Right Triangles and Quadrilaterals
|
|
- Hugh Knight
- 7 years ago
- Views:
Transcription
1 CHATER. RIGHT TRIANGLE AND UADRILATERAL Choose always the way that seems the best, however rough it may be; custom will soon render it easy and agreeable. ythagoras CHATER Right Triangles and uadrilaterals.1 The ythagorean Theorem In a right triangle, the side of the triangle opposite the right angle is called the hypotenuse and the other two sides are called the legs of the triangle. We also often use the terms legs and hypotenuse to refer the lengths of the legs and hypotenuse of a right triangle. leg In this section, we explore one of the most famous math theorems, the ythagorean Theorem, which is a powerful relationship among the sides of a right triangle. We ll start by walking through one of the many proofs of the ythagorean Theorem. ( ythagorean is pronounced puh-thag-uh-ree-uhn. ) leg hypotenuse roblems roblem.1: Four identical right triangles with legs of lengths and are attached to the sides of square W as shown, such that W = = R = = and = = R = W =. (a) Explain why \W = 180, and why R is a square. (b) What is the area of R? (c) Find the area of W. W (d) Find W. R
2 .1. THE THAGOREAN THEOREM roblem.: In this problem, we follow in the steps of the previous problem to prove the ythagorean Theorem. We start again with four copies of a right triangle, attached to the sides of square W as shown at the right. Let the lengths of the legs of each triangle be a and b, as shown, and let the hypotenuse of each right triangle have length c. a W b c (a) Find the area of W in terms of c. (b) Find the area of R in terms of a and b. R (c) Find the area of W in terms of a and b. (d) how that a + b = c. roblem.: Find the missing side lengths in each of the three triangles shown below. B C 5 A T 5 8 U p 15 roblem.: Must the hypotenuse of a right triangle be the longest side of the triangle? Why or why not? roblem.5: In roblems.1 and., we have seen two right triangles in which all three side lengths are integers. Can you find any more right triangles in which all three side lengths are integers? Hints: roblem.6: (a) Find the hypotenuse of a right triangle whose legs are and. (b) Find the hypotenuse of a right triangle whose legs are 5 and 5. (c) Find the hypotenuse of a right triangle whose legs are 011 and 011. (d) Find the hypotenuse of a right triangle whose legs are and roblem.: The length of one leg of a right triangle is 10 and the triangle s hypotenuse has length 50. What is the length of the other leg?
3 CHATER. RIGHT TRIANGLE AND UADRILATERAL roblem.1: Four identical right triangles with legs of lengths and are attached to the sides of square W as shown, such that W = = R = = and = = R = W =. (a) Explain why \W = 180, and why R is a square. (b) What is the area of R? (c) Find the area of W. W (d) Find W. R olution for roblem.1: (a) Back on page 0, we learned that the acute angles of a right triangle add to 90. Therefore, in right triangle W we have \W + \W = 90. ince triangles W and W are identical, we have \W = \W. ubstituting this into the equation above gives \W + \W = 90. We are told that W is a square, so \W = 90, and we have \W = \W + \W + \W = \W \W = 90 + (\W + \W) = = 180. Therefore, \W is a straight angle. This means that W is on. imilarly, each vertex of W is on one of the sides of quadrilateral R. Each side of R has length + =, and each angle of R is the right angle of one of the triangles. o, all the sides of R are congruent, and all the angles of R are congruent, which means R is a square. (b) ince R is a square with side length, its area is = 9. (c) Each right triangle has area ()()/ = 6 square units. Removing the four right triangles from R leaves W, so we have [W] = [R] (6) = 9 = 5. (d) The area of W is the square of its side length. Because the area of W is 5, its side length must be p 5, which equals 5.
4 .1. THE THAGOREAN THEOREM roblem.: In this problem, we follow in the steps of the previous problem to prove the ythagorean Theorem. We start again with four copies of a right triangle, attached to the sides of square W as shown at the right. Let the lengths of the legs of each triangle be a and b, as shown, and let the hypotenuse of each right triangle have length c. a W b c (a) Find the area of W in terms of c. (b) Find the area of R in terms of a and b. R (c) Find the area of W in terms of a and b. (d) how that a + b = c. olution for roblem.: (a) ince W is a square with side length c, its area is c. (b) As in the previous problem, R is a square, and the vertices of W are on the sides of R. Each side of R has length a + b, so the area of R is (a + b). We can expand (a + b) with the distributive property: [R] = (a + b) = (a + b)(a + b) = a(a + b) + b(a + b) = a + ab + ba + b = a + ab + ab + b = a + ab + b. (c) The area of each of the right triangles is ab/, so the four right triangles together have area (ab/) = ab. We can find the area of W in terms of a and b by subtracting the areas of the triangles from the area of R: [W] = [R] (ab/) = a + ab + b ab = a + b. (d) In part (a), we found that [W] = c, and in part (c), we found that [W] = a + b. Equating our expressions for [W], we have a + b = c. 5
5 CHATER. RIGHT TRIANGLE AND UADRILATERAL Important: The ythagorean Theorem tells us that in any right triangle, the sum of the squares of the legs equals the square of the hypotenuse. o, in the diagram to the right, we have a + b = c. A b C a c B Our work in roblem. is the same as the work we did in roblem.1, except that we replaced the numbers in roblem.1 with variables a, b, and c in roblem.. Concept: pecific examples can sometimes be used as guides to discover proofs. The ythagorean Theorem also works in reverse. By this, we mean that if the side lengths of a triangle satisfy the ythagorean Theorem, then the triangle must be a right triangle. o, for example, if we have a triangle with side lengths,, and 5, then we know that the triangle must be a right triangle because + = 5. Let s get a little practice using the ythagorean Theorem. roblem.: Find the missing side lengths in each of the three triangles shown below. B C 5 A T 5 8 U p 15 olution for roblem.: What s wrong with this solution: Bogus olution: Applying the ythagorean Theorem to ABC gives + 5 = BC. Therefore, we find BC = = 6. Taking the square root gives us BC = p 6. 6
6 .1. THE THAGOREAN THEOREM This solution is incorrect because it applies the ythagorean Theorem incorrectly. ide BC is a leg, not the hypotenuse. Applying the ythagorean Theorem to ABC correctly gives AC + BC = AB. B 5 ubstituting AC = and AB = 5 gives us + BC = 5, so 9 + BC = 65. ubtracting 9 from both sides gives BC = 56. Taking the square root of 56 gives BC =. (Note that ( ) = 56 too, but we can t have a negative length, so BC cannot be.) C A WARNING!! j Be careful when applying the ythagorean Theorem. Make sure you correctly identify which sides are the legs and which is the hypotenuse. Applying the ythagorean Theorem to TU gives T + TU = U, so we have = U from the side lengths given in the problem. Therefore, we have U = = 89. Taking the square root gives us U = p 89. In, the ythagorean Theorem gives us T 5 8 U + =, so + ( p 15) =. This gives us =, so = 6 and = 8. p 15 WARNING!! j A common mistake when using the ythagorean Theorem to find the hypotenuse length of a right triangle is forgetting that the hypotenuse is squared in the equation, too. One quick way to avoid this error is to consider the three side lengths after finding the hypotenuse. For example, suppose a right triangle has legs of lengths and. The hypotenuse clearly can t be + = 5, because the lengths,, and 5 don t satisfy the Triangle Inequality. Taking the square root of 5, we see that the hypotenuse should be 5, not 5. roblem.: Must the hypotenuse of a right triangle be the longest side of the triangle? Why or why not?
7 CHATER. RIGHT TRIANGLE AND UADRILATERAL olution for roblem.: es. The square of the hypotenuse equals the sum of the squares of the legs. The sum of any two positive numbers is greater than both of the numbers being added. o, the square of the hypotenuse must be greater than the square of each leg. Therefore, the hypotenuse must be longer than each leg. roblem.5: In roblems.1 and., we have seen two right triangles in which all three side lengths are integers. Can you find any more right triangles in which all three side lengths are integers? olution for roblem.5: There are lots and lots of right triangles in which all three side lengths are integers! To search for some, we can list the first 0 positive perfect squares: 1,, 9, 16, 5, 6, 9, 6, 81, 100, 1, 1, 169, 196, 5, 56, 89,, 61, 00. Then, we look for pairs of squares that add up to another square. We immediately see 9+16 = 5, which is + = 5. We already saw this example in roblem.1. We also see = 169, which is 5 + = 1. o, a right triangle with legs of lengths 5 and has a hypotenuse with length 1. We also find = 89, which is = 1. This gives us a right triangle with 8 and 15 as the legs and 1 as the hypotenuse. A ythagorean triple is a group of three positive integers that satisfy the equation a +b = c. o, for example, {,, 5} is a ythagorean triple, as are {5,, 1} and {8, 15, 1}. There are lots of interesting patterns in ythagorean triples. ee if you can find more ythagorean triples, and look for patterns that you can use to find more ythagorean triples. We can find one such important pattern by looking back at our list of squares: 1,, 9, 16, 5, 6, 9, 6, 81, 100, 1, 1, 169, 196, 5, 56, 89,, 61, 00. We find that = 100, which is = 10. Here, the side lengths are double those of the first triangle we saw with sides of lengths,, and 5. We might wonder if tripling these three side lengths also gives us another right triangle. Indeed, we see that 9 + = 15, since = 5. Let s investigate further. roblem.6: (a) Find the hypotenuse of a right triangle whose legs are and. (b) Find the hypotenuse of a right triangle whose legs are 5 and 5. (c) Find the hypotenuse of a right triangle whose legs are 011 and 011. (d) Find the hypotenuse of a right triangle whose legs are and olution for roblem.6: (a) The legs have lengths and 16. Letting the hypotenuse be c, the ythagorean Theorem gives us c = + 16 = = 00. 8
8 .1. THE THAGOREAN THEOREM Taking the square root gives us c = 0. Notice that 0 = 5. (b) The legs have lengths 15 and 0. Letting the hypotenuse be c, the ythagorean Theorem gives us c = = = 65. Taking the square root gives us c = 5. Notice that 5 = 5 5. (c) The legs have lengths 60 and 80. Um, squaring those doesn t look like much fun. Let s see if we can find a more clever way to solve this problem. We know that a right triangle with legs and has hypotenuse 5. In part (a), we saw that if the legs of a right triangle are and, then the hypotenuse is 5. In part (b), we saw that if the legs of a right triangle are 5 and 5, then the hypotenuse is 5 5. It looks like there s a pattern! Concept: earching for patterns is a powerful problem-solving strategy. It appears that if the legs of a right triangle are x and x, then the hypotenuse is 5x. We can test this guess with the ythagorean Theorem. uppose the legs of a right triangle are x and x. Then, the sum of the squares of the legs is ince (x) + (x) = x + x = 9x + 16x = 5x. (5x) = 5 x = 5x, we have (x) + (x) = (5x), which means that the length of the hypotenuse is indeed 5x. This means that we don t have to square 60 and 80! A right triangle with legs of lengths 011 and 011 has a hypotenuse with length = (d) There s nothing in our explanation in part (c) that requires x to be a whole number; it 1 can be a fraction, too! o, in a right triangle with legs of length and 1, the hypotenuse has length 5 = Our work in roblem.6 is an example of why knowing common ythagorean triples is useful. Any time we have a right triangle in which the legs have ratio :, then we know that all three sides of the triangle are in the ratio : : 5. As we saw in the final two parts of roblem.6, this can allow us to find the hypotenuse quickly without using the ythagorean Theorem directly. We can also sometimes use this approach to quickly find the length of a leg when we know the lengths of the other leg and the hypotenuse. roblem.: The length of one leg of a right triangle is 10 and the triangle s hypotenuse has length 50. What is the length of the other leg? 9
9 CHATER. RIGHT TRIANGLE AND UADRILATERAL olution for roblem.: We find the ratio of the given leg length to the hypotenuse length, hoping it will match the corresponding ratio in one of the ythagorean triples we know. We have 10 : 50 = : = : 5, so the ratio of the given leg to the hypotenuse is : This reminds us of the {,, 5} ythagorean triple that we saw in roblem.. ince the ratio of the known leg to the hypotenuse is : 5, we know that all three sides are in the ratio : : 5. The first leg is 0 and the hypotenuse is 5 0, so the other leg of the right triangle is 0 = 0. Important: If we multiply all three side lengths of a right triangle by the same positive number, then the three new side lengths also satisfy the ythagorean Theorem. In other words, if side lengths a, b, and c satisfy a + b = c, then (na) + (nb) = (nc), for any positive number n. Exercises.1.1 Find the missing side lengths below: 15 U R T V.1. Bill walks 1 mile south, then mile east, and finally 1 mile south. How many miles is he, in a direct line, from his starting point? (ource: AMC 8).1. Find a formula for the length of a diagonal of a rectangle with length l and width w..1. The bases of a 9-foot pole and a 15-foot pole are 5 feet apart, and both poles are perpendicular to the ground. The ground is flat between the two poles. What is the length of the shortest rope that can be used to connect the tops of the two poles?.1.5 A square, a rectangle, a right triangle, and a semicircle are combined to form the figure at the right. Find the area of the whole figure in square units..1.6? Find the hypotenuse of a right triangle whose legs have lengths and
Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
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 informationGeometry: Classifying, Identifying, and Constructing Triangles
Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral
More informationMODERN APPLICATIONS OF PYTHAGORAS S THEOREM
UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented
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 informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More informationhttp://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
More informationAlgebra 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:
More informationBasic Lesson: Pythagorean Theorem
Basic Lesson: Pythagorean Theorem Basic skill One leg of a triangle is 10 cm and other leg is of 24 cm. Find out the hypotenuse? Here we have AB = 10 and BC = 24 Using the Pythagorean Theorem AC 2 = AB
More informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationGeometry 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
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationHeron 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,
More informationSet 4: Special Congruent Triangles Instruction
Instruction Goal: To provide opportunities for students to develop concepts and skills related to proving right, isosceles, and equilateral triangles congruent using real-world problems Common Core Standards
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationSquare Roots and the Pythagorean Theorem
4.8 Square Roots and the Pythagorean Theorem 4.8 OBJECTIVES 1. Find the square root of a perfect square 2. Use the Pythagorean theorem to find the length of a missing side of a right triangle 3. Approximate
More information9 Right Triangle Trigonometry
www.ck12.org CHAPTER 9 Right Triangle Trigonometry Chapter Outline 9.1 THE PYTHAGOREAN THEOREM 9.2 CONVERSE OF THE PYTHAGOREAN THEOREM 9.3 USING SIMILAR RIGHT TRIANGLES 9.4 SPECIAL RIGHT TRIANGLES 9.5
More informationSAT Math Facts & Formulas Review Quiz
Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions
More information8-2 The Pythagorean Theorem and Its Converse. Find x.
1 8- The Pythagorean Theorem and Its Converse Find x. 1. hypotenuse is 13 and the lengths of the legs are 5 and x.. equaltothesquareofthelengthofthehypotenuse. The length of the hypotenuse is x and the
More informationLaw of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.
Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where
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 informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More informationPythagorean Theorem: Proof and Applications
Pythagorean Theorem: Proof and Applications Kamel Al-Khaled & Ameen Alawneh Department of Mathematics and Statistics, Jordan University of Science and Technology IRBID 22110, JORDAN E-mail: kamel@just.edu.jo,
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More information43 Perimeter and Area
43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study
More information6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationPythagoras Theorem. Page I can... 1... identify and label right-angled triangles. 2... explain Pythagoras Theorem. 4... calculate the hypotenuse
Pythagoras Theorem Page I can... 1... identify and label right-angled triangles 2... eplain Pythagoras Theorem 4... calculate the hypotenuse 5... calculate a shorter side 6... determine whether a triangle
More informationConjectures. 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
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationSQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationDigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
More informationCircles in Triangles. This problem gives you the chance to: use algebra to explore a geometric situation
Circles in Triangles This problem gives you the chance to: use algebra to explore a geometric situation A This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units,
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationSECTION 1-6 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More informationWarm-up Theorems about triangles. Geometry. Theorems about triangles. Misha Lavrov. ARML Practice 12/15/2013
ARML Practice 12/15/2013 Problem Solution Warm-up problem Lunes of Hippocrates In the diagram below, the blue triangle is a right triangle with side lengths 3, 4, and 5. What is the total area of the green
More informationThree-Dimensional Figures or Space Figures. Rectangular Prism Cylinder Cone Sphere. Two-Dimensional Figures or Plane Figures
SHAPE NAMES Three-Dimensional Figures or Space Figures Rectangular Prism Cylinder Cone Sphere Two-Dimensional Figures or Plane Figures Square Rectangle Triangle Circle Name each shape. [triangle] [cone]
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More information2nd Semester Geometry Final Exam Review
Class: Date: 2nd Semester Geometry Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of an amusement park created a circular
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationThe 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
More information2004 Solutions Ga lois Contest (Grade 10)
Canadian Mathematics Competition An activity of The Centre for Education in Ma thematics and Computing, University of W aterloo, Wa terloo, Ontario 2004 Solutions Ga lois Contest (Grade 10) 2004 Waterloo
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
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 informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationCSU 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
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More information12. 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
More informationPostulate 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
More informationHeron Triangles. by Kathy Temple. Arizona Teacher Institute. Math Project Thesis
Heron Triangles by Kathy Temple Arizona Teacher Institute Math Project Thesis In partial fulfillment of the M.S. Degree in Middle School Mathematics Teaching Leadership Department of Mathematics University
More informationLesson 18 Pythagorean Triples & Special Right Triangles
Student Name: Date: Contact Person Name: Phone Number: Teas Assessment of Knowledge and Skills Eit Level Math Review Lesson 18 Pythagorean Triples & Special Right Triangles TAKS Objective 6 Demonstrate
More informationGeometry 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
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Thursday, August 16, 2012 8:30 to 11:30 a.m.
INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name
More 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 majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationSAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?
SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationChapter 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
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More informationMATH STUDENT BOOK. 8th Grade Unit 6
MATH STUDENT BOOK 8th Grade Unit 6 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular
More informationThe GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
More informationConjectures 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:
More informationName: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given
More informationGeometry - Semester 2. Mrs. Day-Blattner 1/20/2016
Geometry - Semester 2 Mrs. Day-Blattner 1/20/2016 Agenda 1/20/2016 1) 20 Question Quiz - 20 minutes 2) Jan 15 homework - self-corrections 3) Spot check sheet Thales Theorem - add to your response 4) Finding
More informationAlgebra 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?
More information/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
More information2006 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
More informationPOTENTIAL 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
More informationof 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
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationLesson 33: Example 1 (5 minutes)
Student Outcomes Students understand that the Law of Sines can be used to find missing side lengths in a triangle when you know the measures of the angles and one side length. Students understand that
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More informationTime needed: each worksheet will take approximately 1 hour to complete
Pythagoras Theorem Teacher s Notes Subject: Mathematics Topic: Pythagoras theorem Level: Pre-intermediate, intermediate Time needed: each worksheet will take approximately 1 hour to complete Learning objectives:
More informationSection 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,
More informationGeometry 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
More informationThe 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
More informationVeterans Upward Bound Algebra I Concepts - Honors
Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More informationArea. 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.
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
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 information7. 080207a, P.I. A.A.17
Math A Regents Exam 080 Page 1 1. 08001a, P.I. A.A.6 On a map, 1 centimeter represents 40 kilometers. How many kilometers are represented by 8 centimeters? [A] 48 [B] 30 [C] 5 [D] 80. 0800a, P.I. G.G.38
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationPre-Algebra Lesson 6-1 to 6-3 Quiz
Pre-lgebra Lesson 6-1 to 6-3 Quiz Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the area of the triangle. 17 ft 74 ft Not drawn to scale a. 629 ft
More informationPowerScore Test Preparation (800) 545-1750
Question 1 Test 1, Second QR Section (version 2) Two triangles QA: x QB: y Geometry: Triangles Answer: Quantity A is greater 1. The astute student might recognize the 0:60:90 and 45:45:90 triangle right
More informationApplications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More information