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


 Clemence Marshall
 3 years ago
 Views:
Transcription
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 s Sketchpad. A cartesian coordinate representation and a number of fascinating applications of inversion are also presented. Definition Let O be the center of a fixed circle of radius r in the Euclidean plane. Let P be any point in the plane other than O. An inversion in circle C(O, r), I(O, r), is a function such that if I (O,r) (P ) = P then P OP and (OP )(OP ) = r 2. Here P is called the inverse of P, O is called the center of inversion, and r is called the radius of inversion, and r 2 is called its power. It follows from the above definition that to each point P of the plane, other than O, there corresponds a unique inverse point P. To make the inversion a transformation of the plane, we add to the plane a single ideal point Ω defined to be the inverse of the center of inversion. The point Ω is considered to lie on every line in the plane. 7.1 Constructing The Inverse of a Point: Given the circle of inversion C (O,r) and a point P, how do you construct the inverse of P? If P is inside the circle of inversion: (See Figure 7.1) Draw the ray OP. Draw a perpendicular to OP at P. This intersects the circle of inversion in two points, label one of them Q. Connect the center of the circle O to Q. Draw a perpendicular to OQ from Q. The intersection of this perpendicular with OP is P, the inverse of P. 52
2 Q O P P. Figure 7.1: Inverse of a Point Inside the Circle of Inversion Note that OP Q OQP. (OQ) 2 = r 2. Hence OQ OP = OP OQ. Therefore, OP OP = If P is outside the circle of inversion: (See Figure 7.2) Let M be the midpoint of the segment OP. Construct a circle centered at M of radius MO. It intersects the circle of inversion C (O,r) in two points, Q and R and goes through O and P. Construct the segment QR. The intersection of QR and OP is the point P, the inverse of P. Q... P O.... M P. R Figure 7.2: Inverse of a point Outside the Circle of Inversion Note that OP Q OQP. (OQ) 2 = r 2. Hence OQ OP = OP OQ. Therefore, OP OP = 53
3 Using Sketchpad: Inverting a Line: To invert a straight line in the circle of inversion C (O,r) follow the following steps: 1. Construct the circle of inversion C and the line l. 2. Construct an arbitrary point P on the line l. 3. Construct the ray OP. 4. Construct the intersection of OP and the circle, call it Q. 5. Construct the segment OP. 6. Mark O as the center of dilation. 7. Mark the ratio r. OP 8. Dilate Q by the ratio r centered at O. The image of Q is P the inverse OP image of P. 9. Hide everything except the circle, its center, the point determining its radius, P and the straight line l and the two point determining it. 10. Select everything and create a tool and call it invcirc. 11. Apply the tool a few times and use arc through three points to determine the image of the line. Inverting a Circle: To invert a circle C 1 in the circle of inversion C (O,r), replace l by C 1 in the steps above. 7.2 Inversion Using Coordinates: Theorem An inversion about x 2 + y 2 = r 2 is given by (x, y) (x, y xr 2 ) = ( x 2 + y, yr 2 2 x 2 + y ) 2 Proof. Since (x, y), (x, y ) and (0, 0) are collinear, we have y = y. Now x x d((0, 0), (x, y)) d((0, 0), (x, y )) = r 2, hence x 2 + y 2 x 2 + y 2 = r 2 and (x 2 +y 2 ) = r4 2 y2 x (x 2 + y 2 ) x 2 x 2 + y 2 (x 2 +y 2 ) = r 4. Hence x 2 = r4 x 2 + y 2 y 2 and x 2 = r4 y 2 (x 2 + y 2 ) x 2 + y 2 = r4 x 2 + y y2 x 2 2 x. Hence 2 x 2 + y2 x 2 = r4 x 2 x 2 + y and x 2 (x 2 + y 2 ) = r4 2 x 2 x 2 + y. Hence 2 x 2 = r 4 x 2 (x 2 + y 2 ) 2 and x = r2 x x 2 + y 2. Similarly, we can show that y = r2 y x 2 + y 2. Exercise 1: What is the image of (x 1) 2 + y 2 x 2 + y 2 = 1? = 1 under an inversion in 54
4 Y X Figure 7.3: Inverse of (x 1) 2 + y 2 = 1 in x 2 + y 2 = 1 Answer. Well, x = x and y = y. Hence x x = = 1 and x 2 +y 2 x 2 +y 2 x 2 +1 (x 1) 2 2 y = y. Hence the image of a circle going through the center of the circle of 2x inversion is a line going through the points of intersection of the two circles. Exercise 2: What is the image of x = 1 2 under an inversion in x2 + y 2 = 1? Y X Figure 7.4: Inverse of x = 1 2 in x2 + y 2 = 1 Answer: (x 1) 2 + y 2 = 1 Exercise: What is the image of x = 1 under an inversion in x 2 + y 2 = 1? Answer: (x 1/2) 2 + y 2 = 1/4. Exercise 3: What is the image of x 2 +2x+y 2 = 0 under an inversion in x 2 +y 2 = 1? Answer: x = 1/2. 55
5 Theorem If two circles are orthogonal, (their tangents at the points of intersection are perpendicular), and if a diameter AB of one circle meets the other circle in the points C and D, then OP 2 = OC OD. P A O C B O D Figure 7.5: Orthogonal Circles are Inverses Proof. OP O is a right triangle, hence (OP ) 2 + (P O ) 2 = (OO ) 2. But OO = OC + CO = OC + P O, hence (OP ) 2 + (P O ) 2 = (OC + P O ) 2. Hence (OP ) 2 = (OC) OC P O, which implies that (OP ) 2 = OC (OC + 2P O ). Hence (OP ) 2 = OC OD. Theorem A circle orthogonal to the circle of inversion inverts into itself, and, a circle through a pair of inverse points is orthogonal to the circle of inversion. T P O P A Figure 7.6: Orthogonal Circles are Inverses Proof. Given the circle of inversion C (O,r) and an orthogonal circle centered at A. Let T be one of the points of intersection of the two circles. Now if a line through O meets this orthogonal circle at P and P then OP OP = OT 2 = R 2. Hence P and P are inverse points. Theorem If P P and Q, Q are pairs of inverse points with respect to some circle C (O,r), Then P Q = P Q r2 OP OQ. Proof. If O, P, Q are noncollinear then, OP = OQ and P OQ = Q OP, OQ OP hence OP Q OQ P. Hence P Q = OQ. Hence P Q = OQ P Q OQ = r2 P Q. P Q OP OP OQ OP OQ 56
6 Q. Q... O P P Figure 7.7: P Q P Q = r2 OP OQ 7.3 Applications of Inversion: Given three noncoaxial concurrent circles, construct a circle C tangent to all three circles. Figure 7.8: A Circle Tangent To Three NonCoaxial Circles Solution: Invert the circles about a unit circle centered at the point of concurrency of the circles creating a triangle. Now construct the inscribed circle and invert this circle in the circle of inversion to create the required circle. Ptolemy s Theorem: In a cyclic convex quadrilateral, the product of the diagonals is equal to the sum of the products of the two pairs of opposite sides. Proof: Invert the circle and the convex quadrilateral about a circle centered at one of the vertices of the quadrilateral, say A. Now B D = B C + C D. Hence, r 2 BC AB AC + CD AC AD = BD AB AD 57 r2 r2
7 B B A C C D D Figure 7.9: Ptolemy s Theorem Hence, BC AD + CD AB = BD AC. Homework Find the image of the objects below under the specified inversion. (See Figures 7.10 and 7.11 ) 2. Prove that the inverse of the circumcircle C c of a triangle ABC with respect to the incircle C i, as a circle of inversion, is the nine point circle of the triangle XY Z determined by the points of contact of C i with the sides of ABC. 58
8 Figure 7.10: Inversion  HW 59
9 Figure 7.11: Inversion  HW 60
The Inversion Transformation
The Inversion Transformation A nonlinear transformation The transformations of the Euclidean plane that we have studied so far have all had the property that lines have been mapped to lines. Transformations
More informationChapter 3. Inversion and Applications to Ptolemy and Euler
Chapter 3. Inversion and Applications to Ptolemy and Euler 2 Power of a point with respect to a circle Let A be a point and C a circle (Figure 1). If A is outside C and T is a point of contact of a tangent
More informationDEFINITIONS. 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 informationAngles in a Circle and Cyclic Quadrilateral
130 Mathematics 19 Angles in a Circle and Cyclic Quadrilateral 19.1 INTRODUCTION You must have measured the angles between two straight lines, let us now study the angles made by arcs and chords in a circle
More informationChapter 6 Notes: Circles
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
More informationGeometry in a Nutshell
Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where
More informationChapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
More informationGeometry: Euclidean. Through a given external point there is at most one line parallel to a
Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,
More informationProjective Geometry. Stoienescu Paul. International Computer High School Of Bucharest,
Projective Geometry Stoienescu Paul International Computer High School Of Bucharest, paul98stoienescu@gmail.com Abstract. In this note, I will present some olympiad problems which can be solved using projective
More informationGeometry A Solutions. Written by Ante Qu
Geometry A Solutions Written by Ante Qu 1. [3] Three circles, with radii of 1, 1, and, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to
More informationProjective Geometry  Part 2
Projective Geometry  Part 2 Alexander Remorov alexanderrem@gmail.com Review Four collinear points A, B, C, D form a harmonic bundle (A, C; B, D) when CA : DA CB DB = 1. A pencil P (A, B, C, D) is the
More informationNCERT. Choose the correct answer from the given four options:
CIRCLES (A) Main Concepts and Results The meaning of a tangent and its point of contact on a circle. Tangent is perpendicular to the radius through the point of contact. Only two tangents can be drawn
More informationLesson 13: Proofs in Geometry
211 Lesson 13: Proofs in Geometry Beginning with this lesson and continuing for the next few lessons, we will explore the role of proofs and counterexamples in geometry. To begin, recall the Pythagorean
More informationExercise Set 3. Similar triangles. Parallel lines
Exercise Set 3. Similar triangles Parallel lines Note: The exercises marked with are more difficult and go beyond the course/examination requirements. (1) Let ABC be a triangle with AB = AC. Let D be an
More information1 Solution of Homework
Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,
More informationCasey s Theorem and its Applications
Casey s Theorem and its Applications Luis González Maracaibo. Venezuela July 011 Abstract. We present a proof of the generalized Ptolemy s theorem, also known as Casey s theorem and its applications in
More informationPower of a Point Solutions
ower of a oint Solutions Yufei Zhao Trinity ollege, ambridge yufei.zhao@gmail.com pril 2011 ractice problems: 1. Let Γ 1 and Γ 2 be two intersecting circles. Let a common tangent to Γ 1 and Γ 2 touch Γ
More informationIMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao.
ircles Yufei Zhao yufeiz@mit.edu 1 Warm up problems 1. Let and be two segments, and let lines and meet at X. Let the circumcircles of X and X meet again at O. Prove that triangles O and O are similar.
More informationInscribed (Cyclic) Quadrilaterals and Parallelograms
Lesson Summary: This lesson introduces students to the properties and relationships of inscribed quadrilaterals and parallelograms. Inscribed quadrilaterals are also called cyclic quadrilaterals. These
More information1. Determine all real numbers a, b, c, d that satisfy the following system of equations.
altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c
More informationAngle Bisectors in a Triangle
346/ ANGLE BISECTORS IN A TRIANGLE Angle Bisectors in a Triangle I. F. Sharygin In this article, we have collected some geometric facts which are directly or tangentially related to the angle bisectors
More informationClass10 th (X) Mathematics Chapter: Tangents to Circles
Class10 th (X) Mathematics Chapter: Tangents to Circles 1. Q. AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches
More informationQuadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid
Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More informationBASIC GEOMETRY GLOSSARY
BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that
More informationChapter 12. The Straight Line
302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,
More informationWhat is inversive geometry?
What is inversive geometry? Andrew Krieger July 18, 2013 Throughout, Greek letters (,,...) denote geometric objects like circles or lines; small Roman letters (a, b,... ) denote distances, and large Roman
More information1916 New South Wales Leaving Certificate
1916 New South Wales Leaving Certificate Typeset by AMSTEX New South Wales Department of Education PAPER I Time Allowed: Hours 1. Prove that the radical axes of three circles taken in pairs are concurrent.
More information206 MATHEMATICS CIRCLES
206 MATHEMATICS 10.1 Introduction 10 You have studied in Class IX that a circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). You have
More informationAre right spherical triangles wrong?
Are right spherical triangles wrong? Emily B. Dryden Bucknell University Bowdoin College April 13, 2010 Basic objects Unit sphere: set of points that are 1 unit from the origin in R 3 Straight lines great
More informationMath 531, Exam 1 Information.
Math 531, Exam 1 Information. 9/21/11, LC 310, 9:059:55. Exam 1 will be based on: Sections 1A  1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)
More informationABC is the triangle with vertices at points A, B and C
Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry  symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the
More informationLogic Rule 0 No unstated assumptions may be used in a proof. Logic Rule 1 Allowable justifications.
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and NonEuclidean Geometries, 4th Ed by Marvin Jay Greenberg (Revised: 18 Feb 2011) Logic Rule 0 No unstated assumptions may be
More informationCongruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key
Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More information61. Pascal s Hexagon Theorem.
. Pascal s Hexagon Theorem. Prove that the three points of intersection of the opposite sides of a hexagon inscribed in a conic section lie on a straight line. Hexagon has opposite sides,;, and,. Pascal
More informationIMO Geomety Problems. (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition:
IMO Geomety Problems (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points A and B in S, the perpendicular bisector
More informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationMA 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 informationThe measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures
8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 20132014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to
More informationAdvanced Euclidean Geometry
dvanced Euclidean Geometry What is the center of a triangle? ut what if the triangle is not equilateral?? Circumcenter Equally far from the vertices? P P Points are on the perpendicular bisector of a line
More informationChapter 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 informationThe Area of a Triangle Using Its Semiperimeter and the Radius of the Incircle: An Algebraic and Geometric Approach
The Area of a Triangle Using Its Semiperimeter and the Radius of the Incircle: An Algebraic and Geometric Approach Lesson Summary: This lesson is for more advanced geometry students. In this lesson,
More informationInversion in a Circle
Inversion in a Circle Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles March 23, 2011 Abstract This article will describe the geometric tool of inversion in a circle, and will demonstrate
More informationMath 3372College Geometry
Math 3372College Geometry Yi Wang, Ph.D., Assistant Professor Department of Mathematics Fairmont State University Fairmont, West Virginia Fall, 2004 Fairmont, West Virginia Copyright 2004, Yi Wang Contents
More informationSection 91. Basic Terms: Tangents, Arcs and Chords Homework Pages 330331: 118
Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,
More informationSTRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2
STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of xaxis in anticlockwise direction, then the value of tan θ is called the slope
More informationINVERSION WITH RESPECT TO A CIRCLE
INVERSION WITH RESPECT TO A CIRCLE OLGA RADKO (DEPARTMENT OF MATHEMATICS, UCLA) 1. Introduction: transformations of the plane Transformation of the plane is a rule which specifies where each of the points
More informationA Nice Theorem on Mixtilinear Incircles
A Nice Theorem on Mixtilinear Incircles Khakimboy Egamberganov Abstract There are three mixtilinear incircles and three mixtilinear excircles in an arbitrary triangle. In this paper, we will present many
More information4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.
14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right
More informationMA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
More information3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?
1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, coordinate geometry (which connects algebra
More informationChapter 6 Quiz. Section 6.1 Circles and Related Segments and Angles
Chapter 6 Quiz Section 6.1 Circles and Related Segments and Angles (1.) TRUE or FALSE: The center of a circle lies in the interior of the circle. For exercises 2 4, use the figure provided. (2.) In O,
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationThree Lemmas in Geometry
Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology yufei.zhao@gmail.com 1 iameter of incircle T Lemma 1. Let the incircle of triangle
More informationGeometry Unit 1. Basics of Geometry
Geometry Unit 1 Basics of Geometry Using inductive reasoning  Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture an unproven statement that is based
More informationEquation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1
Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows : gradient = vertical horizontal horizontal A B vertical
More informationwww.sakshieducation.com
LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c
More informationAssignment 3. Solutions. Problems. February 22.
Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.
More informationOrthic Quadrilaterals of a Convex Quadrilateral
Forum Geometricorum Volume 10 (2010) 79 91. FORUM GEOM ISSN 15341178 Orthic Quadrilaterals of a Convex Quadrilateral Maria Flavia Mammana, Biagio Micale, and Mario Pennisi Abstract. We introduce the orthic
More informationMATHEMATICS Grade 12 EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014
EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014 Checklist Make sure you learn proofs of the following theorems: The line drawn from the centre of a circle perpendicular to a chord bisects the chord The angle
More informationThe Euler Line in Hyperbolic Geometry
The Euler Line in Hyperbolic Geometry Jeffrey R. Klus Abstract In Euclidean geometry, the most commonly known system of geometry, a very interesting property has been proven to be common among all triangles.
More informationMathematics 3301001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3
Mathematics 3301001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3 The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs
More informationName Geometry Exam Review #1: Constructions and Vocab
Name Geometry Exam Review #1: Constructions and Vocab Copy an angle: 1. Place your compass on A, make any arc. Label the intersections of the arc and the sides of the angle B and C. 2. Compass on A, make
More informationChapter 1. The Medial Triangle
Chapter 1. The Medial Triangle 2 The triangle formed by joining the midpoints of the sides of a given triangle is called the medial triangle. Let A 1 B 1 C 1 be the medial triangle of the triangle ABC
More informationHigher Geometry Problems
Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
More information3.1 Triangles, Congruence Relations, SAS Hypothesis
Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)
More informationLemmas in Euclidean Geometry 1
Lemmas in uclidean Geometry 1 Yufei Zhao yufeiz@mit.edu 1. onstruction of the symmedian. Let be a triangle and Γ its circumcircle. Let the tangent to Γ at and meet at. Then coincides with a symmedian of.
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationSituation: Proving Quadrilaterals in the Coordinate Plane
Situation: Proving Quadrilaterals in the Coordinate Plane 1 Prepared at the University of Georgia EMAT 6500 Date Last Revised: 07/31/013 Michael Ferra Prompt A teacher in a high school Coordinate Algebra
More informationGeometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles
Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.
More informationCHAPTER 1. LINES AND PLANES IN SPACE
CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given
More informationof one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
2901 Clint Moore Road #319, Boca Raton, FL 33496 Office: (561) 4592058 Mobile: (949) 5108153 Email: HappyFunMathTutor@gmail.com www.happyfunmathtutor.com GEOMETRY THEORUMS AND POSTULATES GEOMETRY POSTULATES:
More informationTopics Covered on Geometry Placement Exam
Topics Covered on Geometry Placement Exam  Use segments and congruence  Use midpoint and distance formulas  Measure and classify angles  Describe angle pair relationships  Use parallel lines and transversals
More informationBC AB = AB. The first proportion is derived from similarity of the triangles BDA and ADC. These triangles are similar because
150 hapter 3. SIMILRITY 397. onstruct a triangle, given the ratio of its altitude to the base, the angle at the vertex, and the median drawn to one of its lateral sides 398. Into a given disk segment,
More informationChapter 1. Foundations of Geometry: Points, Lines, and Planes
Chapter 1 Foundations of Geometry: Points, Lines, and Planes Objectives(Goals) Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in
More information1. 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 informationProblems and Solutions, INMO2011
Problems and Solutions, INMO011 1. Let,, be points on the sides,, respectively of a triangle such that and. Prove that is equilateral. Solution 1: c ka kc b kb a Let ;. Note that +, and hence. Similarly,
More informationSMT 2014 Geometry Test Solutions February 15, 2014
SMT 014 Geometry Test Solutions February 15, 014 1. The coordinates of three vertices of a parallelogram are A(1, 1), B(, 4), and C( 5, 1). Compute the area of the parallelogram. Answer: 18 Solution: Note
More information1. Find the length of BC in the following triangles. It will help to first find the length of the segment marked X.
1 Find the length of BC in the following triangles It will help to first find the length of the segment marked X a: b: Given: the diagonals of parallelogram ABCD meet at point O The altitude OE divides
More informationMost popular response to
Class #33 Most popular response to What did the students want to prove? The angle bisectors of a square meet at a point. A square is a convex quadrilateral in which all sides are congruent and all angles
More informationGEOMETRY. 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: 14 Lesson 1.2: SWBAT: Use
More informationAngles 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 informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More information51 Reteaching ( ) Midsegments of Triangles
51 Reteaching Connecting the midpoints of two sides of a triangle creates a segment called a midsegment of the triangle. Point X is the midpoint of AB. Point Y is the midpoint of BC. Midsegments of Triangles
More information0810ge. Geometry Regents Exam 0810
0810ge 1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify
More informationIsosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors
Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 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 informationVisualizing Triangle Centers Using Geogebra
Visualizing Triangle Centers Using Geogebra Sanjay Gulati Shri Shankaracharya Vidyalaya, Hudco, Bhilai India http://mathematicsbhilai.blogspot.com/ sanjaybhil@gmail.com ABSTRACT. In this paper, we will
More informationOne of the most rewarding accomplishments of working with
Zhonghong Jiang and George E. O Brien & Proof & Mathe One of the most rewarding accomplishments of working with preservice secondary school mathematics teachers is helping them develop conceptually connected
More informationCollinearity and concurrence
Collinearity and concurrence PoShen Loh 23 June 2008 1 Warmup 1. Let I be the incenter of ABC. Let A be the midpoint of the arc BC of the circumcircle of ABC which does not contain A. Prove that the
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfdprep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationSIMSON S THEOREM MARY RIEGEL
SIMSON S THEOREM MARY RIEGEL Abstract. This paper is a presentation and discussion of several proofs of Simson s Theorem. Simson s Theorem is a statement about a specific type of line as related to a given
More informationSTRAIGHT LINE COORDINATE GEOMETRY
STRAIGHT LINE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) The points P and Q have coordinates ( 7,3 ) and ( 5,0), respectively. a) Determine an equation for the straight line PQ, giving the answer
More informationSolutions to Practice Problems
Higher Geometry Final Exam Tues Dec 11, 57:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles
More informationStraight Line. Paper 1 Section A. O xy
PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of
More informationInstitute of Mathematics University of the PhilippinesDiliman. 1. In, the sum of the measures of the three angles of a triangle is π.
Institute of Mathematics University of the Philippinesiliman Math 140: Introduction to Modern Geometries Second Exam MHX 14 ebruary 2008 Name: Student I..: I. illin the blanks. illup the blank in each
More informationSan Jose Math Circle April 25  May 2, 2009 ANGLE BISECTORS
San Jose Math Circle April 25  May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors
More informationAxiom A.1. Lines, planes and space are sets of points. Space contains all points.
73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach
More information