t s time we revisit our friend, the equation of a line: y = mx + b
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1 CH PARALLEL AND PERPENDICULAR LINES Introduction I t s time we revisit our friend, the equation of a line: mx + b SLOPE -INTERCEPT To be precise, b is not the -intercept; b is the -coordinate of the -intercept. The -intercept is properl written (0, b). Parallel Lines Let s begin b assuming that throughout this chapter we will never be referring to horizontal or vertical lines -- these kinds of lines are covered in detail in the chapter Special Lines. Let s graph the two lines x + and x on the same grid:
2 Notice that the two lines appear to be parallel. Now, what do the equations of the two lines have in common? The formulas show that each line has a slope of. Since slope is a measure of steepness, does it seem reasonable that if two lines are parallel, then the are equall steep, and therefore the must have the same slope? Parallel lines have the same slope. For a simple example: Suppose Line has a slope of 9, and also suppose that Line is parallel to Line. We can then deduce (conclude b logic) that the slope of Line is also 9. EXAMPLE : Find the slope of an line which is parallel to the line x. Solution: An line which is parallel to the line x must have the same slope as the line x. So, if we can compute the slope of this line, we will have the slope of an line parallel to it. The easiest wa to find the slope of the line is to convert it to mx + b form: x x + x The slope of the given line is, and so we conclude that an line which is parallel to the line x must have a slope of Ch Parallel and Perpendicular Lines
3 EXAMPLE : Find the equation of the line which is parallel to the line x +, and which passes through the point (6, ). Solution: We re looking for an unknown line mx + b The slope of our unknown line was not given to us, but we know that it s the same as the slope of the given line, since the two lines are parallel. Solving the given line for ields the line x +, whose slope is clearl. So the slope of our unknown line is also. At this point in the problem we can write our line as x + b Plugging the given point (6, ) into this equation allows us to find b: (6) + b 8 + b 0 b and we re done; our line is x + 0 Homework. a. A given line has a slope of. What is the slope of an line that is parallel to the given line? b. A given line has a slope of. What is the slope of an line that is parallel to the given line? Ch Parallel and Perpendicular Lines
4 . a. What is the slope of an line that is parallel to the line x 9? b. What is the slope of an line that is parallel to the line x + 9?. a. Prove that the lines x and x 6 are parallel. b. Prove that the lines x and x + 0 are not parallel.. Find the equation of the line which is parallel to the given line, and which passes through the given point: a. x + ; (, ) b. x ; (, ) c. x + ; (, 9) d. x 8; (, ) e. x ; (, ) f. x + 0; (8, 0) Perpendicular Lines Parallel lines have the same slope -- certainl perpendicular lines do not! But is there some relationship between the slopes of perpendicular lines? Let s see if we can discover one with an example. In the following grid, Line is perpendicular to Line, and points have been labeled so that we can easil calculate m and m, the slopes of the two lines. Line (, ) Line (, 0) (0, ) (9, ) Ch Parallel and Perpendicular Lines
5 First we compute the slope of Line : m ( ) 6 x 0 Next we compute the slope of Line : m 0 ( ) x 9 6 There are two things to note regarding these two slopes of the two perpendicular lines. First, one slope is positive while the other is negative. This makes sense because as we move from left to right, Line is increasing while Line is decreasing. Second, the slope of Line, m, is kind of a big number (the line s prett steep), while the slope of Line, m, (ignoring the minus sign) is a relativel small number (the line s not ver steep). Specificall, the two slopes have opposite signs, and the are also (ignoring the minus sign) reciprocals of each other. In other words, when looking at the slopes of two perpendicular lines, each of the slopes is the opposite reciprocal of the other. Perpendicular lines have slopes that are opposite reciprocals of each other. For example, if a line has a slope of, then an perpendicular line must have a slope of. And consider the line x +. Since its slope is, it follows that the slope of an perpendicular line must be. Some books sa that the slopes of two perpendicular lines are negative reciprocals of each other. Ch Parallel and Perpendicular Lines
6 6 EXAMPLE : Find the equation of the line which is perpendicular to the line x, and which passes through the point (, ). Solution: We re looking for an unknown line mx + b The slope of our unknown line was not given to us, but we know that it s the opposite reciprocal of the slope of the given line, since the two lines are perpendicular. To determine the slope of the given line, we solve for : x x + x x, the given point the unknown line the given line telling us that the slope of the given line is. So the slope of our unknown line is the opposite reciprocal of that, which is. At this point in the problem we can write our line as x b Plugging the given point (, ) into this equation allows us to find b: ( ) b b b and we re done; our line is x Ch Parallel and Perpendicular Lines
7 Final Note: We ve learned that the slopes of two perpendicular lines are opposite reciprocals of each other. But some books sa that two lines are perpendicular if their slopes have a product of. Do both of these rules mean the same thing? Yes -- assume that the product of their slopes is : mm Solving for m gives us the equation m, m which sas that one slope is the opposite reciprocal of the other. Homework. A given line has a slope of. What is the slope of an line that is perpendicular to the given line? 6. Prove that the lines x 0 and x + are perpendicular.. Prove that the lines x + 0 and x + 9 are not perpendicular. 8. Prove that the lines x 0 and x + are not perpendicular. 9. Find the slope of an line which is perpendicular to the given line: a. x 9 b. x 0 c. x + 0 d. x 0 Ch Parallel and Perpendicular Lines
8 8 0. Find the equation of the line which is perpendicular to the given line, and which passes through the given point: a. x + ; (, ) b. x ; (, ) c. x + ; (, 9) d. x 8; (, ) e. x ; (, ) f. x + 0; (8, 0) Review Problems. The slopes of two parallel lines are.. The slopes of two perpendicular lines are.. The slope of a line is. What is the slope of an parallel line?. The slope of a line is. What is the slope of an perpendicular line? 9. T/F: The lines x and x are parallel. 6. T/F: The lines x and x + 0 are perpendicular.. Find the equation of the line which is parallel to x 9 and passes through the point (, 0). 8. Find the equation of the line which is perpendicular to x 9 and passes through the point (, ). 9. Find the equation of the line which is parallel to the line x and which passes through the point (, ). 0. Find the equation of the line which is perpendicular to the line x and which passes through the point (, ). Ch Parallel and Perpendicular Lines
9 9. Which one of the following lines is parallel to the line x? a. x b. x c. d. x e. x. Which one of the following lines is perpendicular to the line x? a. x b. x c. d. x e. x Solutions. a. b.. a. b.. a. Each line has a slope of. Same slope parallel lines. b. The slopes are and /. Different slopes non-parallel lines.. a. x b. d. x 6 e. x c. x x 9 f. x The slopes are and, which are opposite reciprocals of each other.. The slopes are and, which are not opposite reciprocals of each other. (The re reciprocals, but not opposites.) Ch Parallel and Perpendicular Lines
10 0 8. The slopes are and, which are not opposite reciprocals of each other. (The re opposites, but not reciprocals.) 9. a. b. c. d. 0. a. d. x b. x c. x e. x 9 f. x 6 x. equal. opposite reciprocals.. 9. T 6. T. x x 9. x 0.. c.. b. x 9 Upon the subject of education... I can onl sa that I view it as the most important subject which we as a people ma be engaged in. Abraham Lincoln Ch Parallel and Perpendicular Lines
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Chapter : January 26 January 30 Section.7: Inequalities As a diagnostic quiz, I want you to go through the first ten problems of the Chapter Test on page 32. These will test your knowledge of Sections.
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Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has
( ) c. m = 0, 1 2, 3 4
G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are:
Equation. A mathematical sentence formed by setting two expressions equal to each other. Example 1: 3 6 = 18 Example 2: 7 + x = 12
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GK- Mathematics Resources for Some Math Questions: Kaplan et al (2015). Cliff Notes FTCE General Knowledge Test, 3 rd Edition Mander, E. (2015). FTE General Knowledge Test with Online Practice, 3 rd Edition
c) Words: The cost of a taxicab is $2.00 for the first 1/4 of a mile and $1.00 for each additional 1/8 of a mile.
Functions Definition: A function f, defined from a set A to a set B, is a rule that associates with each element of the set A one, and onl one, element of the set B. Eamples: a) Graphs: b) Tables: 0 50
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Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations
Along the way, you learned many manipulative skills using the Properties of Real Numbers.
A LOOK at Algebra ============================= In a nutshell, you have learned how to: 1. solve linear equations and inequalities. solve quadratic equations and inequalities 3. solve systems of linear
Algebra II. Slide 1 / 261. Slide 2 / 261. Slide 3 / 261. Linear, Exponential and Logarithmic Functions. Table of Contents
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An Introduction to Systems of Equations
LESSON 17 An Introduction to Sstems of Equations LEARNING OBJECTIVES Toda I am: completing the Desmos activit Sstems of Two Linear Equations. So that I can: write and solve a sstem of two linear equations
5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5).
Rewrite using rational eponents. 2 1. 2. 5 5. 8 4 4. 4 5. Find the slope intercept equation of the line parallel to y = + 1 through the point (4, 5). 6. Use the limit definition to find the derivative