5-7 the Pythagorean Theorem 5-7 the Pythagorean Theorem
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Geometry Course Outline
GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid -
Right Triangles and the Pythagorean Theorem Related?
Activity Assess 9-6 EXPLORE & REASON Right Triangles and Consider △ ABC with altitude CD‾ as shown. the Pythagorean B Theorem D PearsonRealize.com A 45 C 5√2 I CAN… prove the Pythagorean Theorem using A. What is the area of △ ABC? Of △ACD? Explain your answers. similarity and establish the relationships in special right B. Find the lengths of AD‾ and AB‾ . triangles. C. Look for Relationships Divide the length of the hypotenuse of △ ABC VOCABULARY by the length of one of its sides. Divide the length of the hypotenuse of △ACD by the length of one of its sides. Make a conjecture that explains • Pythagorean triple the results. ESSENTIAL QUESTION How are similarity in right triangles and the Pythagorean Theorem related? Remember that the Pythagorean Theorem and its converse describe how the side lengths of right triangles are related. THEOREM 9-8 Pythagorean Theorem If a triangle is a right triangle, If... △ABC is a right triangle. then the sum of the squares of the B lengths of the legs is equal to the square of the length of the hypotenuse. c a A C b 2 2 2 PROOF: SEE EXAMPLE 1. Then... a + b = c THEOREM 9-9 Converse of the Pythagorean Theorem 2 2 2 If the sum of the squares of the If... a + b = c lengths of two sides of a triangle is B equal to the square of the length of the third side, then the triangle is a right triangle. c a A C b PROOF: SEE EXERCISE 17. Then... △ABC is a right triangle. -
Inner Product Spaces Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 2, 2007)
MAT067 University of California, Davis Winter 2007 Inner Product Spaces Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 2, 2007) The abstract definition of vector spaces only takes into account algebraic properties for the addition and scalar multiplication of vectors. For vectors in Rn, for example, we also have geometric intuition which involves the length of vectors or angles between vectors. In this section we discuss inner product spaces, which are vector spaces with an inner product defined on them, which allow us to introduce the notion of length (or norm) of vectors and concepts such as orthogonality. 1 Inner product In this section V is a finite-dimensional, nonzero vector space over F. Definition 1. An inner product on V is a map ·, · : V × V → F (u, v) →u, v with the following properties: 1. Linearity in first slot: u + v, w = u, w + v, w for all u, v, w ∈ V and au, v = au, v; 2. Positivity: v, v≥0 for all v ∈ V ; 3. Positive definiteness: v, v =0ifandonlyifv =0; 4. Conjugate symmetry: u, v = v, u for all u, v ∈ V . Remark 1. Recall that every real number x ∈ R equals its complex conjugate. Hence for real vector spaces the condition about conjugate symmetry becomes symmetry. Definition 2. An inner product space is a vector space over F together with an inner product ·, ·. Copyright c 2007 by the authors. These lecture notes may be reproduced in their entirety for non- commercial purposes. 2NORMS 2 Example 1. V = Fn n u =(u1,...,un),v =(v1,...,vn) ∈ F Then n u, v = uivi. -
Pythagorean Theorem Word Problems Ws #1 Name ______
Pythagorean Theorem word problems ws #1 Name __________________________ Solve each of the following. Please draw a picture and use the Pythagorean Theorem to solve. Be sure to label all answers and leave answers in exact simplified form. 1. The bottom of a ladder must be placed 3 feet from a wall. The ladder is 12 feet long. How far above the ground does the ladder touch the wall? 2. A soccer field is a rectangle 90 meters wide and 120 meters long. The coach asks players to run from one corner to the corner diagonally across the field. How far do the players run? 3. How far from the base of the house do you need to place a 15’ ladder so that it exactly reaches the top of a 12’ wall? 4. What is the length of the diagonal of a 10 cm by 15 cm rectangle? 5. The diagonal of a rectangle is 25 in. The width is 15 in. What is the area of the rectangle? 6. Two sides of a right triangle are 8” and 12”. A. Find the the area of the triangle if 8 and 12 are legs. B. Find the area of the triangle if 8 and 12 are a leg and hypotenuse. 7. The area of a square is 81 cm2. Find the perimeter of the square. 8. An isosceles triangle has congruent sides of 20 cm. The base is 10 cm. What is the area of the triangle? 9. A baseball diamond is a square that is 90’ on each side. -
Geometry Ch 5 Exterior Angles & Triangle Inequality December 01, 2014
Geometry Ch 5 Exterior Angles & Triangle Inequality December 01, 2014 The “Three Possibilities” Property: either a>b, a=b, or a<b The Transitive Property: If a>b and b>c, then a>c The Addition Property: If a>b, then a+c>b+c The Subtraction Property: If a>b, then a‐c>b‐c The Multiplication Property: If a>b and c>0, then ac>bc The Division Property: If a>b and c>0, then a/c>b/c The Addition Theorem of Inequality: If a>b and c>d, then a+c>b+d The “Whole Greater than Part” Theorem: If a>0, b>0, and a+b=c, then c>a and c>b Def: An exterior angle of a triangle is an angle that forms a linear pair with an angle of the triangle. A In ∆ABC, exterior ∠2 forms a linear pair with ∠ACB. The other two angles of the triangle, ∠1 (∠B) and ∠A are called remote interior angles with respect to ∠2. 1 2 B C Theorem 12: The Exterior Angle Theorem An Exterior angle of a triangle is greater than either remote interior angle. Find each of the following sums. 3 4 26. ∠1+∠2+∠3+∠4 2 1 6 7 5 8 27. ∠1+∠2+∠3+∠4+∠5+∠6+∠7+ 9 12 ∠8+∠9+∠10+∠11+∠12 11 10 28. ∠1+∠5+∠9 31. What does the result in exercise 30 indicate about the sum of the exterior 29. ∠3+∠7+∠11 angles of a triangle? 30. ∠2+∠4+∠6+∠8+∠10+∠12 1 Geometry Ch 5 Exterior Angles & Triangle Inequality December 01, 2014 After proving the Exterior Angle Theorem, Euclid proved that, in any triangle, the sum of any two angles is less than 180°. -
Indirect Proof and Inequalities in One Triangle
6.5 Indirect Proof and Inequalities in One Triangle EEssentialssential QQuestionuestion How are the sides related to the angles of a triangle? How are any two sides of a triangle related to the third side? Comparing Angle Measures and Side Lengths Work with a partner. Use dynamic geometry software. Draw any scalene △ABC. a. Find the side lengths and angle measures of the triangle. 5 Sample C 4 Points Angles A(1, 3) m∠A = ? A 3 B(5, 1) m∠B = ? C(7, 4) m∠C = ? 2 Segments BC = ? 1 B AC = ? AB = 0 ? 01 2 34567 b. Order the side lengths. Order the angle measures. What do you observe? ATTENDING TO c. Drag the vertices of △ABC to form new triangles. Record the side lengths and PRECISION angle measures in a table. Write a conjecture about your fi ndings. To be profi cient in math, you need to express A Relationship of the Side Lengths numerical answers with of a Triangle a degree of precision Work with a partner. Use dynamic geometry software. Draw any △ABC. appropriate for the content. a. Find the side lengths of the triangle. b. Compare each side length with the sum of the other two side lengths. 4 C Sample 3 Points A A(0, 2) 2 B(2, −1) C 1 (5, 3) Segments 0 BC = ? −1 01 2 3456 AC = ? −1 AB = B ? c. Drag the vertices of △ABC to form new triangles and repeat parts (a) and (b). Organize your results in a table. Write a conjecture about your fi ndings. -
CS261 - Optimization Paradigms Lecture Notes for 2013-2014 Academic Year
CS261 - Optimization Paradigms Lecture Notes for 2013-2014 Academic Year ⃝c Serge Plotkin January 2014 1 Contents 1 Steiner Tree Approximation Algorithm 6 2 The Traveling Salesman Problem 10 2.1 General TSP . 10 2.1.1 Definitions . 10 2.1.2 Example of a TSP . 10 2.1.3 Computational Complexity of General TSP . 11 2.1.4 Approximation Methods of General TSP? . 11 2.2 TSP with triangle inequality . 12 2.2.1 Computational Complexity of TSP with Triangle Inequality . 12 2.2.2 Approximation Methods for TSP with Triangle Inequality . 13 3 Matchings, Edge Covers, Node Covers, and Independent Sets 22 3.1 Minumum Edge Cover and Maximum Matching . 22 3.2 Maximum Independent Set and Minimum Node Cover . 24 3.3 Minimum Node Cover Approximation . 24 4 Intro to Linear Programming 27 4.1 Overview . 27 4.2 Transformations . 27 4.3 Geometric interpretation of LP . 29 4.4 Existence of optimum at a vertex of the polytope . 30 4.5 Primal and Dual . 32 4.6 Geometric view of Linear Programming duality in two dimensions . 33 4.7 Historical comments . 35 5 Approximating Weighted Node Cover 37 5.1 Overview and definitions . 37 5.2 Min Weight Node Cover as an Integer Program . 37 5.3 Relaxing the Linear Program . 37 2 5.4 Primal/Dual Approach . 38 5.5 Summary . 40 6 Approximating Set Cover 41 6.1 Solving Minimum Set Cover . 41 7 Randomized Algorithms 46 7.1 Maximum Weight Crossing Edge Set . 46 7.2 The Wire Routing Problem . 48 7.2.1 Overview . -
Use the Converse of the Pythagorean Theorem
Use the Converse of the Pythagorean Theorem Goal • Use the Converse of the Pythagorean Theorem to determine if a triangle is a right triangle. Your Notes THEOREM 7.2: CONVERSE OF THE PYTHAGOREAN THEOREM If the square of the length of the longest side of a triangle is equal aI to the sum of the squares of the c A lengths of the other two sides, then the triangle is a triangle. If c2 = a 2 + b2, then AABC is a triangle. Verify right triangles Tell whether the given triangle is a right triangle. a. b. 24 6 9 Solution Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c 2 = a 2 + b 2. a. ( )2 ? 2 + 2 • The triangle a right triangle. b. 2? 2 + 2 The triangle a right triangle. 174 Lesson 7.2 • Geometry Notetaking Guide Copyright @ McDougal Littell/Houghton Mifflin Company. Your Notes THEOREM 7.3 A If the square of the length of the longest b c side of a triangle is less than the sum of B the squares of the lengths of the other C a two sides, then the triangle ABC is an triangle. If c 2 < a 2 + b2 , then the triangle ABC is THEOREM 7.4 A If the square of the length of the longest bN side of a triangle is greater than the sum of C B the squares of the lengths of the other two a sides, then the triangle ABC is an triangle. -
The Pythagorean Theorem and Area: Postulates Into Theorems Paul A
Humanistic Mathematics Network Journal Issue 25 Article 13 8-1-2001 The Pythagorean Theorem and Area: Postulates into Theorems Paul A. Kennedy Texas State University Kenneth Evans Texas State University Follow this and additional works at: http://scholarship.claremont.edu/hmnj Part of the Mathematics Commons, Science and Mathematics Education Commons, and the Secondary Education and Teaching Commons Recommended Citation Kennedy, Paul A. and Evans, Kenneth (2001) "The Pythagorean Theorem and Area: Postulates into Theorems," Humanistic Mathematics Network Journal: Iss. 25, Article 13. Available at: http://scholarship.claremont.edu/hmnj/vol1/iss25/13 This Article is brought to you for free and open access by the Journals at Claremont at Scholarship @ Claremont. It has been accepted for inclusion in Humanistic Mathematics Network Journal by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. The Pythagorean Theorem and Area: Postulates into Theorems Paul A. Kennedy Contributing Author: Department of Mathematics Kenneth Evans Southwest Texas State University Department of Mathematics, Retired San Marcos, TX 78666-4616 Southwest Texas State University [email protected] San Marcos, TX 78666-4616 Considerable time is spent in high school geometry equal to the area of the square on the hypotenuse. building an axiomatic system that allows students to understand and prove interesting theorems. In tradi- tional geometry classrooms, the theorems were treated in isolation with some of -
The Pythagorean Theorem
6.2 The Pythagorean Theorem How are the lengths of the sides of a right STATES triangle related? STANDARDS MA.8.G.2.4 MA.8.A.6.4 Pythagoras was a Greek mathematician and philosopher who discovered one of the most famous rules in mathematics. In mathematics, a rule is called a theorem. So, the rule that Pythagoras discovered is called the Pythagorean Theorem. Pythagoras (c. 570 B.C.–c. 490 B.C.) 1 ACTIVITY: Discovering the Pythagorean Theorem Work with a partner. a. On grid paper, draw any right triangle. Label the lengths of the two shorter sides (the legs) a and b. c2 c a a2 b. Label the length of the longest side (the hypotenuse) c. b b2 c . Draw squares along each of the three sides. Label the areas of the three squares a 2, b 2, and c 2. d. Cut out the three squares. Make eight copies of the right triangle and cut them out. a2 Arrange the fi gures to form 2 two identical larger squares. c b2 e. What does this tell you about the relationship among a 2, b 2, and c 2? 236 Chapter 6 Square Roots and the Pythagorean Theorem 2 ACTIVITY: Finding the Length of the Hypotenuse Work with a partner. Use the result of Activity 1 to fi nd the length of the hypotenuse of each right triangle. a. b. c 10 c 3 24 4 c. d. c 0.6 2 c 3 0.8 1 2 3 ACTIVITY: Finding the Length of a Leg Work with a partner. -
Inequalities for C-S Seminorms and Lieb Functions
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector LINEAR ALGEBRA AND ITS APPLiCATIONS ELSEVIER Linear Algebra and its Applications 291 (1999) 103-113 - Inequalities for C-S seminorms and Lieb functions Roger A. Horn a,., Xingzhi Zhan b.l ;1 Department (?(Mathematics, Uni('('rsify of Utah, Salt L(lke CifY, UT 84103. USA h Institute of Mathematics. Pekins; University, Beijing 10087J. People's Republic of C/WICt Received 3 July 1998: accepted 3 November 1998 Submitted by R. Bhatia Abstract Let AI" be the space of 1/ x 11 complex matrices. A seminorm /I ." on NI" is said to be a C-S seminorm if IIA-AII = IIAA"II for all A E Mil and IIAII ~ IIBII whenever A, B, and B-A are positive semidefinite. If II . II is any nontrivial C-S seminorm on AI", we show that 111/'111 is a unitarily invariant norm on M,n which permits many known inequalities for unitarily invariant norms to be generalized to the setting of C-S seminorrns, We prove a new inequality for C-S seminorms that includes as special cases inequalities of Bhatia et al., for unitarily invariant norms. Finally. we observe that every C-S seminorm belongs to the larger class of Lieb functions, and we prove some new inequalities for this larger class. © 1999 Elsevier Science Inc. All rights reserved. 1. Introduction Let M" be the space of 11 x Jl complex matrices and denote the matrix ab solute value of any A E M" by IA I=(A* A)1/2. -
Simplex-Tree Based Kinematics of Foldable Objects As Multi-Body Systems Involving Loops
Simplex-Tree Based Kinematics of Foldable Objects as Multi-body Systems Involving Loops Li Han and Lee Rudolph Department of Mathematics and Computer Science Clark University, Worcester, MA 01610 Email: [email protected], [email protected] Abstract— Many practical multi-body systems involve loops. Smooth manifolds are characterized by the existence of local Studying the kinematics of such systems has been challenging, coordinates, but although some manifolds are equipped with partly because of the requirement of maintaining loop closure global coordinates (like the joint angle parameters, in the constraints, which have conventionally been formulated as highly nonlinear equations in joint parameters. Recently, novel para- cases of a serial chain without closure constraints or [7, 8] a meters defined by trees of triangles have been introduced for a planar loop satisfying the technical “3 long links” condition), broad class of linkage systems involving loops (e.g., spatial loops typically a manifold has neither global coordinates nor any with spherical joints and planar loops with revolute joints); these standard atlas of local coordinate charts. Thus, calculations parameters greatly simplify kinematics related computations that depend on coordinates are often difficult to perform. and endow system configuration spaces with highly tractable piecewise convex geometries. In this paper, we describe a more Some recent kinematic work uses new parameters that are general approach for multi-body systems, with loops, that allow not conventional joint parameters. One series of papers [9, construction trees of simplices. We illustrate the applicability and 10, 11, 12, 13] presented novel formulations and techniques, efficiency of our simplex-tree based approach to kinematics by a including distance geometry, linear programming and flag study of foldable objects.