Collinear Points that lie on the same line.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Collinear Points that lie on the same line."

Transcription

1 Point - A point has no dimension. It is represented by a dot. It represents a fixed point in space. Plane - A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end. Through any three points not on the same line, there is exactly one plane. You can use three points that are not all on the same line to name a plane. Coplanar - Points that lie in the same plane. Points S, P, T and V are coplanar. They all lie in plane R. Line - A line has one dimension. It is represented by a line with two arrowheads, but it extends without end. Through any two points, there is EXACTLY one line. You can use any two points on a line to name it. Collinear Points that lie on the same line. Points S, P, and T are collinear. Points P and Q are also collinear. Endpoint A point that is the beginning or end of a segment or ray. Segment A piece of a line which consists of two endpoints. Written as AB or BA Ray A piece of a line which consists of one endpoint and extends indefinitely in the other direction. AB starts a point A and runs through point B BA starts a point B and runs through point A So BA and AB are different rays. 2

2 Opposite Rays Two rays that share a common endpoint and extend in opposite directions. Theorem - A rule that can be proved. CA and CB are opposite rays Postulate A rule that is accepted without proof. (also called axiom) Distance The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B. Distance of a segment is written as: AB or the length of AB Congruent Segments Line segments that have the same length. Construction A geometric drawing that uses a limited set of tools, usually and compass and a straightedge (a ruler without marks) or AB = CD Angle Two different rays with the same endpoint. This angle can be named BAC CAB A Point A is the vertex of the angle. (Segment) Bisector A point, ray, line, line segment, or plane that intersects the segment at its midpoint. Interior/Exterior of an Angle A point is in the interior of an angle if it is between points that lie on each side of the angle. Vertex The common endpoint of two different rays that form an angle. 3

3 Congruent Angles Two angles are congruent if they have the same measure. Vertical Angles Two angles whose sides form two pairs of opposite rays. Adjacent Angles Two angles that share a common vertex and side, but have no common interior points. Acute/Right/Obtuse/Straight Linear Pair Two adjacent angles are a linear pair if their noncommon sides are opposite rays. The angles in a linear pair are supplementary angles. Euclidian Geometry aka flat or parabolic geometry. Named after the greek mathematician Euclid. Geometry based on the assumption of a flat surface. Non-Euclidean Geometry Geometry that is not based on a flat surface. Some types of non-euclidean geometry: Hyperbolic Geometry Elliptic Geometry Complementary/Supplementary Complementary angles the sum of their measures is 90 Supplementary angles the sum of their measures is 180 4

4 1.1 Vocabulary Grid Write down a definition, example or sketch a picture of each word. After you have filled in as many as possible walk around the room and find a different person for each box to help you fill it in. As you fill out others sheet give an educated guess if you do not know a term. You have 8 minutes to get all the boxes filled. Point Intersection Segment Endpoint Line Coplanar Plane Ray Defined Term Opposite Ray Undefined Term Collinear 5

5 Section 1.1 Points, Lines, and Planes Ordered Pair (x,y) Quadrant II Quadrant I x-coordinate y-coordinate Quadrant III Quadrant IV Collinear Points: points that lie on the same line Name three collinear points: Name three points that are not collinear: Ex 1: Find the coordinates of three points that lie on the graph of y=3x+5 and one point that doesn t lie on the line x 3x + 5 y Ex 2: Do points R (4,8) and S (-3,-7) lie on the line with the equation y=2x-1? Undefined Terms No formal definition, but it is understood. The terms point, line, and plane are undefined terms. Point: An exact position in space; Represented by a dot 6

6 Line: Collection of points along a straight path extending infinitely in opposite directions. Plane: Flat surface extending infinitely in all directions. Coplanar: Points that lie in the same plane. Noncoplanar: Points that do not lie in the same plane. Practice: Defined Terms: Terms that can be described using known words such as point or line. Line Segment: Part of a line consisting of two endpoints and all points on the line between the endpoints. Ray: Part of a line with one endpoint that extends infinitely in the opposite direction. 7

7 Practice: Intersections Two or more geometric figures intersect if they have or more points in common. Perpendicular Lines: Two lines that intersect to form a right angle. (opposite reciprocal slope) Parallel Lines: Two coplanar lines that never intersect. (same slope) 8

8 Section 1.2 Segments & Congruence The length of a segment is the distance between its two endpoints. The length of ST can be written as Postulate/Axiom A rule that is accepted without proof Ruler Postulate: The numbers on a ruler are a real-life example of a number line. The distance between two points on a number line is the absolute value of the difference of the coordinates. Example 1: What is the distance from X to Y? X Y Distance from X to Y can be written as or When three points are collinear, you can say that one point is between the two others. Example 2: Find PQ, QR, and PR if P is located at -3, Q is located at 1, and R is located at 6. Segment Addition Postulate: 9

9 Example 3: Find LM if L is between N and M, NL = 6x-5, LM = 2x+3, and NM = 30 Example 4: Find the length of MN if N is between M and P, MN=3x+2, NP=18, MP=5x 10

10 Congruent Same value Congruent Segments: Line segments that have the same length. Example 5: Find AB if K is between A and B, AK=2x+10, KB=5x+4, and AK is congruent to KB. 11

11 You will need a compass and a ruler to do the following. Use the following segments to connect them on the dotted line provided: A B B C D M R T A D Add segments AB and BC : AB=, BC=, AC= + = Add segments AD and DM : AD=, DM=, AM= + = 12

12 Section 1.3 Midpoint and Distance Formulas (Day 1) Midpoint (of a segment) The point that divides the segment into two congruent segments. Segment Bisector A point, ray, line, line segment, or plane that intersects the segment at its midpoint. To determine the midpoint on a number line, add the two endpoints and divide by 2. Midpoint Formula: x 1 + x 2 2 Example 1 Midpoint X A Y B What is the coordinate of the midpoint of segment AB? What is the coordinate of the midpoint of segment XY? Example 2 Midpoint: Line RM bisects segment LK at point D. Find DK if LK is

13 Example 3 - Midpoint: Point M is the midpoint of segment VW. Find the length of segment VM. Example 4 - Midpoint: If P is the midpoint of segment CD, PC=x and PD=5x-4, find the value of x and the length of CD. Coordinate Plane Midpoint Formula: Example 5 Coordinate Plane Midpoint: The endpoint of segment RS are R(1,-3) and S(4,2). Find the midpoint. Example 6 Missing Endpoint: The midpoint of segment JK is M(2,1). One endpoint is J(1,4). Find the coordinates of endpoint K. 14

14 Coordinate Plane Distance Formula Example 7 - Distance: Find the length of RS if R(1,-3) and S(4,2). Example 8 - Distance: Determine if the two segments AB and LK are congruent. A(4,6) B(7,2) L(-1,-6) K(4,-6) 15

15 1.3 Using the Midpoint and Distance Formulas (Day 2) Midpoint Formula Distance Formula The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints. Find the midpoint and the distance for the following segments: 1. (2,3) and (2, 7) 2. ( 1,5) and (5,5) 3. (2,4) and (4,8) Find the midpoint and the distance for the following segments: 4. (4,5) and (9,17) 5. ( 2,8) and ( 6, 4) 6. (2,3) and (11,16) 16

16 Given the following endpoint and midpoint, find the missing endpoint and the distance: 7. Endpoint: (4,-4) 8. Endpoint: (-3, 0) 9. Endpoint: (4, 4) Midpoint: (1,-1) Midpoint: (0, 1) Midpoint (-1, -2) Endpoint Endpoint Endpoint Distance Distance Distance 17

17 Section 1.4 Measure and Classify Angles An is formed by two different rays with the same endpoint. Name angles with three letters. The vertex must be in the middle. None of the angles should be named < X, because all three angles have X as their vertex. Angles are measured in. If an angle is 32 degrees, it can be written two different ways: Angles are measured with a protractor. The image to the right demonstrates how a protractor is used. 18

18 Classifying Angles Angle Addition Postulate Example 1 Find Angle Measures: Given that find Example 2 - Find Angle Measures: 19

19 Congruent Angles Angles that have the same measure. Angle Bisector A ray that divides and angle into to congruent angles. Since < XYW and < ZYW are congruent, ray YW is the angle bisector of < XYZ. Example 3 Double an Angle Measure: Example 4 Challenge 20

20 Use the steps and copy the following segment: Use the steps above and copy the following angle right next to it not under: 21

21 Section 1.5 Angle Pair Relationships Adjacent Angles Two angles that share a common vertex and side. Complementary Angles Two angles whose sum is 90 Supplementary Angles Two angles whose sum is 180 Sum is 90 Sum is 180 Example 1: Find the measure of the complement and supplement of 1. m 1 = 22 Example 2: What are the measures of angles 2 and 3? 22

22 Example 3: Linear Pair Vertical Angles Vertical angles are always congruent!! Example 4: FGH and HGJ form a linear pair. Find the measures of the angles if m FHG=(12x+1) and m HGJ=(4x-9). (Draw a picture to help you) Example 5: Find the values of x and y. 23

23 1.5 Class Practice 1) Tell whether the angles are supplementary, complementary, vertical, bisecting. For each identify if adjacent or not. 2) Find m 2=, 3= 3) Solve for x and y: Equation: Equation: Equation: 4) Use one of the learned concepts ( Complementary, supplementary, vertical, angle addition), set up an equation and solve it. 24

24 Section 1.6 Classify Polygons Polygon A closed plane figure with the following properties: 1. It is formed by three or more sides (have to be straight) 2. No lines in the interior of the shape Vertex (plural: vertices) A polygon can be named by listing the vertices in consecutive order. Two possible names for this polygon are ABCDE and Convex and Concave All interior angles are At least one 25

25 Example 1: Naming Polygons A polygon is named by the number of sides. 26

26 Equilateral All sides congruent Equiangular All angles congruent Regular both equilateral and equiangular Example 2: Example 3: Find the value of x in the regular polygon. Section 1.7 Find Perimeter, Circumference and Area Perimeter: Distance around a figure Circumference: Distance around a circle Area: Amount of surface covered by a figure 27

27 Example 1: Find the perimeter and area of the rectangular basketball court shown. Example 2: Find the approximate circumference and area of the patch shown. 28

28 Example 3: The base of a triangle is 28 meters. Its area is 308 square meters. Find the height of the triangle. Example 4: Find the area and perimeter of the triangle. (Use distance formula to find the length of each side) 1.7 Class Practice 1) 29

29 Use the distance (length) formula to find out the length of each one QR, QS, and RS QR= QS= RS= Perimeter= 2) Read this solved problem and solve the next one in a similar way: 30

30 h= 3) a) Circumference rend ring= Circumference yellow ring= b) --This Page is intentionally left blank 31

31 --This Page is intentionally left blank-- 32

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

More information

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle. Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.

More information

Final Review Geometry A Fall Semester

Final Review Geometry A Fall Semester Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over

More information

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures. Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.

More information

1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above? 1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

More information

GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd: GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use

More information

1.1 Identify Points, Lines, and Planes

1.1 Identify Points, Lines, and Planes 1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms - These words do not have formal definitions, but there is agreement aboutwhat they mean.

More information

Mathematics Geometry Unit 1 (SAMPLE)

Mathematics Geometry Unit 1 (SAMPLE) Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This

More information

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle. DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

More information

Geometry Review Flash Cards

Geometry Review Flash Cards point is like a star in the night sky. However, unlike stars, geometric points have no size. Think of them as being so small that they take up zero amount of space. point may be represented by a dot on

More information

Quadrilaterals GETTING READY FOR INSTRUCTION

Quadrilaterals GETTING READY FOR INSTRUCTION Quadrilaterals / Mathematics Unit: 11 Lesson: 01 Duration: 7 days Lesson Synopsis: In this lesson students explore properties of quadrilaterals in a variety of ways including concrete modeling, patty paper

More information

Incenter Circumcenter

Incenter Circumcenter TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles that are between parallel lines, but on opposite sides of a transversal. GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

More information

The Geometry of Piles of Salt Thinking Deeply About Simple Things

The Geometry of Piles of Salt Thinking Deeply About Simple Things The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word

More information

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades. Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

More information

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

More information

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points. 6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

More information

INTRODUCTION TO EUCLID S GEOMETRY

INTRODUCTION TO EUCLID S GEOMETRY 78 MATHEMATICS INTRODUCTION TO EUCLID S GEOMETRY CHAPTER 5 5.1 Introduction The word geometry comes form the Greek words geo, meaning the earth, and metrein, meaning to measure. Geometry appears to have

More information

Show all work for credit. Attach paper as needed to keep work neat & organized.

Show all work for credit. Attach paper as needed to keep work neat & organized. Geometry Semester 1 Review Part 2 Name Show all work for credit. Attach paper as needed to keep work neat & organized. Determine the reflectional (# of lines and draw them in) and rotational symmetry (order

More information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

Elements of Plane Geometry by LK

Elements of Plane Geometry by LK Elements of Plane Geometry by LK These are notes indicating just some bare essentials of plane geometry and some problems to think about. We give a modified version of the axioms for Euclidean Geometry

More information

CHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.

CHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder. TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has

More information

Lesson 2: Circles, Chords, Diameters, and Their Relationships

Lesson 2: Circles, Chords, Diameters, and Their Relationships Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct

More information

15. Appendix 1: List of Definitions

15. Appendix 1: List of Definitions page 321 15. Appendix 1: List of Definitions Definition 1: Interpretation of an axiom system (page 12) Suppose that an axiom system consists of the following four things an undefined object of one type,

More information

Line Segments, Rays, and Lines

Line Segments, Rays, and Lines HOME LINK Line Segments, Rays, and Lines Family Note Help your child match each name below with the correct drawing of a line, ray, or line segment. Then observe as your child uses a straightedge to draw

More information

Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

More information

Geometry - Semester 2. Mrs. Day-Blattner 1/20/2016

Geometry - Semester 2. Mrs. Day-Blattner 1/20/2016 Geometry - Semester 2 Mrs. Day-Blattner 1/20/2016 Agenda 1/20/2016 1) 20 Question Quiz - 20 minutes 2) Jan 15 homework - self-corrections 3) Spot check sheet Thales Theorem - add to your response 4) Finding

More information

Rays and Angles Examples

Rays and Angles Examples Rays and Angles Examples 1. In geometry, an angle is defined in terms of two rays that form the angle. You can think of a ray as a segment that is extended indefinitely in one direction. Rays have exactly

More information

Cabri Geometry Application User Guide

Cabri Geometry Application User Guide Cabri Geometry Application User Guide Preview of Geometry... 2 Learning the Basics... 3 Managing File Operations... 12 Setting Application Preferences... 14 Selecting and Moving Objects... 17 Deleting

More information

CIRCUMFERENCE AND AREA OF A CIRCLE

CIRCUMFERENCE AND AREA OF A CIRCLE CIRCUMFERENCE AND AREA OF A CIRCLE 1. AC and BD are two perpendicular diameters of a circle with centre O. If AC = 16 cm, calculate the area and perimeter of the shaded part. (Take = 3.14) 2. In the given

More information

MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry. Figure 1: Lines in the Poincaré Disk Model

MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry. Figure 1: Lines in the Poincaré Disk Model MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry Put your name here: Score: Instructions: For this lab you will be using the applet, NonEuclid, created by Castellanos, Austin, Darnell,

More information

a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a

a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a Bellwork a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.

More information

Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain

Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent

More information

Mathematics Notes for Class 12 chapter 10. Vector Algebra

Mathematics Notes for Class 12 chapter 10. Vector Algebra 1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

More information

Geo, Chap 4 Practice Test, EV Ver 1

Geo, Chap 4 Practice Test, EV Ver 1 Class: Date: Geo, Chap 4 Practice Test, EV Ver 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (4-3) In each pair of triangles, parts are congruent as

More information

GEOMETRIC FIGURES, AREAS, AND VOLUMES

GEOMETRIC FIGURES, AREAS, AND VOLUMES HPTER GEOMETRI FIGURES, RES, N VOLUMES carpenter is building a deck on the back of a house. s he works, he follows a plan that he made in the form of a drawing or blueprint. His blueprint is a model of

More information

Lesson 17. Introduction to Geometry. Objectives

Lesson 17. Introduction to Geometry. Objectives Student Name: Date: Contact Person Name: Phone Number: Lesson 17 Introduction to Geometry Objectives Understand the definitions of points, lines, rays, line segments Classify angles and certain relationships

More information

Advanced GMAT Math Questions

Advanced GMAT Math Questions Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of

More information

Basic Geometry Review For Trigonometry Students. 16 June 2010 Ventura College Mathematics Department 1

Basic Geometry Review For Trigonometry Students. 16 June 2010 Ventura College Mathematics Department 1 Basic Geometry Review For Trigonometry Students 16 June 2010 Ventura College Mathematics Department 1 Undefined Geometric Terms Point A Line AB Plane ABC 16 June 2010 Ventura College Mathematics Department

More information

PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

More information

@12 @1. G5 definition s. G1 Little devils. G3 false proofs. G2 sketches. G1 Little devils. G3 definition s. G5 examples and counters

@12 @1. G5 definition s. G1 Little devils. G3 false proofs. G2 sketches. G1 Little devils. G3 definition s. G5 examples and counters Class #31 @12 @1 G1 Little devils G2 False proofs G3 definition s G4 sketches G5 examples and counters G1 Little devils G2 sketches G3 false proofs G4 examples and counters G5 definition s Jacob Amanda

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

Geometry Notes PERIMETER AND AREA

Geometry Notes PERIMETER AND AREA Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

More information

2009 Chicago Area All-Star Math Team Tryouts Solutions

2009 Chicago Area All-Star Math Team Tryouts Solutions 1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total

More information

Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

More information

Tutorial 1: The Freehand Tools

Tutorial 1: The Freehand Tools UNC Charlotte Tutorial 1: The Freehand Tools In this tutorial you ll learn how to draw and construct geometric figures using Sketchpad s freehand construction tools. You ll also learn how to undo your

More information

SURFACE AREA AND VOLUME

SURFACE AREA AND VOLUME SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has

More information

Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1) Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

More information

Common Core State Standards for Mathematics Accelerated 7th Grade

Common Core State Standards for Mathematics Accelerated 7th Grade A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting

More information

Questions. Strategies August/September Number Theory. What is meant by a number being evenly divisible by another number?

Questions. Strategies August/September Number Theory. What is meant by a number being evenly divisible by another number? Content Skills Essential August/September Number Theory Identify factors List multiples of whole numbers Classify prime and composite numbers Analyze the rules of divisibility What is meant by a number

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155 Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

More information

Lesson 1 Section 2.5 Angle Relationships

Lesson 1 Section 2.5 Angle Relationships Creator: Heather McNeill Grade: 10 th grade Course: Geometry Honors Length: 50 minutes Lesson 1 Section 2.5 Angle Relationships 1. Prior Knowledge, Skills, and Dispositions: In this lesson, students should

More information

Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A. Tuesday, August 13, 2002 8:30 to 11:30 a.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A. Tuesday, August 13, 2002 8:30 to 11:30 a.m. The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A Tuesday, August 13, 2002 8:30 to 11:30 a.m., only Print Your Name: Print Your School s Name: Print your name and the

More information

The GED math test gives you a page of math formulas that

The GED math test gives you a page of math formulas that Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding

More information

NEW MEXICO Grade 6 MATHEMATICS STANDARDS

NEW MEXICO Grade 6 MATHEMATICS STANDARDS PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

More information

Perimeter. 14ft. 5ft. 11ft.

Perimeter. 14ft. 5ft. 11ft. Perimeter The perimeter of a geometric figure is the distance around the figure. The perimeter could be thought of as walking around the figure while keeping track of the distance traveled. To determine

More information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

More information

Sample Test Questions

Sample Test Questions mathematics College Algebra Geometry Trigonometry Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about

More information

Surface Area and Volume Cylinders, Cones, and Spheres

Surface Area and Volume Cylinders, Cones, and Spheres Surface Area and Volume Cylinders, Cones, and Spheres Michael Fauteux Rosamaria Zapata CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable

More information

Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes

Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical

More information

Objectives. Cabri Jr. Tools

Objectives. Cabri Jr. Tools ^Åíáîáíó=NO Objectives To learn how to construct all types of triangles using the Cabri Jr. application To reinforce the difference between a construction and a drawing Cabri Jr. Tools fåíêççìåíáçå `çåëíêìåíáåö

More information

C relative to O being abc,, respectively, then b a c.

C relative to O being abc,, respectively, then b a c. 2 EP-Program - Strisuksa School - Roi-et Math : Vectors Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 2. Vectors A

More information

Quick Reference ebook

Quick Reference ebook This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

More information

Using the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º.

Using the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º. Using the Quadrant Eye Piece Protractor Handle You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements 90º. Plumb Bob ø

More information

12 Surface Area and Volume

12 Surface Area and Volume 12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

More information

Lateral and Surface Area of Right Prisms

Lateral and Surface Area of Right Prisms CHAPTER A Lateral and Surface Area of Right Prisms c GOAL Calculate lateral area and surface area of right prisms. You will need a ruler a calculator Learn about the Math A prism is a polyhedron (solid

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

Illinois State Standards Alignments Grades Three through Eleven

Illinois State Standards Alignments Grades Three through Eleven Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other

More information

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50 Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

More information

Study Guide Workbook. Contents Include: 96 worksheets one for each lesson

Study Guide Workbook. Contents Include: 96 worksheets one for each lesson Workbook Contents Include: 96 worksheets one for each lesson To the Teacher: Answers to each worksheet are found in Glencoe s Geometry: Concepts and Applications Chapter Resource Masters and also in the

More information

Geometric Patterns. Introduction. Geometric Patterns 1 ~ Border Patterns. Geometric Patterns 2 ~ Tile Patterns

Geometric Patterns. Introduction. Geometric Patterns 1 ~ Border Patterns. Geometric Patterns 2 ~ Tile Patterns Geometric Patterns Geometric Patterns Introduction In working with shapes, whether mathematically or artistically, it is necessary to have a good feeling for how more complex shapes can be made from simpler

More information

CHAPTER FIVE. 5. Equations of Lines in R 3

CHAPTER FIVE. 5. Equations of Lines in R 3 118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

More information

GEOMETRY (Common Core)

GEOMETRY (Common Core) GEOMETRY (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY (Common Core) Thursday, January 28, 2016 9:15 a.m. to 12:15 p.m., only Student Name: School Name:

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Thursday, August 16, 2012 8:30 to 11:30 a.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Thursday, August 16, 2012 8:30 to 11:30 a.m. INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name

More information

Review Sheet for Test 1

Review Sheet for Test 1 Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And

More information

GeoGebra. 10 lessons. Gerrit Stols

GeoGebra. 10 lessons. Gerrit Stols GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter

More information

What You ll Learn. Why It s Important

What You ll Learn. Why It s Important These students are setting up a tent. How do the students know how to set up the tent? How is the shape of the tent created? How could students find the amount of material needed to make the tent? Why

More information

Gymnázium, Brno, Slovanské nám. 7, SCHEME OF WORK Mathematics SCHEME OF WORK. http://agb.gymnaslo. cz

Gymnázium, Brno, Slovanské nám. 7, SCHEME OF WORK Mathematics SCHEME OF WORK. http://agb.gymnaslo. cz SCHEME OF WORK Subject: Mathematics Year: Third grade, 3.X School year:../ List of topics Topics Time period 1. Revision (functions, plane geometry) September 2. Constructive geometry in the plane October

More information

Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

More information

NON-EUCLIDEAN GEOMETRIES

NON-EUCLIDEAN GEOMETRIES Chapter 3 NON-EUCLIDEN GEOMETRIES In the previous chapter we began by adding Euclid s Fifth Postulate to his five common notions and first four postulates. This produced the familiar geometry of the Euclidean

More information

2014 Chapter Competition Solutions

2014 Chapter Competition Solutions 2014 Chapter Competition Solutions Are you wondering how we could have possibly thought that a Mathlete would be able to answer a particular Sprint Round problem without a calculator? Are you wondering

More information

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

More information

Factoring, Solving. Equations, and Problem Solving REVISED PAGES

Factoring, Solving. Equations, and Problem Solving REVISED PAGES 05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

More information

Name Date Class G 4 J 4. 14. If f(x) = 3x 5, what is f( 2)? F 6x + 10 H 3x 7

Name Date Class G 4 J 4. 14. If f(x) = 3x 5, what is f( 2)? F 6x + 10 H 3x 7 1 Choose the best answer. 1. Evaluate 6(18 4). 4 A 21 C 26 B 23 D 102 2. Evaluate the expression 3(4a 5) b for a = 6 and b = 3. F 54 H 64 G 60 J 70 3. Evaluate 3 r for r = 10. A 13 C 7 B 7 D 13 4. In which

More information

Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade)

Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade) Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade) Teacher: School Phone: Email: Kim Schnakenberg 402-443- 3101 kschnakenberg@esu2.org Course Descriptions: Both Concept and Application

More information

Solutions Manual for How to Read and Do Proofs

Solutions Manual for How to Read and Do Proofs Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve

More information

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

More information

Section 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.

Section 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t. . The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

angle attribute Figure 1 4 right angles opposite sides parallel Lesson 14 5 Lesson 14 4 Vocab

angle attribute Figure 1 4 right angles opposite sides parallel Lesson 14 5 Lesson 14 4 Vocab Vocab angle attribute 006_055_G4_VisV_101741.indd Page F46 2/13/10 11:27:36 PM u-s014 /Volumes/114/GO00403/GO00403_Natl_Visual_Voca_Cards_G4%0 Figure 1 4 right angles opposite sides parallel endpoint hexagon

More information

A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

More information

Support Materials for Core Content for Assessment. Mathematics

Support Materials for Core Content for Assessment. Mathematics Support Materials for Core Content for Assessment Version 4.1 Mathematics August 2007 Kentucky Department of Education Introduction to Depth of Knowledge (DOK) - Based on Norman Webb s Model (Karin Hess,

More information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

SAT Subject Math Level 2 Facts & Formulas

SAT Subject Math Level 2 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

Two vectors are equal if they have the same length and direction. They do not

Two vectors are equal if they have the same length and direction. They do not Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

More information