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1 Page 1 of Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 > 3-8 Slopes of 3-8 Slopes of Teks Focus TEKS ()(C) Determine an equation of a line parallel or perpendicular to a given line that passes through a given point. TEKS (1)(B) Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Additional TEKS (1)(G), ()(B) Vocabulary Formulate create with careful effort and purpose. You can formulate a plan or strategy to solve a problem. Strategy a plan or method for solving a problem Reasonableness the quality of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense. ESSENTIAL UNDERSTANDING You can determine whether two lines are parallel or perpendicular by comparing their slopes. take note Key Concept Slopes of Parallel Lines If two nonvertical lines are parallel, then their slopes are equal. If the slopes of two distinct nonvertical lines are equal, then the lines are parallel. Any two vertical lines or any two horizontal lines are parallel. take note Key Concept Slopes of Perpendicular Lines If two nonvertical lines are perpendicular, then the product of their slopes is 1. If the slopes of two lines have a product of 1, then the lines are perpendicular. Any horizontal line and vertical line are perpendicular. Problem 1 Verifying Parallelism Are lines l 1 and l parallel? Explain. Step 1 Find the slope of each line. 5 ( 4) 9 slope of l 1 = = = ( 4) 7 7 slope of l = = = 3 ( 1) Step Compare the slopes. Think Can you tell from the diagram whether the lines are parallel? No. The lines may look parallel, but you only see a small portion of their graphs. Compare their slopes to know for sure. Since 3 7, l 1 and l are not parallel.
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3 Page 1 of 1 Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 > 3-8 Slopes of Problem TEKS Process Standard (1)(G) Writing Equations of Parallel Lines What is an equation of the line parallel to y = 3x 5 that contains ( 1, 8)? Think Write Identify the slope of the given line. y = 3x 5 You now know the slope of the new line and that it passes through ( 1, 8). Use point-slope form to write the equation. y y 1 = m(x x 1) y 8 = 3(x Substitute 3 for m and ( 1, 8) for (x 1, y 1) and simplify. ( 1)) y 8 = 3(x + 1) Problem 3 Plan How does the given line help you? Parallel lines have the same slope. Once you know the slope of the given line, you know the slope you need to write an equation. Verifying Perpendicularity Lines l 1 and l are neither horizontal nor vertical. Are they perpendicular? Explain. Plan Can you tell from the diagram whether the lines are perpendicular? No. You can tell that the lines intersect, but not necessarily at right angles. So you need to compare their slopes. Step 1 Find the slope of each line. ( 4) 6 3 m 1 = slope of l 1 = = = ( 3) 6 m = slope of l = = = 4 ( 5) 9 3 Step Find the product of the slopes. 3 m 1 m = = 1 3 Lines l 1 and l are perpendicular because the product of their slopes is 1. Page 130 Copyright 016 Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy Terms of Use Rights and Permissions
4 Page 1 of 1 Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 > 3-8 Slopes of Problem 4 TEKS Process Standard (1)(B) Writing Equations of Perpendicular Lines Sports The baseball field below is on a coordinate grid with home plate at the origin. A batter hits a ground ball along the line shown. The player at (110, 70) runs along a path perpendicular to the path of the baseball. What is an equation of the line on which the player runs? Think How is this similar to writing equations of parallel lines? You follow the same process. The only difference here is that the slopes of perpendicular lines have product 1. Step 1 Find the slope of the baseball's path. y m = y x Points (30, 10) and (60, 0) are on the baseball's path. x 1 = = = Step Find the slope of a line perpendicular to the baseball's path. m 1 m = 1 The product of the slopes of lines is m = 1 Substitute for m m = 3 Multiply each side by 3. Step 3 Write an equation of the line on which the player runs. The slope is 3, and a point on the line is (110, 70). y y 1 = m(x x 1) Point-slope form y 70 = 3(x 110) Substitute 3 for m and (110, 70) for (x 1, y 1). PearsonTEXAS.com Think Which linear equation form should you use? You know the slope. The player is located at a point on the line. Use point-slope form. Page 131 Copyright 016 Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy Terms of Use Rights and Permissions
5 Page 1 of Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 > 3-8 Slopes of PRACTICE and APPLICATION EXERCISES For Exercises 1 and, are lines l 1 and l parallel? Explain. Scan page for a Virtual Nerd tutorial video. 1. For additional. support when completing your homework, go to PearsonTEXAS.com. Write an equation of the line parallel to the given line that contains C. 3. C(0, 3); y = x C(6, 0); y = 1 x C(, 4); y = x C(6, ); y = x +6 For Exercises 7 and 8, are lines l 1 and l perpendicular? Explain Write an equation of the line perpendicular to the given line that contains P. 9. P(6, 6); y = x P(4, 0); y = x P(4, 4); y = x 8 STEM 1. Apply Mathematics (1)(A) City planners want to construct a bike path perpendicular to Bruckner Boulevard at point P. An equation of the Bruckner Boulevard line is y = x. Find an equation of the line for the bike path. 3 4
6 Page of Rewrite each equation in slope-intercept form, if necessary. Then determine whether the lines are parallel. Explain. 13. y = x + 6 x + y = y 7x = 6 y + 7x = x + 4y = 1 6x + y = x + 5y = 1 10y = 4x Analyze Mathematical Relationships (1)(F) Line l 1 contains ( 4, 1) and (, 5), and line l contains (3, 0) and ( 3, k). What value of k makes l 1 and l parallel? 18. Connect Mathematical Ideas (1)(F) Write equations for two perpendicular lines that have the same y-intercept and do not pass through the origin. 19. Can the y-intercepts of two nonvertical parallel lines be the same? Explain. Page 13 Copyright 016 Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy Terms of Use Rights and Permissions
7 Page 1 of Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 > 3-8 Slopes of Use slopes to determine whether the opposite sides of quadrilateral ABCD are parallel. 0. A(0, ), B(3, 4), C(, 7), D( 1, 5). A(1, 1), B(5, 3), C(7, 1), D(3, 0) 1. A( 3, 1), B(1, ), C(0, 3), D( 4, 0) 3. A(1, 0), B(4, 0), C(3, 3), D( 1, 3) 4. Analyze Mathematical Relationships (1)(F) Are opposite sides of hexagon RSTUVW at the right parallel? Justify your answer. 5. Which line is perpendicular to 3y + x = 1? A. 6x 4y = 4 B. y + 3x = C. x + 3y = 6 D. y = x + 6 Rewrite each equation in slope-intercept form, if necessary. Then determine whether the lines are perpendicular. Explain. 6. y = x 7 y x = 0 7. y = 3 x = 8. x 7y = 4 4y = 7x 9. Apply Mathematics (1)(A) A community recently converted an old railroad corridor into a recreational trail. The graph at the right shows a map of the trail on a coordinate grid. The community plans to construct a path to connect the trail to a parking lot. The new path will be perpendicular to the recreational trail. a. Write an equation of the line representing the new path. b. What are the coordinates of the point at which the path will meet the recreational trail? 30. Justify Mathematical Arguments (1)(G) Is a triangle with vertices G(3, ), H(8, 5), and K(0, 10) a right triangle? Justify your answer. 31. A triangle has vertices L( 5, 6), M(, 3), and N(4, 5). Write an equation for the line perpendicular to LM that contains point N. TEXAS Test Practice 3. ΔABC is right with right angle C. The slope of AC is. What is the slope of BC?
8 Page of 33. What is the value of x in the diagram at the right? 34. The perimeter of a square is 0 ft. What is the area of the square in square feet? PearsonTEXAS.com Page 133 Copyright 016 Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy Terms of Use Rights and Permissions
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