2.8 Distance and Midpoint Formulas; Circles

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1 Section.8 Distance and Midpoint Formulas; Circles 9 Eercises are based on the following cartoon. B.C. b permission of Johnn Hart and Creators Sndicate, Inc. 89. Assuming that there is no such thing as metric crickets, I modeled the information in the first frame of the cartoon with the function where Tn is the temperature, in degrees Fahrenheit, and n is the number of cricket chirps per minute. 90. I used the function in Eercise 89 and found an equation for T - n, which epresses the number of cricket chirps per minute as a function of Fahrenheit temperature. In Eercises 9 94, determine whether each statement is true or false. If the statement is false, make the necessar change(s) to produce a true statement. 9. The inverse of 5, 4,, 76 is 5, 7,, The function f = 5 is one-to-one. 9. If f =, then 94. The domain of f is the same as the range of f If f = and g = + 5, find f g - and g - f Show that is its own inverse. Tn = n , f - =. f = Freedom 7 was the spacecraft that carried the first American into space in 96. Total flight time was 5 minutes and the spacecraft reached a maimum height of 6 miles. Consider a function, s, that epresses Freedom 7 s height, st, in miles, after t minutes. Is s a one-to-one function? Eplain our answer. 98. If f = 6, and f is one-to-one, find satisfing 8 + f - - = 0. Group Eercise 99. In Tom Stoppard s pla Arcadia, the characters dream and talk about mathematics, including ideas involving graphing, composite functions, smmetr, and lack of smmetr in things that are tangled, msterious, and unpredictable. Group members should read the pla. Present a report on the ideas discussed b the characters that are related to concepts that we studied in this chapter. Bring in a cop of the pla and read appropriate ecerpts. Preview Eercises Eercises 00 0 will help ou prepare for the material covered in the net section. 00. Let, = 7, and, =, -. Find Epress the answer in simplified radical form. 0. Use a rectangular coordinate sstem to graph the circle with center, - and radius. 0. Solve b completing the square: = 0. Objectives Section Find the distance between two points. Find the midpoint of a line segment. Write the standard form of a circle s equation. Give the center and radius of a circle whose equation is in standard form. Convert the general form of a circle s equation to standard form..8 Distance and Midpoint Formulas; Circles I t s a good idea to know our wa around a circle. Clocks, angles, maps, and compasses are based on circles. Circles occur everwhere in nature: in ripples on water, patterns on a moth s wings, and cross sections of trees. Some consider the circle to be the most pleasing of all shapes. The rectangular coordinate sstem gives us a unique wa of knowing a circle. It enables us to translate a circle s geometric definition into an algebraic equation. To do this, we must first develop a formula for the distance between an two points in rectangular coordinates.

2 94 Chapter Functions and Graphs Find the distance between two points. P (, ) Figure.6 d P (, ) (, ) The Distance Formula Using the Pthagorean Theorem, we can find the distance between the two points P, and P, in the rectangular coordinate sstem. The two points are illustrated in Figure.6. The distance that we need to find is represented b d and shown in blue. Notice that the distance between the two points on the dashed horizontal line is the absolute value of the difference between the -coordinates of the points. This distance, ƒ - ƒ, is shown in pink. Similarl, the distance between the two points on the dashed vertical line is the absolute value of the difference between the -coordinates of the points. This distance, ƒ - ƒ, is also shown in pink. Because the dashed lines are horizontal and vertical, a right triangle is formed. Thus, we can use the Pthagorean Theorem to find the distance d. Squaring the lengths of the triangle s sides results in positive numbers, so absolute value notation is not necessar. d = d = ; Appl the Pthagorean Theorem to the right triangle in Figure.6. Appl the square root propert. d = This result is called the distance formula. Because distance is nonnegative, write onl the principal square root. The Distance Formula The distance, d, between the points, and, in the rectangular coordinate sstem is d = To compute the distance between two points, find the square of the difference between the -coordinates plus the square of the difference between the -coordinates. The principal square root of this sum is the distance. When using the distance formula, it does not matter which point ou call, and which ou call,. EXAMPLE Using the Distance Formula Find the distance between -, 4 and, -. Solution We will let, = -, 4 and, =, -. d = = Use the distance formula. Substitute the given values. (, 4) Distance is units. 4 5 (, ) Figure.6 Finding the distance between two points = = 6+6 = 5 Caution: This does not equal = 4 # = L 7. Perform operations inside grouping smbols: - - = + = 4 and = -6. Square 4 and -6. Add. 5 = 4 # = 4 = The distance between the given points is units, or approimatel 7. units. The situation is illustrated in Figure.6.

3 Section.8 Distance and Midpoint Formulas; Circles 95 Check Point Find the distance between -4, 9 and, -. Find the midpoint of a line segment. The Midpoint Formula The distance formula can be used to derive a formula for finding the midpoint of a line segment between two given points. The formula is given as follows: The Midpoint Formula Consider a line segment whose endpoints are, and,. The coordinates of the segment s midpoint are +, +. To find the midpoint, take the average of the two -coordinates and the average of the two -coordinates. Stud Tip The midpoint formula requires finding the sum of coordinates. B contrast, the distance formula requires finding the difference of coordinates: Midpoint: Sum of coordinates Distance: Difference of coordinates a + +, b A - B +A - B It s eas to confuse the two formulas. Be sure to use addition, not subtraction, when appling the midpoint formula. EXAMPLE Using the Midpoint Formula ( 8, 4) 7 (, 5) Midpoint Figure.6 Finding a line segment s midpoint (, 6) Find the midpoint of the line segment with endpoints, -6 and -8, -4. Solution To find the coordinates of the midpoint, we average the coordinates of the endpoints. +( 8) 6+( 4) a b a 7 0 Midpoint=, =, b a 7 =, 5b Average the -coordinates. Average the -coordinates. Figure.6 illustrates that the point A - 7, -5B is midwa between the points, -6 and -8, -4. Check Point Find the midpoint of the line segment with endpoints (, ) and 7, -.

4 96 Chapter Functions and Graphs Circles Our goal is to translate a circle s geometric definition into an equation. We begin with this geometric definition. Definition of a Circle A circle is the set of all points in a plane that are equidistant from a fied point, called the center. The fied distance from the circle s center to an point on the circle is called the radius. Center (h, k) Figure.64 is our starting point for obtaining a circle s equation. We ve placed the circle into a rectangular coordinate sstem. The circle s center is h, k and its radius is r. We let, represent the coordinates of an point on the circle. What does the geometric definition of a circle tell us about the point, in Figure.64? The point is on the circle if and onl if its distance from the center is r. We can use the distance formula to epress this idea algebraicall: Radius: r (, ) An point on the circle The distance between (, ) and (h, k) is alwas r. A-hB +A-kB = r. Figure.64 A circle centered at h, k with radius r Squaring both sides of 4 - h + - k = r ields the standard form of the equation of a circle. Write the standard form of a circle s equation. The Standard Form of the Equation of a Circle The standard form of the equation of a circle with center h, k and radius r is - h + - k = r. EXAMPLE Finding the Standard Form of a Circle s Equation (, ) Write the standard form of the equation of the circle with center (0, 0) and radius. Graph the circle. Solution The center is (0, 0). Because the center is represented as h, k in the standard form of the equation, h = 0 and k = 0. The radius is, so we will let r = in the equation. (0, 0) - h + - k = r = This is the standard form of a circle s equation. Substitute 0 for h, 0 for k, and for r. + = 4 + = 4 Simplif. Figure.65 + = 4 The graph of The standard form of the equation of the circle is + = 4. Figure.65 shows the graph. Check Point Write the standard form of the equation of the circle with center (0, 0) and radius 4.

5 Section.8 Distance and Midpoint Formulas; Circles 97 Technolog = 4 To graph a circle with a graphing utilit, first solve the equation for. + = 4 = 4 - = 4- is not a function of. = 4 Graph the two equations = 4 - and = -4 - in the same viewing rectangle. The graph of = 4 - is the top semicircle because is alwas positive. The graph of = -4 - is the bottom semicircle because is alwas negative. Use a setting so that the circle looks like a circle. (Man graphing utilities have problems connecting the two semicircles because the segments directl to the left and to the right of the center become nearl vertical.) Eample and Check Point involved circles centered at the origin. The standard form of the equation of all such circles is + = r, where r is the circle s radius. Now, let s consider a circle whose center is not at the origin. EXAMPLE 4 Finding the Standard Form of a Circle s Equation Write the standard form of the equation of the circle with center -, and radius 4. Solution The center is -,. Because the center is represented as h, k in the standard form of the equation, h = - and k =. The radius is 4, so we will let r = 4 in the equation. - h + - k = r This is the standard form of a circle s equation = 4 Substitute - for h, for k, and 4 for r = 6 Simplif. The standard form of the equation of the circle is = 6. Check Point 4 Write the standard form of the equation of the circle with center 0, -6 and radius 0. ZOOM SQUARE Give the center and radius of a circle whose equation is in standard form. EXAMPLE 5 Using the Standard Form of a Circle s Equation to Graph the Circle a. Find the center and radius of the circle whose equation is = 9. b. Graph the equation. c. Use the graph to identif the relation s domain and range. Solution a. We begin b finding the circle s center, h, k, and its radius, r. We can find the values for h, k, and r b comparing the given equation to the standard form of the equation of a circle, - h + - k = r = 9 A-B +A-( 4)) = This is ( h), with h =. This is ( k), with k = 4. This is r, with r =. We see that h =, k = -4, and r =. Thus, the circle has center h, k =, -4 and a radius of units.

6 98 Chapter Functions and Graphs Range (, 4) Domain (, ) (, 4) (, 7) Figure.66 The graph of = 9 (5, 4) b. To graph this circle, first locate the center, -4. Because the radius is, ou can locate at least four points on the circle b going out three units to the right, to the left, up, and down from the center. The points three units to the right and to the left of, -4 are 5, -4 and -, -4, respectivel. The points three units up and down from, -4 are, - and, -7, respectivel. Using these points, we obtain the graph in Figure.66. c. The four points that we located on the circle can be used to determine the relation s domain and range. The points -, -4 and 5, -4 show that values of etend from - to 5, inclusive: Domain = -, 54. The points, -7 and, - show that values of etend from -7 to -, inclusive: Stud Tip Range = -7, -4. It s eas to make sign errors when finding h and k, the coordinates of a circle s center, h, k. Keep in mind that h and k are the numbers that follow the subtraction signs in a circle s equation: (-) +(+4) =9 (-) +(-( 4)) =9. The number after the subtraction is : h =. The number after the subtraction is 4: k = 4. Check Point 5 a. Find the center and radius of the circle whose equation is b. Graph the equation. c. Use the graph to identif the relation s domain and range. If we square - and + 4 in the standard form of the equation in Eample 5, we obtain another form for the circle s equation = = = = 0 The General Form of the Equation of a Circle The general form of the equation of a circle is where D, E, and F are real numbers = 4. This is the standard form of the equation in Eample 5. Square - and + 4. Combine constants and rearrange terms. Subtract 9 from both sides. This result suggests that an equation in the form + + D + E + F = 0 can represent a circle. This is called the general form of the equation of a circle. + + D + E + F = 0,

7 Section.8 Distance and Midpoint Formulas; Circles 99 Convert the general form of a circle s equation to standard form. We can convert the general form of the equation of a circle to the standard form - h + - k = r. We do so b completing the square on and. Let s see how this is done. EXAMPLE 6 Converting the General Form of a Circle s Equation to Standard Form and Graphing the Circle Stud Tip To review completing the square, see Section.5, pages (, ) Figure.67 The graph of = Write in standard form and graph: = 0. Solution Because we plan to complete the square on both and, let s rearrange the terms so that -terms are arranged in descending order, -terms are arranged in descending order, and the constant term appears on the right = = ( +4+4)+( -6+9)= = 6 This is the given equation. Rewrite in anticipation of completing the square. Complete the square on : # and 4 = = 4, so add 4 to both sides. Complete the square on : and - -6 = - = 9, so add 9 to both sides. Factor on the left and add on the right. This last equation is in standard form.we can identif the circle s center and radius b comparing this equation to the standard form of the equation of a circle, - h + - k = r = 6 A-( )B +A-) =6 This is ( h), with h =. Remember that numbers added on the left side must also be added on the right side. This is ( k), with k =. This is r, with r = 6. We use the center, h, k = -,, and the radius, r = 6, to graph the circle. The graph is shown in Figure.67. Technolog To graph = 0, the general form of the circle s equation given in Eample 6, rewrite the equation as a quadratic equation in = 0 Now solve for using the quadratic formula, with a =, b = -6, and c = = -b ; b - 4ac a = --6 ; # # = 6 ; 4 Because we will enter these equations, there is no need to simplif. Enter and = = Use a ZOOM SQUARE setting. The graph is shown on the right.

8 00 Chapter Functions and Graphs Check Point 6 Write in standard form and graph: = 0. Eercise Set.8 Practice Eercises In Eercises 8, find the distance between each pair of points. If necessar, round answers to two decimals places.. (, ) and (4, 8). (5, ) and (8, 5). 4, - and -6, 4., - and -, 5 5. (0, 0) and -, 4 6. (0, 0) and, , -6 and, , - and, , - and (4, ) 0. 0, - and (4, ). (.5, 8.) and -0.5, 6.. (.6,.) and.6, A0, - B and A 5, 0B 4. A0, - B and A 7, 0B 5. A, 5B and A -, 45B 6. A, 6B and A -, 56B 7. a 7 and, 6, 5 b 5 b 8. a - and a 4, 6 4, - 7 b 7 b In Eercises 9 0, find the midpoint of each line segment with the given endpoints. 9. (6, 8) and (, 4) 0. (0, 4) and (, 6). -, -8 and -6, -. -4, -7 and -, -. -, -4 and 6, , - and -8, 6 5. a and a -, -, b b 6. a - and a - 5, - 4 5, 7 5 b 5 b 7. A8, 5B and A -6, 75B 8. A7, -6B and A, -B 9. A 8, -4B and A, 4B 0. A 50, -6B and A, 6B In Eercises 40, write the standard form of the equation of the circle with the given center and radius.. Center (0, 0), r = 7. Center (0, 0), r = 8. Center (, ), r = 5 4. Center, -, r = 4 5. Center -, 4, r = 6. Center -, 5, r = 7. Center -, -, r = 8. Center -5, -, r = 5 9. Center -4, 0, r = Center -, 0, r = 6 In Eercises 4 5, give the center and radius of the circle described b the equation and graph each equation. Use the graph to identif the relation s domain and range = = = = = = = = = = = = 6 In Eercises 5 64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation = = = = = = = = = = = = 0 Practice Plus In Eercises 65 66, a line segment through the center of each circle intersects the circle at the points shown. a. Find the coordinates of the circle s center. b. Find the radius of the circle. c. Use our answers from parts (a) and (b) to write the standard form of the circle s equation (7, ) (, 9) (, 6) (5, 4)

9 Section.8 Distance and Midpoint Formulas; Circles 0 In Eercises 67 70, graph both equations in the same rectangular coordinate sstem and find all points of intersection. Then show that these ordered pairs satisf the equations = 6 - = 4 + = 9 - = = 4 = = 9 = - Application Eercises The cell phone screen shows coordinates of si cities from a rectangular coordinate sstem placed on North America b long-distance telephone companies. Each unit in this sstem represents 0. mile. Source: Peter H. Dana San Francisco Chicago New Orleans Denver Houston Boston (8495, 870) (5985, 49) (8448, 65) (7490, 588) (896, 54) (44, 4) In Eercises 7 7, use this information to find the distance, to the nearest mile, between each pair of cities. 7. Boston and San Francisco 7. New Orleans and Houston 7. A rectangular coordinate sstem with coordinates in miles is placed with the origin at the center of Los Angeles. The figure indicates that the Universit of Southern California is located.4 miles west and.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake s epicenter is estimated to be approimatel 0 miles from the universit. Write the standard form of the equation for the set of points that could be the epicenter of the quake. Los Angeles (0, 0) 74. The Ferris wheel in the figure has a radius of 68 feet. The clearance between the wheel and the ground is 4 feet. The rectangular coordinate sstem shown has its origin on the ground directl below the center of the wheel. Use the coordinate sstem to write the equation of the circular wheel. Origin Writing in Mathematics Ground 68 feet Clearance 4 feet 75. In our own words, describe how to find the distance between two points in the rectangular coordinate sstem. 76. In our own words, describe how to find the midpoint of a line segment if its endpoints are known. 77. What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation. 78. Give an eample of a circle s equation in standard form. Describe how to find the center and radius for this circle. 79. How is the standard form of a circle s equation obtained from its general form? 80. Does = 0 represent the equation of a circle? If not, describe the graph of this equation. 8. Does = -5 represent the equation of a circle? What sort of set is the graph of this equation? 8. Write and solve a problem about the fling time between a pair of cities shown on the cell phone screen for Eercises 7 7. Do not use the pairs in Eercise 7 or Eercise 7. Begin b determining a reasonable average speed, in miles per hour, for a jet fling between the cities. USC (.4,.7) 0 miles Technolog Eercises In Eercises 8 85, use a graphing utilit to graph each circle whose equation is given = = = 0

10 0 Chapter Functions and Graphs Critical Thinking Eercises Make Sense? In Eercises 86 89, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 86. I ve noticed that in mathematics, one topic often leads logicall to a new topic: 94. Show that the points A, + d, B, + d, and C6, 6 + d are collinear (lie along a straight line) b showing that the distance from A to B plus the distance from B to C equals the distance from A to C. 95. Prove the midpoint formula b using the following procedure. a. Show that the distance between, and +, + is equal to the distance between Pthagorean Theorem leads to Distance Formula leads to Equations of Circles, and +, +. b. Use the procedure from Eercise 94 and the distances from part (a) to show that the points,, +, +, and, are collinear. 87. To avoid sign errors when finding h and k, I place parentheses around the numbers that follow the subtraction signs in a circle s equation. 88. I used the equation = -4 to identif the circle s center and radius. 89. M graph of = 6 is m graph of + = 6 translated two units right and one unit down. In Eercises 90 9, determine whether each statement is true or false. If the statement is false, make the necessar change(s) to produce a true statement. 90. The equation of the circle whose center is at the origin with radius 6 is + = The graph of = 6 is a circle with radius 6 centered at -, The graph of = 5 is a circle with radius 5 centered at 4, The graph of = -6 is a circle with radius 6 centered at, Find the area of the donut-shaped region bounded b the graphs of = 5 and = A tangent line to a circle is a line that intersects the circle at eactl one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is + = 5 at the point, -4. Preview Eercises Eercises will help ou prepare for the material covered in the first section of the net chapter. In Eercises 98 99, solve each quadratic equation b the method of our choice = = Use the graph of f = to graph g = + +. Chapter Summar Summar, Review, and Test DEFINITIONS AND CONCEPTS EXAMPLES. Basics of Functions and Their Graphs a. A relation is an set of ordered pairs. The set of first components is the domain of the relation and the set of E., p. 98 second components is the range. b. A function is a correspondence from a first set, called the domain, to a second set, called the range, such that E., p. 0 each element in the domain corresponds to eactl one element in the range. If an element in a relation s domain corresponds to more than one element in the range, the relation is not a function. c. Functions are usuall given in terms of equations involving and, in which is the independent variable E., p. 0; and is the dependent variable. If an equation is solved for and more than one value of can be obtained E. 4, p. 0 for a given, then the equation does not define as a function of. If an equation defines a function, the value of the function at, f, often replaces. d. The graph of a function is the graph of its ordered pairs. E. 5, p. 04 e. The vertical line test for functions: If an vertical line intersects a graph in more than one point, the graph E. 6, p. 05 does not define as a function of.

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