0_00.qd 78 /8/05 Chapter 0 8:5 AM Page 78 Topics in Analtic Geometr 0. Lines What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and a line. Wh ou should learn it The inclination of a line can be used to measure heights indirectl. For instance, in Eercise 5 on page 7, the inclination of a line can be used to determine the change in elevation from the base to the top of the Johnstown Inclined Plane. Inclination of a Line In Section., ou learned that the graph of the linear equation m b is a nonvertical line with slope m and -intercept 0, b. There, the slope of a line was described as the rate of change in with respect to. In this section, ou will look at the slope of a line in terms of the angle of inclination of the line. Ever nonhorizontal line must intersect the -ais. The angle formed b such an intersection determines the inclination of the line, as specified in the following definition. Definition of Inclination The inclination of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line. (See Figure 0..) =0 = AP/Wide World Photos Horizontal Line FIGURE 0. Vertical Line Acute Angle Obtuse Angle The inclination of a line is related to its slope in the following manner. Inclination and Slope If a nonvertical line has inclination and slope m, then m tan. The HM mathspace CD-ROM and Eduspace for this tet contain additional resources related to the concepts discussed in this chapter. For a proof of the relation between inclination and slope, see Proofs in Mathematics on page 80.
0_00.qd /8/05 8:5 AM Page 79 Section 0. Lines 79 Eample Finding the Inclination of a Line FIGURE 0. + =. Find the inclination of the line. The slope of this line is m. So, its inclination is determined from the equation tan. From Figure 0., it follows that < arctan <. This means that 0.588 0.588.55. The angle of inclination is about.55 radians or about.. Now tr Eercise 9. The Angle Between Two Lines = Two distinct lines in a plane are either parallel or intersecting. If the intersect and are nonperpendicular, their intersection forms two pairs of opposite angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines. As shown in Figure 0., ou can use the inclinations of the two lines to find the angle between the two lines. If two lines have inclinations and, where < and <, the angle between the two lines is. You can use the formula for the tangent of the difference of two angles tan tan FIGURE 0. tan tan tan tan to obtain the formula for the angle between two lines. Angle Between Two Lines If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan m m m m.
0_00.qd /8/05 8:5 AM Page 70 70 Chapter 0 Topics in Analtic Geometr Eample Finding the Angle Between Two Lines FIGURE 0. + = 0 79.70 = 0 Find the angle between the two lines. Line : 0 Line : 0 The two lines have slopes of m and m, respectivel. So, the tangent of the angle between the two lines is tan m m m. m Finall, ou can conclude that the angle is arctan.9 radians 79.70 as shown in Figure 0.. Now tr Eercise 7. The Distance Between a Point and a Line (, ) (, ) d Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular line segment joining the point and the line, as shown in Figure 0.5. Distance Between a Point and a Line The distance between the point, and the line A B C 0 is d A B C A B. FIGURE 0.5 Remember that the values of A, B, and C in this distance formula correspond to the general equation of a line, A B C 0. For a proof of the distance between a point and a line, see Proofs in Mathematics on page 80. = + (, ) 5 FIGURE 0. Eample Finding the Distance Between a Point and a Line Find the distance between the point, and the line. The general form of the equation is 0. So, the distance between the point and the line is d 8.58 units. 5 The line and the point are shown in Figure 0.. Now tr Eercise 9.
0_00.qd /8/05 8:5 AM Page 7 5 B (0, ) h C (5, ) A (, 0) 5 FIGURE 0.7 Eample An Application of Two Distance Formulas Figure 0.7 shows a triangle with vertices A, 0, B0,, and C5,. a. Find the altitude h from verte B to side AC. b. Find the area of the triangle. a. To find the altitude, use the formula for the distance between line AC and the point 0,. The equation of line AC is obtained as follows. Slope: Equation: m 0 5 8 0 Section 0. Lines 7 Point-slope form Activities. Find the inclination of the line 5 0. Answer: radian or 5.. Find the angle between the lines 5 and. Answer: radians or 8.87. Find the distance between the point, and the line 0. Answer: d 5 unit 0.89.9 Multipl each side b. 0 General form So, the distance between this line and the point 0, is Altitude h 0 7 units. b. Using the formula for the distance between two points, ou can find the length of the base AC to be b 5 0 Distance Formula 8 Simplif. 8 Simplif. 7 units. Simplif. Finall, the area of the triangle in Figure 0.7 is A bh 7 7 square units. Now tr Eercise 5. Formula for the area of a triangle Substitute for b and h. Simplif. Group Activit: Graphing Utilit Put our students in groups of two. Ask each group to write a graphing calculator program that finds the angle between two lines based on user input. Ask each group to demonstrate the program on an eample or eercise in this section. W RITING ABOUT MATHEMATICS Inclination and the Angle Between Two Lines Discuss wh the inclination of a line can be an angle that is larger than, but the angle between two lines cannot be larger than. Decide whether the following statement is true or false: The inclination of a line is the angle between the line and the -ais. Eplain.
0_00.qd /8/05 :05 AM Page 7 7 Chapter 0 Topics in Analtic Geometr 0. Eercises The HM mathspace CD-ROM and Eduspace for this tet contain step-b-step solutions to all odd-numbered eercises. The also provide Tutorial Eercises for additional help. VOCABULARY CHECK: Fill in the blanks.. The of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line.. If a nonvertical line has inclination and slope m, then m.. If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan.. The distance between the point, and the line A B C 0 is given b d. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.eduspace.com. In Eercises 8, find the slope of the line with inclination... = = In Eercises 9, find the inclination degrees) of the line. 9. 8 0 0. 5 9 0. 5 0. 0 0 (in radians and.. = = In Eercises, find the angle between the lines. (in radians and degrees).. 5. radians. radians.7 5.88 7. radians 8. radians In Eercises 9, find the inclination degrees) of the line with a slope of m. 9. m 0. m. m. m. m. m 5 (in radians and In Eercises 5 8, find the inclination (in radians and degrees) of the line passing through the points. 5.,, 0, 8., 8,, 7., 0, 0, 0 8. 0, 00, 50, 0 5. 0. 7. 7 8. 5 9. 8 0. 5 5 5 5
0_00.qd /8/05 8:5 AM Page 7 Section 0. Lines 7. 0.05 0.0 0. 0.07 0.0 0.. 0.0 0.05 0.9 0.0 0.0 0.5 Angle Measurement In Eercises, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles... (, ) 5.. (, ) (, ) (, ) (, ) (, 0) (, ) (, ) (, ) (, ) (, 0) (, ) In Eercises 9 and 50, find the distance between the parallel lines. 9. 50. 5 5. Road Grade A straight road rises with an inclination of 0.0 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road. mi 0 0. radian In Eercises 7, find the distance between the point and the line. Point 7. 0, 0 8. 0, 0 9., 0.,.,. 0, 8. 0, 8., Line 0 0 0 0 0 0 5. Road Grade A straight road rises with an inclination of 0.0 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road. 5. Pitch of a Roof A roof has a rise of feet for ever horizontal change of 5 feet (see figure). Find the inclination of the roof. 5 ft ft In Eercises 5 8, the points represent the vertices of a triangle. (a) Draw triangle ABC in the coordinate plane, (b) find the altitude from verte B of the triangle to side AC, and (c) find the area of the triangle. 5.. 7. 8. A 0, 0, B,, C, 0 A 0, 0, B, 5, C 5, A,, B,, C 5, 0 A, 5, B, 0, C,
0_00.qd /8/05 8:5 AM Page 7 7 Chapter 0 Topics in Analtic Geometr 5. Conveor Design A moving conveor is built so that it rises meter for each meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveor. (c) The conveor runs between two floors in a factor. The distance between the floors is 5 meters. Find the length of the conveor. 55. Truss Find the angles and shown in the drawing of the roof truss. Snthesis ft α True or False? In Eercises 57 and 58, determine whether the statement is true or false. Justif our answer. 57. A line that has an inclination greater than radians has a negative slope. β Model It 9 ft ft ft 5. Inclined Plane The Johnstown Inclined Plane in Johnstown, Pennslvania is an inclined railwa that was designed to carr people to the hilltop communit of Westmont. It also proved useful in carring people and vehicles to safet during severe floods. The railwa is 89.5 feet long with a 70.9% uphill grade (see figure). 89.5 ft Not drawn to scale (a) Find the inclination of the railwa. (b) Find the change in elevation from the base to the top of the railwa. (c) Using the origin of a rectangular coordinate sstem as the base of the inclined plane, find the equation of the line that models the railwa track. (d) Sketch a graph of the equation ou found in part (c). 58. To find the angle between two lines whose angles of inclination and are known, substitute and for m and m, respectivel, in the formula for the angle between two lines. 59. Eploration Consider a line with slope m and -intercept 0,. (a) Write the distance d between the origin and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the origin and the line. (d) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. 0. Eploration Consider a line with slope m and -intercept 0,. (a) Write the distance d between the point, and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the point and the line. (d) Is it possible for the distance to be 0? If so, what is the slope of the line that ields a distance of 0? (e) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. Skills Review In Eercises, find all -intercepts and -intercepts of the graph of the quadratic function.. f 7. f 9. f 5 5. f 5. f 7. f 9 In Eercises 7 7, write the quadratic function in standard form b completing the square. Identif the verte of the function. 7. f 8. f 9. f 5 7 70. f 8 5 7. 7. f f 8 In Eercises 7 7, graph the quadratic function. 7. f 7. f 75. g 7. g 8