# Selected practice exam solutions (part 5, item 2) (MAT 360)

Size: px
Start display at page:

Transcription

1 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, On one side of an angle A, the segments AB and AC are marked, and on the other side the segments AB = AB and AC = AC. Prove that the lines BC and B C meet on the bisector of the angle A. The reflection in the bisector exchanges the rays forming the angle. Since the reflection sends A to A, the congruence AB = AB implies that B is sent to B and the congruence AC = AC implies that B is sent to B. The line B C is sent to the line BC, and vice-versa. The intersection point of B C/BC, call it O, is sent to the intersection point of BC /B C, which is the same point. Then O lies on the line of symmetry of the reflection, the bisector of the angle A. 91. A median drawn to a side of a triangle is smaller than the semisum of the other two sides. The triangle has sides a, b, c, and the median bisects c. Following the hint, double the median by prolonging it past the side it bisects. The segments drawn from the new endpoint to the original vertices are also a and b by the Side Angle Side congruence test. Then by the triangle inequality, m < a + b m < (a + b)/ 9. The sum of the medians of a triangle is smaller than its perimeter but greater than its semi-perimeter. According to exercise 90 (homework ), each median of a triangle is smaller than its semiperimeter. By addition, we know that the sum of the medians is smaller than 3/ of the perimeter, but according to this exercise this sum is actually even smaller than the perimeter itself. This follows by adding up the 3 inequalities in exercise 91. Let x, y, z be the medians cutting sides a, b, c. Then 1

2 x < b + c y < a + c z < a + b x + y + z < a + b + c This is an upper bound. For the lower bound: Again let the medians be x, y, z corresponding to sides a, b, c. For each median, select a side of the original triangle which is smaller (by choosing the side across from the acute angle formed by the median at the point of bisection). The list of 3 sides obtained this way cannot be the full list {a, b, c}, since then the sum of the medians would be larger than the perimeter. Then at least one side is missing from this list; suppose that c is missing. x + y + z < a + a + b = a + (a + b) < a + c = a + c + a + c < a + c + b = a + b + c 94. The sum of segments connecting a point inside a triangle with its vertices is less than the semiperimeter of the triangle. Let x, y, z be the segments emanating toward sides a, b, c. According to the triangle inequality, x + y < c x + z < b y + z < a By addition, (x + y + z) < a + b + c, that is x + y + z < (a + b + c)/. 95. Given an acute angle XOY and an interior point A. Find a point B on the side OX and a point C on the side OY such that the perimeter of the triangle ABC is minimal. The construction is exercise 19. For an illustration, see the solution page for exercises First, let s see that if ABC is minimal, then OB = OC. If not, assume OB < OC, and let B and C be the reflections in the bisector of angle XOY. Note that B and C are both on the same side as O, the same side of the perpendiculars from A to the angle sides; otherwise all 3 sides of triangle ABC can be decreased by selecting new B and C closer to O. Then angle OBA = OB A is obtuse. According to the inequalities related to slants, sides BC = B C and BA = B A will decrease if new B and B are selected between the old B, B and C, C. The third side of the triangles ABC and AB C will remain the same. This is a contradiction, so OB = OC for any minimal solution.

3 (Rest of proof: Prove that OB = OC and ABC minimal perimeter implies that angles OAB and OAC are congruent to the original angle XOY?) 103. A line and a circle can have at most common points. Suppose that a line and a circle both contain 3 distinct points X, Y, Z, and assume that Y is between X and Z. Let C be the center of the circle. The triangles XCY, XCZ, Y CZ are all isosceles with vertex C, so the angles CY X and CY Z are congruent. That means CY X and CY Z are both half a straight angle, and therefore all the angles formed at the points X, Y, Z between the given line and the ray toward C are right. No triangle has right angles. This proves that a line and a circle can commonly contain no more than points Find the geometric locus of vertices A of triangles ABC with the given base BC and B > C. This exercise appears in homework 3. Pick a point A not on the line BC. If A lies on the perpendicular bisector of BC, then triangle ABC is isosceles, so angle B and angle C are equal. Otherwise, A lies on one or the other side of the perpendicular bisector. If it lies on the same side as B, then it lies on the opposite side from C, so line AC meets the perpendicular at a point X between A and C. BX and CX are congruent, since points along the perpendicular bisector of a segment are equidistant from the endpoints. Angle XBC is contained in angle ABC, so it is lesser. But angle XBC is congruent to ACB, by Side Angle Side triangle congruence XBC = XCB. So angle B is greater than angle C. Similarly, were A picked on the other side of the perpendicular, the roles would be reversed, and angle C would be greater than B. Therefore the geometric locus in question is all points, not lying on the line BC or its perpendicular bisector, lying on the same side of this bisector as C Divide the plane by infinite straight lines into five parts, using as few lines as possible. The plane can be divided in n + 1 parts by n parallel lines, so we can divide it in 5 parts with 4 parallel lines. Can we do it with fewer lines? No. Consider a configuration with 3 or fewer lines. If all are parallel, there are only 3 regions formed. Then some pair intersect. If there are only these lines, then 4 regions are formed (which is not 5). Then there must be a third line. If this third line is parallel to one or the other of the first two, then 6 regions are formed. Then, instead, the third line must be skew to both other lines. If the two resulting intersection points are distinct from each other and from the remaining intersection, then these 3 intersection points cannot be collinear; if they were, all lines would be the same. Instead, these 3 points are non-collinear, and hence they form a triangle, and in this case = 7 regions are formed Prove that in a convex polygon, one of the angles between the bisectors of two consecutive angles is congruent to the semisum of these two angles. 3

4 (178,179,180,181 this is really one problem) 178. Midpoints of the sides of a quadrilateral are the vertices of a parallelogram. Determine under what conditions this parallelogram will be (a) a rectangle (b) a rhombus (c) a square In a right triangle, the median to a hypotenuse is congruent to a half of it Conversely, if a median is congruent to a half of the side it bisects, then the triangle is right In a right triangle, the median and the altitude drawn to the hypotenuse make an angle congruent to the difference of the acute angles. There are two basic facts. First, the altitude of a right trangle drawn to the hypotenuse divides the right angle into the same angles as the acute angles of the original triangle. This is because two new right triangles are formed, and right triangles have complementary acute angles. Second, the median of a right triangle divides the triangle into isosceles triangles whose bases are the original legs. For, consider one of these new triangles, T. Its altitude dropped from the original hypotenuse midpoint is parallel to a leg of the original triangle. Parallel lines cutting out congruent segments on one line cut out congruent segments on all lines, so the foot of this altitude divides a leg of the original triangle in half. By the Side Angle Side congruence theorem, this altitude divides T into congruent triangles, making T isosceles. By inspecting which two sides are congruent as a result, this already proves 179. For 180, the converse: If the median is congruent to half of the side it bisects, it divides the triangle into two isosceles triangles T and T. The vertex angles of are supplementary, which implies that the base angles are complementary, which implies that the triangle is right. Combining these two facts, the angle in question between the median and altitude drawn to the altitude of a right triangle is contained in an angle congruent to one of the acute angles A of the right triangle, such that the remainined angle is congruent to the other acute angle B; it is congruent to the difference of these angles. This establishes 181. Finally for 178: Consider the parallelogram formed from the midpoints of a quadrilateral. Recall the classification of quadrilaterals (homework 3) by number of axes of symmetry. This classification can be refined by number and degree of rotational symmetries, or other properties (e.g. convexity). In this case the crudest classification, by number of axes and angle type, gives the whole story: 4 axes of symmetry (square): In this case the associated parallelogram is evidently a square; indeed, the associated parallelogram will have at least as many axes of symmetry as the original quadrilateral, and the only quadrilateral having 4 axes of symmetry is the square. axes of symmetry (rectangle or rhombus): In the case of a rectangle or a rhombus, the diagonals of the associated parallelogram are congruent to the original sides. Use the previous results for the triangles formed by casting segments from the diagonals intersection to the original vertices: in the rectangle case, since half of one side is not congruent to half of the other side, the angles of the associated parallelogram are not right. Since this parallelogram nevertheless has axes of symmetry, it is a rhombus. Similarly, the associated parallelogram of a rhombus is a rectangle. 1 axis of symmetry ( kite or special trapezoid ): The associated parallelogram has an axis of symmetry, so it is therefore a rectangle or a rhombus. Indeed, in the kite case, the axis 4

5 of symmetry passes through original vertices, so the associated parallelogram has an axis of symmetry not passing through the vertices; it is a rectangle. In the special trapezoid case, the line of symmetry passes through two midpoints, and hence through two vertices of the associated parallelogram, which is therefore a rhombus says, in an isosceles triangle, the sum of the distances from each point of the base to the lateral sides is constant, namely it is congruent to the altitude dropped to a lateral side. How does this theorem change if the points on the extension of the base are taken instead? If the point of the base lies on the part of the base extension which is opposite the side x of the isosceles triangle, then what is constant now is the the distance to the nearer lateral side extension subtracted from the distance to the side x. This is proved by reproducing the isosceles triangle rotated a complete straight angle around a vertex on the base; the perpendiculars dropped from the base point to the far side and to it s corresponding side on the rotated triangle will coincide. Then the two distances from the base point along this perpendicular will sum to a constant, namely twice an altitude of the original triangle. We combine this fact with the exercise 187, applied to both triangles, and we have the result From the intersection point of the diagonals of a rhombus, perpendiculars are dropped to the sides of the rhombus. Prove that the feet of these perpendiculars are vertices of a rectangle. The diagonals of a rhombus are lines of symmetry of the rhombus. Therefore they are lines of symmetry of the altitudes dropped from the diagonal intersection to the sides. Similarly, they are lines of symmetry of the quadrilateral formed from the resulting 4 feet. Since the original diagonals are perpendicular, and they do not contain the 4 feet, the quadrilateral formed must be a rectangle (by the classification of quadrilaterals, e.g. as in homework 3) Bisectors of the angles of a rectangle cut out a square. The bisectors of the two angles at a smaller side of a rectangle meet at a right angle, since this point is the vertex of an isosceles triangle with half-right base angles. Doing this for both smaller sides, we obtain congruent isosceles right triangles. Translating them to have a common hypotenuse, the resulting quadrilateral has equal sides and all angles are right; it is a square. The words cut out probably refer to the fact that the triangles constructed above lie inside the rectangle, whereas they would not if we used the larger sides instead. This fact require proof, but it also requires more precise definitions of lie inside. 5

6 195. Let A, B, C, and D be the midpoints of the sides CD, DA, AB, and BC of a square. Prove that the segments AA, CC, DD, and BB cut out a square, whose sides are congruent to /5th of any of the segments. We are supposed to think of the smaller, triangular pieces being excised; we are to show that the remaining piece in the center is a square of side /5 of the cutting segments. The shape is evidently a quadrilateral, which is rotationally symmetric of order 4. Therefore it is a square. Indeed, the picture suggests a clear argument. Looking at the large diagonal, we should show that the lower left segment, the upper right segment, and the middle segment joining two smaller segments are all congruent to the square side. Then of the cut segments are congruent to 5 of the square segments. The same argument will work for the upper right and lower left parts: parallel lines cutting one line in equal parts namely the original square cut all other lines in the same ratio. As for the middle two small segments, we find that their sum is the side of the the interior square of a square congruent to the original one: the square formed from sufficiently many translates of the lower-left interior square vertex. 6

### Definitions, 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

### 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

### 39 Symmetry of Plane Figures

39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### 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

### Math 531, Exam 1 Information.

Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. 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)

### Geometry Module 4 Unit 2 Practice Exam

Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### Conjectures for Geometry for Math 70 By I. L. Tse

Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:

### Chapters 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

### Chapter 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

### 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.

### 2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?

MATH 206 - Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of

### Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)

Mathematical Sentence - a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement

### 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,

### QUADRILATERALS CHAPTER 8. (A) Main Concepts and Results

CHAPTER 8 QUADRILATERALS (A) Main Concepts and Results Sides, Angles and diagonals of a quadrilateral; Different types of quadrilaterals: Trapezium, parallelogram, rectangle, rhombus and square. Sum of

### /27 Intro to Geometry Review

/27 Intro to Geometry Review 1. An acute has a measure of. 2. A right has a measure of. 3. An obtuse has a measure of. 13. Two supplementary angles are in ratio 11:7. Find the measure of each. 14. In the

### 11.3 Curves, Polygons and Symmetry

11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### 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

### www.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

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### 3.1 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)

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

### Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

### CHAPTER 8 QUADRILATERALS. 8.1 Introduction

CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three non-collinear points in pairs, the figure so obtained is

### Area. Area Overview. Define: Area:

Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.

### 5.1 Midsegment Theorem and Coordinate Proof

5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects

Lecture 24: Saccheri Quadrilaterals 24.1 Saccheri Quadrilaterals Definition In a protractor geometry, we call a quadrilateral ABCD a Saccheri quadrilateral, denoted S ABCD, if A and D are right angles

### http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

### POTENTIAL REASONS: Definition of Congruence:

Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides

### Visualizing 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

### 1 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,

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

### Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3

Mathematics 3301-001 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

### Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

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,

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### Intermediate Math Circles October 10, 2012 Geometry I: Angles

Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### IMO 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

### Most 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

### Situation: 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

### Session 5 Dissections and Proof

Key Terms for This Session Session 5 Dissections and Proof Previously Introduced midline parallelogram quadrilateral rectangle side-angle-side (SAS) congruence square trapezoid vertex New in This Session

### The Triangle and its Properties

THE TRINGLE ND ITS PROPERTIES 113 The Triangle and its Properties Chapter 6 6.1 INTRODUCTION triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three

### GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic

### 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

### 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

### Postulate 17 The area of a square is the square of the length of a. Postulate 18 If two figures are congruent, then they have the same.

Chapter 11: Areas of Plane Figures (page 422) 11-1: Areas of Rectangles (page 423) Rectangle Rectangular Region Area is measured in units. Postulate 17 The area of a square is the square of the length

### New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### CHAPTER 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

### 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

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

### 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.

### A summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs:

summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs: efinitions: efinition of mid-point and segment bisector M If a line intersects another line segment

### San 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

### Terminology: When one line intersects each of two given lines, we call that line a transversal.

Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal.

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, June 17, 2010 1:15 to 4:15 p.m., only Student Name: School Name: Print your name and the name of your

### Contents. 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..................................

### Analytical Geometry (4)

Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line

### 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

### Unit 2 - Triangles. Equilateral Triangles

Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

### Geometry. Higher Mathematics Courses 69. Geometry

The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and

### Circle 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

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 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, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### 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

### Chapter 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

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### Three 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

### Grade 3 Core Standard III Assessment

Grade 3 Core Standard III Assessment Geometry and Measurement Name: Date: 3.3.1 Identify right angles in two-dimensional shapes and determine if angles are greater than or less than a right angle (obtuse

### 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

### Chapter 7 Quiz. (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter?

Chapter Quiz Section.1 Area and Initial Postulates (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter? (.) TRUE or FALSE: If two plane

### Cumulative Test. 161 Holt Geometry. Name Date Class

Choose the best answer. 1. P, W, and K are collinear, and W is between P and K. PW 10x, WK 2x 7, and PW WK 6x 11. What is PK? A 2 C 90 B 6 D 11 2. RM bisects VRQ. If mmrq 2, what is mvrm? F 41 H 9 G 2

Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes

### Semester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question.

Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Are O, N, and P collinear? If so, name the line on which they lie. O N M P a. No,

### Exercise 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

### 12. Parallels. Then there exists a line through P parallel to l.

12. Parallels Given one rail of a railroad track, is there always a second rail whose (perpendicular) distance from the first rail is exactly the width across the tires of a train, so that the two rails

### GEOMETRY COMMON CORE STANDARDS

1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,

### 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)

### Geometry 8-1 Angles of Polygons

. Sum of Measures of Interior ngles Geometry 8-1 ngles of Polygons 1. Interior angles - The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of a triangle.

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### Lesson 5-3: Concurrent Lines, Medians and Altitudes

Playing with bisectors Yesterday we learned some properties of perpendicular bisectors of the sides of triangles, and of triangle angle bisectors. Today we are going to use those skills to construct special

### CIRCLE 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

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations

The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations Dynamic geometry technology should be used to maximize student learning in geometry. Such technology

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

### Week 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test

Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan

### http://jsuniltutorial.weebly.com/ Page 1

Parallelogram solved Worksheet/ Questions Paper 1.Q. Name each of the following parallelograms. (i) The diagonals are equal and the adjacent sides are unequal. (ii) The diagonals are equal and the adjacent

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

### Heron s Formula. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter

Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula,

### Circle Theorems. This circle shown is described an OT. As always, when we introduce a new topic we have to define the things we wish to talk about.

Circle s circle is a set of points in a plane that are a given distance from a given point, called the center. The center is often used to name the circle. T This circle shown is described an OT. s always,