(0, 4) Figure 12. x + 3. d = c. = b. Figure 13
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1 80 CHAPTER EQUATIONS AND INEQUALITIES Plot both points, and draw a line passing through them as in Figure. Tr It # _, 0 Figure Find the intercepts of the equation and sketch the graph: = _ +. (0, (This handout is an ecerpt from another College Algebra tet. It hast the two parts covering the distance and midpoint formulas and the problems and odd answers. Do problems 7- odd for homework. Using the Distance Formula Derived from the Pthagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pthagorean Theorem, a + b = c, is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hpotenuse. See Figure. (, 7 0 d = c = a (, = b (, 7 Figure The relationship of sides and to side d is the same as that of sides a and b to side c. We use the absolute value smbol to indicate that the length is a positive number because the absolute value of an number is positive. (For eample, =. The smbols and indicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pthagorean Theorem. It follows that the distance formula is given as c = a + b c = a + b d = ( + ( ( + ( We do not have to use the absolute value smbols in this definition because an number squared is positive. the distance formula Given endpoints (, and (,, the distance between two points is given b ( + ( Eample Finding the Distance between Two Points Find the distance between the points (, and (,. Solution Let us first look at the graph of the two points. Connect the points to form a right triangle as in Figure.
2 SECTION. THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 8 (, (, (, Then, calculate the length of d using the distance formula. Figure ( + ( ( ( + ( ( = ( + ( = + = Tr It # Find the distance between two points: (, and (, 9. Eample Finding the Distance between Two Locations Let s return to the situation introduced at the beginning of this section. Tracie set out from Elmhurst, IL, to go to Franklin Park. On the wa, she made a few stops to do errands. Each stop is indicated b a red dot in Figure. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions. Solution The first thing we should do is identif ordered pairs to describe each position. If we set the starting position at the origin, we can identif each of the other points b counting units east (right and north (up on the grid. For eample, the first stop is block east and block north, so it is at (,. The net stop is blocks to the east, so it is at (,. After that, she traveled blocks east and blocks north to (8,. Lastl, she traveled blocks north to (8, 7. We can label these points on the grid as in Figure. Schiller Avenue (8, 7 Bertau Avenue Mannhelm Road McLean Street (0, 0 (, North Avenue (, Wolf Road (8, Figure
3 8 CHAPTER EQUATIONS AND INEQUALITIES Net, we can calculate the distance. Note that each grid unit represents,000 feet. From her starting location to her first stop at (,, Tracie might have driven north,000 feet and then east,000 feet, or vice versa. Either wa, she drove,000 feet to her first stop. Her second stop is at (,. So from (, to (,, Tracie drove east,000 feet. Her third stop is at (8,. There are a number of routes from (, to (8,. Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let s sa she drove east,000 feet and then north,000 feet for a total of,000 feet. Tracie s final stop is at (8, 7. This is a straight drive north from (8, for a total of,000 feet. Net, we will add the distances listed in Table. From/To Number of Feet Driven (0, 0 to (,,000 (, to (,,000 (, to (8,,000 (8, to (8, 7,000 Total,000 Table The total distance Tracie drove is,000 feet, or.8 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points (0, 0 and (8, 7. d = (8 0 + (7 0 = + 9 = 0. units At,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 0,0. feet, or.0 miles. The distance formula results in a shorter calculation because it is based on the hpotenuse of a right triangle, a straight diagonal from the origin to the point (8, 7. Perhaps ou have heard the saing as the crow flies, which means the shortest distance between two points because a crow can fl in a straight line even though a person on the ground has to travel a longer distance on eisting roadwas. Using the Midpoint Formula When the endpoints of a line segment are known, we can find the point midwa between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment, (, and (,, the midpoint formula states how to find the coordinates of the midpoint M. M = ( + + A graphical view of a midpoint is shown in Figure. Notice that the line segments on either side of the midpoint are congruent Figure
4 SECTION. THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 8 Eample 7 Finding the Midpoint of the Line Segment Find the midpoint of the line segment with the endpoints (7, and (9,. Solution Use the formula to find the midpoint of the line segment. Tr It # ( + + = ( = ( 8, + Find the midpoint of the line segment with endpoints (, and ( 8,. Eample 8 Finding the Center of a Circle The diameter of a circle has endpoints (, and (,. Find the center of the circle. Solution The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will ield the center point. ( + + ( +, = (, 8 = (, Access these online resources for additional instruction and practice with the Cartesian coordinate sstem. Plotting Points on the Coordinate Plane ( Find - and -intercepts Based on the Graph of a Line (
5 8 CHAPTER EQUATIONS AND INEQUALITIES. SECTION EXERCISES VERBAL. Is it possible for a point plotted in the Cartesian coordinate sstem to not lie in one of the four quadrants? Eplain.. Describe in our own words what the -intercept of a graph is.. Describe the process for finding the -intercept and the -intercept of a graph algebraicall.. When using the distance formula ( + (, eplain the correct order of operations that are to be performed to obtain the correct answer. ALGEBRAIC For each of the following eercises, find the -intercept and the -intercept without graphing. Write the coordinates of each intercept.. = +. = 7. = 8. = = 9 0. _ = _ + For each of the following eercises, solve the equation for in terms of.. + = 8. =. =. = 7. + = 0. + = 0 For each of the following eercises, find the distance between the two points. Simplif our answers, and write the eact answer in simplest radical form for irrational answers. 7. (, and (, 8. (, and (7, 9. (, 0 and (, 0. (, and (0,. Find the distance between the two points given using our calculator, and round our answer to the nearest hundredth. (9, and (, 7 For each of the following eercises, find the coordinates of the midpoint of the line segment that joins the two given points.. (, and (,. (, and (7,. (, and (, 8. (0, 7 and (, 9. (, 7 and (, GRAPHICAL For each of the following eercises, identif the information requested. 7. What are the coordinates of the origin? 8. If a point is located on the -ais, what is the -coordinate? 9. If a point is located on the -ais, what is the -coordinate? For each of the following eercises, plot the three points on the given coordinate plane. State whether the three points ou plotted appear to be collinear (on the same line. 0. (, (, (, 0. (, (0, (,
6 C- ODD ANSWERS n 8 7n. t + t + 9. b + b b + 7a ab b c c p + p + 9. z + z + z z 8 + ab b _. a + a. d _ d 7. d + 9 d +. 9a 7 a a a b 9. c + c c + c +. Chapter Review Eercises = 7. m 9. Whole. Irrational.. a a _ k k a + ab b. 9p 7. a 9. (a (a + 9. ( +. (h k. (p + (p p + 7. (q p(q + pq + 9p _ 9. (p + ( p. _ +. _. _ m + m _ 9. _ Chapter Practice Test. Rational. =.,, _ q q q. n n + n 8. ( + 9( 9. (c (9c + c + 7. z 9. a + b z b CHAPTER Section.. Answers ma var. Yes. It is possible for a point to be on the -ais or on the -ais and therefore is considered to NOT be in one of the quadrants.. The -intercept is the point where the graph crosses the -ais.. The -intercept is (, 0 and the -intercept is (0,. 7. The -intercept is (, 0 and the -intercept is (0,. 9. The -intercept is (, 0 and the.. =. = -intercept is ( 0, _ 8 9. = _ 7. d = 7 9. d = =. d.97. (, _. (, 7. (0, 0 9. = 0. Not collinear (0, (, (, (, 0 8 (, (0, (, (8, (0,. (,, (,, (, (, (, (0,. (0, (, 0. d = 8.. d = 7. (, 9. = 0, =. = 0.7, = 0. =.7, = 0.. =.8 mi shorter Midpoint of each diagonal is the same point (,. Note this is a characteristic of rectangles, but not other quadrilaterals.. 7 mi. ft Section.. It means the have the same slope.. The eponent of the variable is. It is called a first-degree equation.. If we insert either value into the equation, the make an epression in the equation undefined (zero in the denominator. 7. = 9. = _ 7. =. =. = 7. ; = 9. ; when we solve this we get =, which is ecluded, therefore NO solution. 0; = _. = +. = + 7. = + 9. =. = 7. =. 8 + = 7 7. Parallel Perpendicular
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