ACT Study Modules Topic: Finding Slope from Parallel and Perpendicular Lines

Total Page：16

File Type：pdf, Size：1020Kb

ACT Study Modules Topic: Finding slope from parallel and perpendicular lines Khan Academy Link: https://www.khanacademy.org/math/geometry-home/analytic-geometry- topic/parallel-perpendicular-lines-coordinate-plane/v/classify-lines To determine the slope of parallel lines from an equation: Step one: arrange the equation in the form, = + Step two: determine the slope ( value) Step three: compare the slopes Step four: compare the “ ” values If the b values are different and the slope is the same, then the lines are parallel. Example: 3 + = 9 5 = 3 Add 3 to both sides add 5 to both sides − of the equation− of the− equation = 3 9 = 3 + 5 In the functions, = 3 −9 & = 3 + 5 the slope is 3, and the values are different, making the lines parallel. − To determine the slope of perpendicular lines from an equation: Step one: arrange the equation in the form = + Step two: determine the slope ( value) Step three: compare the slopes If the slopes are negative reciprocals, the lines are perpendicular. Example: 5 = 4 + 8 = 1/5 Add 5 to both sides Subtract 8 from both sides − of the −equation of the equation− = 5 4 = 3 + 5 − Slopes are 5 and -1/5, the slopes are negative reciprocals, making the lines perpendicular. Example: In the standard ( , ) coordinate plane which of the lines are perpendicular to the line = 2 + 2 that passes through the point (0,3) Step one:− determine the slope of the perpendicular line:. The slope of the original line is 2 the negative reciprocal of 2 is 1 − Step two: = + − 2 Step three: to determine1 , substitute the and - values of the point, (0,3) 2 3 = (0) + 1 3 = 2 Step four: write the equation, = + 3 1 2 Practice: Line Equation of a parallel line Equation of a perpendicular line = 2 5 − 2 3 = 12 − 3 = 1 4 − − 5 + 2 = 15 .

Copy Link

Recommended publications

Proofs with Perpendicular Lines
3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units.
[Show full text]
Chapter 1 – Symmetry of Molecules – P. 1
Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules.
[Show full text]
20. Geometry of the Circle (SC)
20. GEOMETRY OF THE CIRCLE PARTS OF THE CIRCLE Segments When we speak of a circle we may be referring to the plane figure itself or the boundary of the shape, called the circumference. In solving problems involving the circle, we must be familiar with several theorems. In order to understand these theorems, we review the names given to parts of a circle. Diameter and chord The region that is encompassed between an arc and a chord is called a segment. The region between the chord and the minor arc is called the minor segment. The region between the chord and the major arc is called the major segment. If the chord is a diameter, then both segments are equal and are called semi-circles. The straight line joining any two points on the circle is called a chord. Sectors A diameter is a chord that passes through the center of the circle. It is, therefore, the longest possible chord of a circle. In the diagram, O is the center of the circle, AB is a diameter and PQ is also a chord. Arcs The region that is enclosed by any two radii and an arc is called a sector. If the region is bounded by the two radii and a minor arc, then it is called the minor sector. www.faspassmaths.comIf the region is bounded by two radii and the major arc, it is called the major sector. An arc of a circle is the part of the circumference of the circle that is cut off by a chord.
[Show full text]
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
[Show full text]
Symmetry in Chemistry - Group Theory
Symmetry in Chemistry - Group Theory Group Theory is one of the most powerful mathematical tools used in Quantum Chemistry and Spectroscopy. It allows the user to predict, interpret, rationalize, and often simplify complex theory and data. At its heart is the fact that the Set of Operations associated with the Symmetry Elements of a molecule constitute a mathematical set called a Group. This allows the application of the mathematical theorems associated with such groups to the Symmetry Operations. All Symmetry Operations associated with isolated molecules can be characterized as Rotations: k (a) Proper Rotations: Cn ; k = 1,......, n k When k = n, Cn = E, the Identity Operation n indicates a rotation of 360/n where n = 1,.... k (b) Improper Rotations: Sn , k = 1,....., n k When k = 1, n = 1 Sn = , Reflection Operation k When k = 1, n = 2 Sn = i , Inversion Operation In general practice we distinguish Five types of operation: (i) E, Identity Operation k (ii) Cn , Proper Rotation about an axis (iii) , Reflection through a plane (iv) i, Inversion through a center k (v) Sn , Rotation about an an axis followed by reflection through a plane perpendicular to that axis. Each of these Symmetry Operations is associated with a Symmetry Element which is a point, a line, or a plane about which the operation is performed such that the molecule's orientation and position before and after the operation are indistinguishable. The Symmetry Elements associated with a molecule are: (i) A Proper Axis of Rotation: Cn where n = 1,.... This implies n-fold rotational symmetry about the axis.
[Show full text]
CHAPTER 3 Parallel and Perpendicular Lines Chapter Outline
www.ck12.org CHAPTER 3 Parallel and Perpendicular Lines Chapter Outline 3.1 LINES AND ANGLES 3.2 PROPERTIES OF PARALLEL LINES 3.3 PROVING LINES PARALLEL 3.4 PROPERTIES OF PERPENDICULAR LINES 3.5 PARALLEL AND PERPENDICULAR LINES IN THE COORDINATE PLANE 3.6 THE DISTANCE FORMULA 3.7 CHAPTER 3REVIEW In this chapter, you will explore the different relationships formed by parallel and perpendicular lines and planes. Different angle relationships will also be explored and what happens to these angles when lines are parallel. You will continue to use proofs, to prove that lines are parallel or perpendicular. There will also be a review of equations of lines and slopes and how we show algebraically that lines are parallel and perpendicular. 114 www.ck12.org Chapter 3. Parallel and Perpendicular Lines 3.1 Lines and Angles Learning Objectives • Identify parallel lines, skew lines, and parallel planes. • Use the Parallel Line Postulate and the Perpendicular Line Postulate. • Identify angles made by transversals. Review Queue 1. What is the equation of a line with slope -2 and passes through the point (0, 3)? 2. What is the equation of the line that passes through (3, 2) and (5, -6). 3. Change 4x − 3y = 12 into slope-intercept form. = 1 = − 4. Are y 3 x and y 3x perpendicular? How do you know? Know What? A partial map of Washington DC is shown. The streets are designed on a grid system, where lettered streets, A through Z run east to west and numbered streets 1st to 30th run north to south.
[Show full text]
11.4 Arcs and Chords
Page 1 of 5 11.4 Arcs and Chords Goal By finding the perpendicular bisectors of two Use properties of chords of chords, an archaeologist can recreate a whole circles. plate from just one piece. This approach relies on Theorem 11.5, and is Key Words shown in Example 2. • congruent arcs p. 602 • perpendicular bisector p. 274 THEOREM 11.4 B Words If a diameter of a circle is perpendicular to a chord, then the diameter bisects the F chord and its arc. E G Symbols If BG&* ∏ FD&* , then DE&* c EF&* and DGs c GFs. D EXAMPLE 1 Find the Length of a Chord In ᭪C the diameter AF&* is perpendicular to BD&* . D Use the diagram to find the length of BD&* . 5 F C Solution E A Because AF&* is a diameter that is perpendicular to BD&* , you can use Theorem 11.4 to conclude that AF&* bisects B BD&* . So, BE ϭ ED ϭ 5. BD ϭ BE ϩ ED Segment Addition Postulate ϭ 5 ϩ 5 Substitute 5 for BE and ED . ϭ 10 Simplify. ANSWER ᮣ The length of BD&* is 10. Find the Length of a Segment 1. Find the length of JM&* . 2. Find the length of SR&* . H P 12 K C S C M N G 15 P J R 608 Chapter 11 Circles Page 2 of 5 THEOREM 11.5 Words If one chord is a perpendicular bisector M of another chord, then the first chord is a diameter. J P K Symbols If JK&* ∏ ML&* and MP&** c PL&* , then JK&* is a diameter.
[Show full text]
Geometry Course Outline Learning Targets Unit 1: Proof, Parallel, And
Geometry Course Outline Learning Targets Unit 1: Proof, Parallel, and Perpendicular Lines 1-1-1 Identify, describe, and name points, lines, line segments, rays and planes using correct notation. 1-1-2 Identify and name angles. 1-2-1 Describe angles and angle pairs. 1-2-2 Identify and name parts of a circle. 2-1-1 Make conjectures by applying inductive reasoning. 2-1-2 Recognize the limits of inductive reasoning. 2-2-1 Use deductive reasoning to prove that a conjecture is true. 2-2-2 Develop geometric and algebraic arguments based on deductive reasoning. 3-1-1 Distinguish between undefined and defined terms. 3-1-2 Use properties to complete algebraic two-column proofs. 3-2-1 Identify the hypothesis and conclusion of a conditional statement. 3-2-2 Give counterexamples for false conditional statements. 3-3-1 Write and determine the truth value of the converse, inverse, and contrapositive of a conditional statement. 3-3-2 Write and interpret biconditional statements. 4-1-1 Apply the Segment Addition Postulate to find lengths of segments. 4-1-2 Use the definition of midpoint to find lengths of segments. 4-2-1 Apply the Angle Addition Postulate to find angle measures. 4-2-2 Use the definition of angle bisector to find angle measures. 5-1-1 Derive the Distance Formula. 5-1-2 Use the Distance Formula to find the distance between two points on the coordinate plane. 5-2-1 Use inductive reasoning to determine the Midpoint Formula. 5-2-2 Use the Midpoint Formula to find the coordinates of the midpoint of a segment on the coordinate plane.
[Show full text]
Chapter 3: Parallel and Perpendicular Lines
Parallel and Perpendicular Lines • Lessons 3-1, 3-2, and 3-5 Identify angle Key Vocabulary relationships that occur with parallel lines and a • parallel lines (p. 126) transversal, and identify and prove lines parallel • transversal (p. 127) from given angle relationships. • slope (p. 139) Use slope to analyze a line • Lessons 3-3 and 3-4 • equidistant (p. 160) and to write its equation. • Lesson 3-6 Find the distance between a point and a line and between two parallel lines. The framework of a wooden roller coaster is composed of millions of feet of intersecting lumber that often form parallel lines and transversals. Roller coaster designers, construction managers, and carpenters must know the relationships of angles created by parallel lines and their transversals to create a safe and stable ride. You will find how measures of angles are used in carpentry and construction in Lesson 3-2. 124 Chapter 3 Parallel and Perpendicular Lines Richard Cummins/CORBIS Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 3. For Lesson 3-1 Naming Segments Name all of the lines that contain the given point. (For review, see Lesson 1-1.) P Q R 1. Q 2. R 3. S 4. T S T For Lessons 3-2 and 3-5 Congruent Angles Name all angles congruent to the given angle. (For review, see Lesson 1-4.) 1 2 8 5. Є2 6. Є5 7 3 4 6 5 7.
[Show full text]
Perpendicular and Parallel Line Segments
Name: Date: r te p a h Perpendicular and C Parallel Line Segments Practice 1 Drawing Perpendicular Line Segments Use a protractor to draw perpendicular line segments. Example Draw a line segment perpendicular to RS through point T. S 0 180 10 1 70 160 20 150 30 140 40 130 50 0 120 180 6 0 10 11 0 170 70 20 100 30 160 80 40 90 80 150 50 70 60 140 100 130 110 120 R T © Marshall Cavendish International (Singapore) Private Limited. Private (Singapore) International © Marshall Cavendish 1. Draw a line segment perpendicular to PQ. P Q 2. Draw a line segment perpendicular to TU through point X. X U T 67 Lesson 10.1 Drawing Perpendicular Line Segments 08(M)MIF2015CC_WBG4B_Ch10.indd 67 4/30/13 11:18 AM Use a drawing triangle to draw perpendicular line segments. Example M L N K 3. Draw a line segment 4. Draw a line segment perpendicular to EF . perpendicular to CD. C E F D © Marshall Cavendish International (Singapore) Private Limited. Private (Singapore) International © Marshall Cavendish 5. Draw a line segment perpendicular to VW at point P. Then, draw another line segment perpendicular to VW through point Q. Q V P W Line Segments 68 and Parallel Chapter 10 Perpendicular 08(M)MIF2015CC_WBG4B_Ch10.indd 68 4/30/13 11:18 AM Name: Date: Practice 2 Drawing Parallel Line Segments Use a drawing triangle and a straightedge to draw parallel line segments. Example Draw a line segment parallel to AB . A B 1. Draw a pair of parallel line segments.
[Show full text]
Tangent Lines to a Circle This Example Will Illustrate How to find the Tangent Lines to a Given Circle Which Pass Through a Given Point
Tangent lines to a circle This example will illustrate how to find the tangent lines to a given circle which pass through a given point. Suppose our circle has center (0; 0) and radius 2, and we are interested in tangent lines to the circle that pass through (5; 3). The picture we might draw of this situation looks like this. (5; 3) We are interested in finding the equations of these tangent lines (i.e., the lines which pass through exactly one point of the circle, and pass through (5; 3)). The key is to find the points of tangency, labeled A1 and A2 in the next figure. (5; 3) A1 A2 The trick to doing this is to introduce variables for the coordinates for one of these points. Let’s say one of these points is (a; b). (Note: I strongly recommend using variables other than x and y here, so that we can easily remember that (a; b) is a point of tangency; x and y are too generic, and we will want to use them later for other purposes (such as expressing our line equations).) Then this is the key idea: Since we have two variables (i.e., unknowns), we want to come up with two equations that these variables satisfy. If we can do that, then we can use algebra to solve for our unknowns. How do we come up with two equations? First, we know that (a; b) is a point on our circle, and so (a; b) satisfies the equation of the circle.
[Show full text]
Lesson 8: Parallel and Perpendicular Lines
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M4 GEOMETRY Lesson 8: Parallel and Perpendicular Lines Student Outcomes . Students recognize parallel and perpendicular lines from slope. Students create equations for lines satisfying criteria of the kind: “Contains a given point and is parallel/perpendicular to a given line.” Lesson Notes This lesson brings together several of the ideas from the previous lessons. In places where ideas from certain lessons are employed, these are identified with the lesson number underlined (e.g., Lesson 6). Classwork Opening (5 minutes) Students begin the lesson with the following activity using geometry software to reinforce the theorem studied in Lesson 6, which states that given points 퐴(푎1,푎2), 퐵(푏1, 푏2), 퐶(푐1, 푐2), and 퐷(푑1, 푑2), ̅퐴퐵̅̅̅ ⊥ ̅퐶퐷̅̅̅ if and only if (푏1 − 푎1)(푑1 − 푐1) + (푏2 − 푎2)(푑2 − 푐2) = 0. Construct two perpendicular segments, and measure the abscissa (the 푥-coordinate) and ordinate (the 푦-coordinate) of each of the endpoints of the segments. (The teacher may extend this activity by asking students to determine whether the points used must be the endpoints. This is easily investigated using the dynamic geometry software by creating free moving points on the segment and watching the sum of the products of the differences as the points slide along the segments.) . Calculate (푏1 − 푎1)(푑1 − 푐1) + (푏2 − 푎2)(푑2 − 푐2). Note the value of the sum, and observe what happens to the sum as students manipulate the endpoints of the perpendicular segments. Lesson 8: Parallel and Perpendicular Lines 81 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds.
[Show full text]