geometry problems From Wikipedia we learn that symmetry (from Greek symmetra measure together ) generally conveys two primary meanings The

More information

Western Cape Education Department. Examination Preparation Learning Resource 2016 GEOMETRY MATHEMATICS

Western Cape Education Department. Examination Preparation Learning Resource 2016 GEOMETRY MATHEMATICS Western ape Education Department Examination Preparation Learning Resource 06 GEOMETRY MTHEMTIS Grade Geometry Questions Grade Theorems on Razzia Ebrahim Senior urriculum Planner for Mathematics E-mail:

More information

The Law of Averages. MARK FLANAGAN School of Electrical, Electronic and Communications Engineering University College Dublin

The Law of Averages. MARK FLANAGAN School of Electrical, Electronic and Communications Engineering University College Dublin The Law of Averages MARK FLANAGAN School of Electrical, Electronic and Communications Engineering University College Dublin Basic Principle of Inequalities: For any real number x, we have 3 x 2 0, with

More information

Click one of the buttons below or press the enter key EXIT

Click one of the buttons below or press the enter key EXIT Click one of the buttons below or press the enter key In geometry, two polygons are similar when one is a replica (scale model) of the other. Consider Dr. Evil and Mini Me from Mike Meyers hit movie Austin

More information

MATHEMATICS: PAPER II

MATHEMATICS: PAPER II NATIONAL SENIOR CERTIFICATE EXAMINATION SUPPLEMENTARY EXAMINATION 2015 MATHEMATICS: PAPER II Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of

More information

Proofs. by Bill Hanlon

Proofs. by Bill Hanlon Proofs by Bill Hanlon Future Reference To prove congruence, it is important that you remember not only your congruence theorems, but know your parallel line theorems, and theorems concerning triangles.

More information

( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.

( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3. Chapter 6 Exercise Set 6.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.

More information

Answers. Chapter 9 A92. Angles Theorem (Thm. 5.6) then XZY. Base Angles Theorem (Thm. 5.6) 5, 2. then WV WZ;

Answers. Chapter 9 A92. Angles Theorem (Thm. 5.6) then XZY. Base Angles Theorem (Thm. 5.6) 5, 2. then WV WZ; 9 9. M, 0. M ( 9, 4) 7. If WZ XZ, then ZWX ZXW ; Base Angles Theorem (Thm..6). M 9,. M ( 4, ) 74. If XZ XY, then XZY Y; Base Angles Theorem (Thm..6). M, 4. M ( 9, ) 7. If V WZV, then WV WZ; Converse of

More information

Algebraic Expressions

Algebraic Expressions Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

More information

Mathematics SL. Mock Exam 2014 PAPER 2. Instructions: The use of graphing calculator is allowed.

Mathematics SL. Mock Exam 2014 PAPER 2. Instructions: The use of graphing calculator is allowed. Mock Exam 2014 Mathematics SL PAPER 2 Instructions: The use of graphing calculator is allowed Show working when possible (even when using a graphing calculator) Give your answers in exact form or round

More information

Course Updates. 2) This week: Electromagnetic Waves +

Course Updates.  2) This week: Electromagnetic Waves + Course Updates http://www.phys.hawaii.edu/~varner/phys272-spr1/physics272.html Reminders: 1) Assignment #11 due Wednesday 2) This week: Electromagnetic Waves + 3) In the home stretch [review schedule]

More information

Department of Systems Innovation, School of Engineering, The University of Tokyo 2011 Entrance Examination and Answer Sheets.

Department of Systems Innovation, School of Engineering, The University of Tokyo 2011 Entrance Examination and Answer Sheets. Examinee s No. Department of Systems Innovation, School of Engineering, The University of Tokyo 2011 Entrance Examination and Answer Sheets Mathematics Monday, August 30, 2010 13:00-15:30 Notice 1. Do

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 0 MEMORANDUM MARKS: 00 This memorandum consists of pages. Mathematics/P DBE/Feb. Mar. 0 QUESTION Time taken to complete task (in seconds)

More information

Shape Booster 6 Similar Shapes

Shape Booster 6 Similar Shapes Shape Booster 6 Similar Shapes Check: 85T) The two triangles are similar. 5cm y x 37.8cm 8cm 43.2cm a) Work out the size of x. b) Work out the size of y. a) x = 27cm b) y = 7cm Learn: Maths Watch Reference

More information

Intermediate Math Circles Wednesday October Problem Set 3

Intermediate Math Circles Wednesday October Problem Set 3 The CETRE for EDUCTI in MTHEMTICS and CMPUTIG Intermediate Math Circles Wednesday ctober 24 2012 Problem Set 3.. Unless otherwise stated, any point labelled is assumed to represent the centre of the circle.

More information

0811ge. Geometry Regents Exam BC, AT = 5, TB = 7, and AV = 10.

0811ge. Geometry Regents Exam BC, AT = 5, TB = 7, and AV = 10. 0811ge 1 The statement "x is a multiple of 3, and x is an even integer" is true when x is equal to 1) 9 2) 8 3) 3 4) 6 2 In the diagram below, ABC XYZ. 4 Pentagon PQRST has PQ parallel to TS. After a translation

More information

Here are some solutions to the sample problems assigned for Chapter 6.8 to 6.11.

Here are some solutions to the sample problems assigned for Chapter 6.8 to 6.11. Lecture 3 Appendi B: Some sample problems from Boas Here are some solutions to the sample problems assigned for Chapter 6.8 to 6.. 6.8: Solution: We want to practice doing closed line integrals of the

More information

Mathematics. Mock Paper. With. Blue Print of Original Paper. on Latest Pattern. Solution Visits:

Mathematics. Mock Paper. With. Blue Print of Original Paper. on Latest Pattern. Solution Visits: 10 th CBSE{SA I} Mathematics Mock Paper With Blue Print of Original Paper on Latest Pattern Solution Visits: www.pioneermathematics.com/latest_updates www.pioneermathematics.com S.C.O. - 36, Sector 40

More information

Distance. Warm Ups. Learning Objectives I can find the distance between two points. Football Problem: Bailey. Watson

Distance. Warm Ups. Learning Objectives I can find the distance between two points. Football Problem: Bailey. Watson Distance Warm Ups Learning Objectives I can find the distance between two points. Football Problem: Bailey Watson. Find the distance between the points (, ) and (4, 5). + 4 = c 9 + 6 = c 5 = c 5 = c. Using

More information

Unit 1 Packet Honors Math 2 1

Unit 1 Packet Honors Math 2 1 Unit 1 Packet Honors Math 2 1 Day 1 Homework Part 1 Graph the image of the figure using the transformation given and write the algebraic rule. translation: < 1, -2 > translation: < 0, 3 > Unit 1 Packet

More information

13. a. right triangular prism

13. a. right triangular prism C H A P T E R 0 SURFACE AREAS AND VOLUMES LESSON 0- pp. 56 568. Surface area measures the boundary of a -dimensional figure, while volume measures the interior.. volume. surface area. volume 5. surface

More information

TUTORIAL 7. Discussion of Quiz 2 Solution of Electrostatics part 1

TUTORIAL 7. Discussion of Quiz 2 Solution of Electrostatics part 1 TUTORIAL 7 Discussion of Quiz 2 Solution of Electrostatics part 1 Quiz 2 - Question 1! Postulations of Electrostatics %&''()(*+&,-$'.)/ : % (1)!! E # $$$$$$$$$$ & # (2)!" E # #! Static Electric field is

More information

MECE 3321 MECHANICS OF SOLIDS CHAPTER 1

MECE 3321 MECHANICS OF SOLIDS CHAPTER 1 MECE 3321 MECHANICS O SOLIDS CHAPTER 1 Samantha Ramirez, MSE WHAT IS MECHANICS O MATERIALS? Rigid Bodies Statics Dynamics Mechanics Deformable Bodies Solids/Mech. Of Materials luids 1 WHAT IS MECHANICS

More information

LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6.2 POLYNOMIAL OPERATIONS I LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order

More information

2012 GCSE Maths Tutor All Rights Reserved

2012 GCSE Maths Tutor All Rights Reserved 2012 GCSE Maths Tutor All Rights Reserved www.gcsemathstutor.com This book is under copyright to GCSE Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents angles

More information

Analytical Geometry- Common Core

Analytical Geometry- Common Core Analytical Geometry- Common Core 1. A B C is a dilation of triangle ABC by a scale factor of ½. The dilation is centered at the point ( 5, 5). Which statement below is true? A. AB = B C A B BC C. AB =

More information

ECE 3318 Applied Electricity and Magnetism Spring 2018 Homework #7

ECE 3318 Applied Electricity and Magnetism Spring 2018 Homework #7 EE 3318 Applied Electricity and Magnetism Spring 2018 Homework #7 Date assigned: Tuesday, March 6, 2018 Date due: Tuesday, March 20, 2018 Do Probs. 1, 2, and 7-12. (You are welcome to do the other problems

More information

NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).

NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system). EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles

More information

Unit 2 Boolean Algebra

Unit 2 Boolean Algebra Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2

More information

ENERGY IN ELECTROSTATICS

ENERGY IN ELECTROSTATICS ENERGY IN ELECTROSTATICS We now turn to the question of energy in electrostatics. The first question to consider is whether or not the force is conservative. You will recall from last semester that a conservative

More information

Created by T. Madas MIXED SURD QUESTIONS. Created by T. Madas

Created by T. Madas MIXED SURD QUESTIONS. Created by T. Madas MIXED SURD QUESTIONS Question 1 (**) Write each of the following expressions a single simplified surd a) 150 54 b) 21 7 C1A, 2 6, 3 7 Question 2 (**) Write each of the following surd expressions as simple

More information

Sage for Dynamical Systems

Sage for Dynamical Systems for Dynamical Systems Benjamin Hutz Department of Mathematics and Computer Science Saint Louis University December 5, 2015 RTG Workshop: Arithmetic Dynamics This work is supported by NSF grant DMS-1415294

More information

Operating C 1 C 1 C 2 and C 2 C 2 C 3, we get = 0, as R 1 and R 3 are identical. Ans: 0

Operating C 1 C 1 C 2 and C 2 C 2 C 3, we get = 0, as R 1 and R 3 are identical. Ans: 0 Q. Write the value of MATHEMATICS y y z z z y y y z z z y Operating R R + R, we get y z y z z y z y ( y z) z y Operating C C C and C C C, we get 0 0 0 0 ( y z) z y y ( y z)( ) z y y 0 0 0 0 = 0, as R and

More information

THE RING OF POLYNOMIALS. Special Products and Factoring

THE RING OF POLYNOMIALS. Special Products and Factoring THE RING OF POLYNOMIALS Special Products and Factoring Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely Special Products - rules

More information

Roll No. CLASS - VIII. 34 th ARYABHATTA INTER-SCHOOL MATHEMATICS COMPETITION Time Allowed: 2 Hours Max. Marks: 100 GENERAL INSTRUCTIONS:

Roll No. CLASS - VIII. 34 th ARYABHATTA INTER-SCHOOL MATHEMATICS COMPETITION Time Allowed: 2 Hours Max. Marks: 100 GENERAL INSTRUCTIONS: Roll No. A Please check that this questionnaire contains 15 printed pages. Code A, B or C given on the right hand top corner of the questionnaire should be written on the answer sheet in the space provided.

More information

visuteach.com PPTC Maths Sample Test

visuteach.com PPTC Maths Sample Test visuteach.com PPTC Maths Sample Test Contents Questions pages 2-3 Answers pages 4-6 Answer sheet page 7 Copy of answer sheet page 8 with correct answers filled in This is a multiple choice test. Answers

More information

11.1 Three-Dimensional Coordinate System

11.1 Three-Dimensional Coordinate System 11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into

More information

Lecture 5. Karnaugh-Map

Lecture 5. Karnaugh-Map Lecture 5 - Lecture 5 Karnaugh-Map Lecture 5-2 Karnaugh-Map Set Logic Venn Diagram K-map Lecture 5-3 K-Map for 2 Variables Lecture 5-4 K-Map for 3 Variables C C C Lecture 5-5 Logic Expression, Truth Table,

More information

Pre-AP Geometry 8-2 Study Guide: Trigonometric Ratios (pp ) Page! 1 of! 14

Pre-AP Geometry 8-2 Study Guide: Trigonometric Ratios (pp ) Page! 1 of! 14 Pre-AP Geometry 8-2 Study Guide: Trigonometric Ratios (pp 541-544) Page! 1 of! 14 Attendance Problems. Write each fraction as a decimal rounded to the nearest hundredths. 2 7 1.! 2.! 3 24 Solve each equation.

More information

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true? chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "

More information

10.1 Tangents to Circles. Geometry Mrs. Spitz Spring 2005

10.1 Tangents to Circles. Geometry Mrs. Spitz Spring 2005 10.1 Tangents to Circles Geometry Mrs. Spitz Spring 2005 Objectives/Assignment Identify segments and lines related to circles. Use properties of a tangent to a circle. Assignment: Chapter 10 Definitions

More information

HW Set #1: Problems #1-8 For #1-4, choose the best answer for each multiple choice question.

HW Set #1: Problems #1-8 For #1-4, choose the best answer for each multiple choice question. Geometry Homework Worksheets: Chapter 2 HW Set #1: Problems #1-8 For #1-4, choose the best answer for each multiple choice question. 1. Which of the following statements is/are always true? I. adjacent

More information

8-2 Trigonometric Ratios

8-2 Trigonometric Ratios 8-2 Trigonometric Ratios Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Write each fraction as a decimal rounded to the nearest hundredth. 1. 2. 0.67 0.29 Solve each equation. 3. 4. x = 7.25

More information

Esser et al. Crystal Structures of R. sphaeroides bc 1

Esser et al. Crystal Structures of R. sphaeroides bc 1 Esser et al. Crystal Structures of R. sphaeroides bc Supplementary Information Trivariate Gaussian Probability Analysis The superposition of six structures results in sextets of 3D coordinates for every

More information

NOVEMBER 2008 MATHEMATICS THIRD PAPER

NOVEMBER 2008 MATHEMATICS THIRD PAPER MATH ISEBE LEMFUNDO LEMPUMA KOLONI EASTERN CAPE EDUCATION DEPARTMENT OOS-KAAP ONDERWYSDEPARTEMENT IIMVIWO ZEBANGA LESHUMI ELINANYE GRADE 11-EXAMINATIONS GRAAD 11-EKSAMEN NOVEMBER 2008 MATHEMATICS THIRD

More information

Good Luck! Mlanie LaRoche-Boisvert - Electromagnetism Electromagnetism and Optics - Winter PH. Electromagnetism and Optics - Winter PH

Good Luck! Mlanie LaRoche-Boisvert - Electromagnetism Electromagnetism and Optics - Winter PH. Electromagnetism and Optics - Winter PH 1 2 Notes: 1. To submit a problem, just click the Submit button under it. The Submit All button is not necessary. 2. A problem accepted as correct by CAPA will be highlighted in green. Once you see this,

More information

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS 00-CE MTH HONG KONG EXMINTIONS UTHORITY HONG KONG CERTIFICTE OF EDUCTION EXMINTION 00 DDITIONL MTHEMTICS 8.0 am.00 am (½ hours) This paper must be answered in English. nswer LL questions in Section and

More information

LESSON 8.1 RATIONAL EXPRESSIONS I

LESSON 8.1 RATIONAL EXPRESSIONS I LESSON 8. RATIONAL EXPRESSIONS I LESSON 8. RATIONAL EXPRESSIONS I 7 OVERVIEW Here is what you'll learn in this lesson: Multiplying and Dividing a. Determining when a rational expression is undefined Almost

More information