{"id":17655,"date":"2021-07-25T22:16:06","date_gmt":"2021-07-25T22:16:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/?post_type=chapter&#038;p=17655"},"modified":"2021-08-21T02:36:05","modified_gmt":"2021-08-21T02:36:05","slug":"1-1-the-distance-and-midpoint-formulas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/1-1-the-distance-and-midpoint-formulas\/","title":{"raw":"Section 1.1 The Distance and Midpoint Formulas","rendered":"Section 1.1 The Distance and Midpoint Formulas"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define the components of the Cartesian coordinate system.<\/li>\r\n \t<li>Plot points on the Cartesian coordinate plane.<\/li>\r\n \t<li>Use the distance formula to find the distance between two points in the plane.<\/li>\r\n \t<li>Use the midpoint formula to find the midpoint between two points.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.\r\n\r\nWhile there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.\r\n\r\nThe Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/> <b>The Cartesian coordinate system with all four quadrants labeled.<\/b>[\/caption]\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]92752[\/ohm_question]\r\n\r\n<\/div>\r\nThe center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042401\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\" \/>\r\n\r\nEach point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of <em>x\u00a0<\/em>and <em>y<\/em>-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/> <b>An illustration of how to plot the point (3,-1).<\/b>[\/caption]\r\n\r\nWhen dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>y-<\/em>axis may be divided and labeled by increments of 2 or 10 or 100. In fact, the axes may represent other units such as years against the balance in a savings account or quantity against cost. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Cartesian Coordinate System<\/h3>\r\nA two-dimensional plane where the\r\n<ul>\r\n \t<li><em>x<\/em>-axis is the horizontal axis<\/li>\r\n \t<li><em>y<\/em>-axis is the vertical axis<\/li>\r\n<\/ul>\r\nA point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/latex], such that <em>x <\/em>is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\r\nPlot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.\r\n[reveal-answer q=\"380739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380739\"]\r\n\r\nTo plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is \u20132, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y <\/em>direction.\r\n\r\nTo plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 3, so move three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y <\/em>direction.\r\n\r\nTo plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]92753[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=7JMXi_FxA2o\r\n<h2>The Distance Formula<\/h2>\r\nDerived from the <strong>Pythagorean Theorem<\/strong>, the <strong>distance formula<\/strong> is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex], is based on a right triangle where <em>a <\/em>and <em>b<\/em> are the lengths of the legs adjacent to the right angle, and <em>c<\/em> is the length of the hypotenuse.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042428\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\" \/>\r\n\r\nThe relationship of sides [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] to side <em>d<\/em> is the same as that of sides <em>a <\/em>and <em>b <\/em>to side <em>c.<\/em> We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex]|-3|=3[\/latex]. ) The symbols [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] indicate that the lengths of the sides of the triangle are positive. To find the length <em>c<\/em>, take the square root of both sides of the Pythagorean Theorem.\r\n<div style=\"text-align: center;\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\rightarrow c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/div>\r\nIt follows that the distance formula is given as\r\n<div style=\"text-align: center;\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\r\nWe do not have to use the absolute value symbols in this definition because any number squared is positive.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Distance Formula<\/h3>\r\nGiven endpoints [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the distance between two points is given by\r\n<div style=\"text-align: center;\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Distance between Two Points<\/h3>\r\nFind the distance between the points [latex]\\left(-3,-1\\right)[\/latex] and [latex]\\left(2,3\\right)[\/latex].\r\n\r\n[reveal-answer q=\"737169\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"737169\"]\r\n\r\nLet us first look at the graph of the two points. Connect the points to form a right triangle.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042430\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\" \/>\r\n\r\nThen, calculate the length of <em>d <\/em>using the distance formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ =\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ =\\sqrt{25+16}\\hfill \\\\ =\\sqrt{41}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<div>Find the distance between two points: [latex]\\left(1,4\\right)[\/latex] and [latex]\\left(11,9\\right)[\/latex].\r\n[reveal-answer q=\"934526\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"934526\"][latex]\\sqrt{125}=5\\sqrt{5}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\nIn the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.\r\n\r\nhttps:\/\/youtu.be\/Vj7twkiUgf0\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Distance between Two Locations<\/h3>\r\nLet\u2019s return to the situation introduced at the beginning of this section.\r\n\r\nTracie set out from Elmhurst, IL to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.\r\n\r\n[reveal-answer q=\"411250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"411250\"]\r\n\r\nThe first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at [latex]\\left(1,1\\right)[\/latex]. The next stop is 5 blocks to the east so it is at [latex]\\left(5,1\\right)[\/latex]. After that, she traveled 3 blocks east and 2 blocks north to [latex]\\left(8,3\\right)[\/latex]. Lastly, she traveled 4 blocks north to [latex]\\left(8,7\\right)[\/latex]. We can label these points on the grid.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042433\/CNX_CAT_Figure_02_01_017.jpg\" alt=\"This is an image of a road map of a city. The point (1, 1) is on North Avenue and Bertau Avenue. The point (5, 1) is on North Avenue and Wolf Road. The point (8, 3) is on Mannheim Road and McLean Street. The point (8, 7) is on Mannheim Road and Schiller Avenue.\" width=\"731\" height=\"480\" \/>\r\n\r\nNext, we can calculate the distance. Note that each grid unit represents 1,000 feet.\r\n<ul>\r\n \t<li>From her starting location to her first stop at [latex]\\left(1,1\\right)[\/latex], Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.<\/li>\r\n \t<li>Her second stop is at [latex]\\left(5,1\\right)[\/latex]. So from [latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex], Tracie drove east 4,000 feet.<\/li>\r\n \t<li>Her third stop is at [latex]\\left(8,3\\right)[\/latex]. There are a number of routes from [latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]. Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let\u2019s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.<\/li>\r\n \t<li>Tracie\u2019s final stop is at [latex]\\left(8,7\\right)[\/latex]. This is a straight drive north from [latex]\\left(8,3\\right)[\/latex] for a total of 4,000 feet.<\/li>\r\n<\/ul>\r\nNext, we will add the distances listed in the table.\r\n<table summary=\"A table with 6 rows and 2 columns. The entries in the first row are: From\/To and Number of Feet Driven. The entries in the second row are: (0, 0) to (1, 1) and 2,000. The entries in the third row are: (1, 1) to (5, 1) and 4,000. The entries in the fourth row are: (5, 1) to (8, 3) and 5,000. The entries in the fourth row are: (8, 3) to (8, 7) and 4,000. The entries in the sixth row are: Total and 15,000.\">\r\n<thead>\r\n<tr>\r\n<th>From\/To<\/th>\r\n<th>Number of Feet Driven<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(0,0\\right)[\/latex] to [latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<td>2,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex]<\/td>\r\n<td>4,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]<\/td>\r\n<td>5,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(8,3\\right)[\/latex] to [latex]\\left(8,7\\right)[\/latex]<\/td>\r\n<td>4,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td>15,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe total distance Tracie drove is 15,000 feet or 2.84 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 [latex]\\left(0,0\\right)[\/latex] and [latex]\\left(8,7\\right)[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left(8 - 0\\right)}^{2}+{\\left(7 - 0\\right)}^{2}}\\hfill \\\\ =\\sqrt{64+49}\\hfill \\\\ =\\sqrt{113}\\hfill \\\\ =10.63\\text{ units}\\hfill \\end{array}[\/latex]<\/div>\r\nAt 1,000 feet per grid unit, the distance between Elmhurst, IL to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point [latex]\\left(8,7\\right)[\/latex]. Perhaps you have heard the saying \"as the crow flies,\" which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Using the Midpoint Formula<\/h2>\r\nWhen the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the <strong>midpoint formula<\/strong>. Given the endpoints of a line segment, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[\/latex].\r\n<div style=\"text-align: center;\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\r\nA graphical view of a midpoint is shown below. Notice that the line segments on either side of the midpoint are congruent.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042436\/CNX_CAT_Figure_02_01_018.jpg\" alt=\"This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted. A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.\" width=\"487\" height=\"290\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Midpoint of the Line Segment<\/h3>\r\nFind the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"788934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"788934\"]\r\n\r\nUse the formula to find the midpoint of the line segment.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&amp;=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&amp;=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].\r\n\r\n[reveal-answer q=\"964077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"964077\"]\r\n\r\n[latex]\\left(-5,\\frac{5}{2}\\right)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]2308[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]110938[\/ohm_question]\r\n\r\n<\/div>\r\n<dl id=\"fs-id1165135315542\" class=\"definition\">\r\n \t<dd><\/dd>\r\n<\/dl>\r\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 1.1 Homework Exercises<\/span><\/h2>\r\nFor each of the following exercises, find the distance between the two points.\u00a0 Simplify your answers, and write the exact answer in simplest radical form for irrational answers.\r\n\r\n1.\u00a0 [latex](-4,1)[\/latex] and\u00a0[latex](3,-4)[\/latex]\r\n\r\n2.\u00a0 [latex](2,-5)[\/latex] and\u00a0[latex](7,4)[\/latex]\r\n\r\n3.\u00a0 [latex](5,0)[\/latex] and\u00a0[latex](5,6)[\/latex]\r\n\r\n4.\u00a0 [latex](-4,3)[\/latex] and\u00a0[latex](10,3)[\/latex]\r\n\r\n5. Find the distance between the two points given your calculator, and round your answer to the nearest hundredth.\u00a0\u00a0[latex](19,12)[\/latex] and\u00a0[latex](41,71)[\/latex]\r\n\r\nFor each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.\r\n\r\n6.\u00a0 [latex](-5,6)[\/latex] and\u00a0[latex](4,2)[\/latex]\r\n\r\n7.\u00a0 [latex](-1,1)[\/latex] and\u00a0[latex](7,-4)[\/latex]\r\n\r\n8.\u00a0 [latex](-5,-3)[\/latex] and\u00a0[latex](-2,-8)[\/latex]\r\n\r\n9.\u00a0 [latex](0,7)[\/latex] and\u00a0[latex](4,-9)[\/latex]\r\n\r\n10.\u00a0 [latex](-43,17)[\/latex] and\u00a0[latex](23,-34)[\/latex]\r\n\r\nFor each of the following exercises, identify the information requested.\r\n\r\n11. What are the coordinates of the origin?\r\n\r\n12. If a point is located on the <em>y<\/em>-axis, what is the <em>x<\/em>-coordinate?\r\n\r\n13. If a point is located on the <em>x<\/em>-axis, what is the <em>y<\/em>-coordinate?\r\n\r\nFor each of the following exercises, plot the three points on the same coordinate plane.\u00a0 State whether the three points you plotted appear to be collinear (on the same line).\r\n\r\n14. [latex](4,1),(-2,3),(5,0)[\/latex]\r\n\r\n15. [latex](-1,2),(0,4),(2,1)[\/latex]\r\n\r\n16. [latex](-3,0),(-3,4),(-3,-3)[\/latex]\r\n\r\n17. Name the coordinates of the points graphed below.\r\n<div align=\"left\">\r\n\r\n<img src=\"http:\/\/www.hutchmath.com\/Images\/PlotGraph.jpg\" alt=\"Graph with three points A, B, C, plotted\" \/>\r\n\r\n18. Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.\r\n\r\na.\u00a0 [latex](-3,4)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 b.\u00a0 [latex](-5,0)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0c.\u00a0 [latex](1,-4)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 d.\u00a0 [latex](-2,7)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0e.\u00a0\u00a0[latex](0,-3)[\/latex]\r\n\r\nFor questions 19 - 23, use the graph in the figure below.\r\n<img src=\"http:\/\/www.hutchmath.com\/Images\/PlotLine.jpg\" alt=\"Graph of line segment connecting (-3,4) and (5,2)\" \/>\r\n\r\n19. Find the distance between the two endpoints using the distance formula. Round to three decimal places.\r\n\r\n20. Find the coordinates of the midpoint of the line segment connecting the two points.\r\n\r\n21. Find the distance that [latex](-3,4)[\/latex] is from the origin.\r\n\r\n22. Find the distance that [latex](5,2)[\/latex] is from the origin.\u00a0 Round to three decimal places.\r\n\r\n23. Which point is closer to the origin?\r\n\r\n24. A man drove 10 mi directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?\r\n\r\n25. If the road was made in the previous exercise, how much shorter would the man's one way trip be everyday?\r\n\r\n26 Given these four points [latex]A=(1,3),\\text{ }B=(-3,5),\\text{ }C=(4,7),\\text{ }D=(5,-4)[\/latex], find the coordinates of the midpoint of line segments [latex]\\overline{AB}[\/latex] and [latex]\\overline{CD}[\/latex].\r\n\r\n27. After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.\r\n\r\n28. The coordinates on a map for San Francisco are [latex](53,17)[\/latex] and those for Sacramento are [latex](123,78)[\/latex]. Note that the coordinates represent miles. Find the distance between the cities to the nearest mile.\r\n\r\n29. If San Jose's coordinates are [latex](76,-12)[\/latex], where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.\r\n\r\n30. A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were [latex](49,64)[\/latex]. One rescue boat is at the coordinates [latex](60,82)[\/latex] and a second Coast Guard craft is at coordinates [latex](58,47)[\/latex]. Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define the components of the Cartesian coordinate system.<\/li>\n<li>Plot points on the Cartesian coordinate plane.<\/li>\n<li>Use the distance formula to find the distance between two points in the plane.<\/li>\n<li>Use the midpoint formula to find the midpoint between two points.<\/li>\n<\/ul>\n<\/div>\n<p>An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.<\/p>\n<p>While there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.<\/p>\n<p>The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>The Cartesian coordinate system with all four quadrants labeled.<\/b><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm92752\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92752&theme=oea&iframe_resize_id=ohm92752&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042401\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\" \/><\/p>\n<p>Each point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of <em>x\u00a0<\/em>and <em>y<\/em>-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>An illustration of how to plot the point (3,-1).<\/b><\/p>\n<\/div>\n<p>When dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>y-<\/em>axis may be divided and labeled by increments of 2 or 10 or 100. In fact, the axes may represent other units such as years against the balance in a savings account or quantity against cost. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Cartesian Coordinate System<\/h3>\n<p>A two-dimensional plane where the<\/p>\n<ul>\n<li><em>x<\/em>-axis is the horizontal axis<\/li>\n<li><em>y<\/em>-axis is the vertical axis<\/li>\n<\/ul>\n<p>A point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/latex], such that <em>x <\/em>is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\n<p>Plot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380739\">Show Solution<\/span><\/p>\n<div id=\"q380739\" class=\"hidden-answer\" style=\"display: none\">\n<p>To plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is \u20132, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y <\/em>direction.<\/p>\n<p>To plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 3, so move three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y <\/em>direction.<\/p>\n<p>To plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm92753\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92753&theme=oea&iframe_resize_id=ohm92753&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Plotting Points on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7JMXi_FxA2o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Distance Formula<\/h2>\n<p>Derived from the <strong>Pythagorean Theorem<\/strong>, the <strong>distance formula<\/strong> is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex], is based on a right triangle where <em>a <\/em>and <em>b<\/em> are the lengths of the legs adjacent to the right angle, and <em>c<\/em> is the length of the hypotenuse.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042428\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\" \/><\/p>\n<p>The relationship of sides [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] to side <em>d<\/em> is the same as that of sides <em>a <\/em>and <em>b <\/em>to side <em>c.<\/em> We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex]|-3|=3[\/latex]. ) The symbols [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] indicate that the lengths of the sides of the triangle are positive. To find the length <em>c<\/em>, take the square root of both sides of the Pythagorean Theorem.<\/p>\n<div style=\"text-align: center;\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\rightarrow c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/div>\n<p>It follows that the distance formula is given as<\/p>\n<div style=\"text-align: center;\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<p>We do not have to use the absolute value symbols in this definition because any number squared is positive.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Distance Formula<\/h3>\n<p>Given endpoints [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the distance between two points is given by<\/p>\n<div style=\"text-align: center;\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Distance between Two Points<\/h3>\n<p>Find the distance between the points [latex]\\left(-3,-1\\right)[\/latex] and [latex]\\left(2,3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q737169\">Show Solution<\/span><\/p>\n<div id=\"q737169\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let us first look at the graph of the two points. Connect the points to form a right triangle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042430\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\" \/><\/p>\n<p>Then, calculate the length of <em>d <\/em>using the distance formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ =\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ =\\sqrt{25+16}\\hfill \\\\ =\\sqrt{41}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<div>Find the distance between two points: [latex]\\left(1,4\\right)[\/latex] and [latex]\\left(11,9\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q934526\">Show Solution<\/span><\/p>\n<div id=\"q934526\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sqrt{125}=5\\sqrt{5}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Example:  Determine the Distance Between Two Points\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Vj7twkiUgf0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Distance between Two Locations<\/h3>\n<p>Let\u2019s return to the situation introduced at the beginning of this section.<\/p>\n<p>Tracie set out from Elmhurst, IL to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q411250\">Show Solution<\/span><\/p>\n<div id=\"q411250\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at [latex]\\left(1,1\\right)[\/latex]. The next stop is 5 blocks to the east so it is at [latex]\\left(5,1\\right)[\/latex]. After that, she traveled 3 blocks east and 2 blocks north to [latex]\\left(8,3\\right)[\/latex]. Lastly, she traveled 4 blocks north to [latex]\\left(8,7\\right)[\/latex]. We can label these points on the grid.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042433\/CNX_CAT_Figure_02_01_017.jpg\" alt=\"This is an image of a road map of a city. The point (1, 1) is on North Avenue and Bertau Avenue. The point (5, 1) is on North Avenue and Wolf Road. The point (8, 3) is on Mannheim Road and McLean Street. The point (8, 7) is on Mannheim Road and Schiller Avenue.\" width=\"731\" height=\"480\" \/><\/p>\n<p>Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.<\/p>\n<ul>\n<li>From her starting location to her first stop at [latex]\\left(1,1\\right)[\/latex], Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.<\/li>\n<li>Her second stop is at [latex]\\left(5,1\\right)[\/latex]. So from [latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex], Tracie drove east 4,000 feet.<\/li>\n<li>Her third stop is at [latex]\\left(8,3\\right)[\/latex]. There are a number of routes from [latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]. Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let\u2019s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.<\/li>\n<li>Tracie\u2019s final stop is at [latex]\\left(8,7\\right)[\/latex]. This is a straight drive north from [latex]\\left(8,3\\right)[\/latex] for a total of 4,000 feet.<\/li>\n<\/ul>\n<p>Next, we will add the distances listed in the table.<\/p>\n<table summary=\"A table with 6 rows and 2 columns. The entries in the first row are: From\/To and Number of Feet Driven. The entries in the second row are: (0, 0) to (1, 1) and 2,000. The entries in the third row are: (1, 1) to (5, 1) and 4,000. The entries in the fourth row are: (5, 1) to (8, 3) and 5,000. The entries in the fourth row are: (8, 3) to (8, 7) and 4,000. The entries in the sixth row are: Total and 15,000.\">\n<thead>\n<tr>\n<th>From\/To<\/th>\n<th>Number of Feet Driven<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\left(0,0\\right)[\/latex] to [latex]\\left(1,1\\right)[\/latex]<\/td>\n<td>2,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]<\/td>\n<td>5,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(8,3\\right)[\/latex] to [latex]\\left(8,7\\right)[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>15,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The total distance Tracie drove is 15,000 feet or 2.84 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 [latex]\\left(0,0\\right)[\/latex] and [latex]\\left(8,7\\right)[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left(8 - 0\\right)}^{2}+{\\left(7 - 0\\right)}^{2}}\\hfill \\\\ =\\sqrt{64+49}\\hfill \\\\ =\\sqrt{113}\\hfill \\\\ =10.63\\text{ units}\\hfill \\end{array}[\/latex]<\/div>\n<p>At 1,000 feet per grid unit, the distance between Elmhurst, IL to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point [latex]\\left(8,7\\right)[\/latex]. Perhaps you have heard the saying &#8220;as the crow flies,&#8221; which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Using the Midpoint Formula<\/h2>\n<p>When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the <strong>midpoint formula<\/strong>. Given the endpoints of a line segment, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\n<p>A graphical view of a midpoint is shown below. Notice that the line segments on either side of the midpoint are congruent.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042436\/CNX_CAT_Figure_02_01_018.jpg\" alt=\"This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted. A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.\" width=\"487\" height=\"290\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Midpoint of the Line Segment<\/h3>\n<p>Find the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q788934\">Show Solution<\/span><\/p>\n<div id=\"q788934\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the formula to find the midpoint of the line segment.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q964077\">Show Solution<\/span><\/p>\n<div id=\"q964077\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-5,\\frac{5}{2}\\right)[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2308\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2308&theme=oea&iframe_resize_id=ohm2308&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm110938\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110938&theme=oea&iframe_resize_id=ohm110938&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n<dd><\/dd>\n<\/dl>\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 1.1 Homework Exercises<\/span><\/h2>\n<p>For each of the following exercises, find the distance between the two points.\u00a0 Simplify your answers, and write the exact answer in simplest radical form for irrational answers.<\/p>\n<p>1.\u00a0 [latex](-4,1)[\/latex] and\u00a0[latex](3,-4)[\/latex]<\/p>\n<p>2.\u00a0 [latex](2,-5)[\/latex] and\u00a0[latex](7,4)[\/latex]<\/p>\n<p>3.\u00a0 [latex](5,0)[\/latex] and\u00a0[latex](5,6)[\/latex]<\/p>\n<p>4.\u00a0 [latex](-4,3)[\/latex] and\u00a0[latex](10,3)[\/latex]<\/p>\n<p>5. Find the distance between the two points given your calculator, and round your answer to the nearest hundredth.\u00a0\u00a0[latex](19,12)[\/latex] and\u00a0[latex](41,71)[\/latex]<\/p>\n<p>For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.<\/p>\n<p>6.\u00a0 [latex](-5,6)[\/latex] and\u00a0[latex](4,2)[\/latex]<\/p>\n<p>7.\u00a0 [latex](-1,1)[\/latex] and\u00a0[latex](7,-4)[\/latex]<\/p>\n<p>8.\u00a0 [latex](-5,-3)[\/latex] and\u00a0[latex](-2,-8)[\/latex]<\/p>\n<p>9.\u00a0 [latex](0,7)[\/latex] and\u00a0[latex](4,-9)[\/latex]<\/p>\n<p>10.\u00a0 [latex](-43,17)[\/latex] and\u00a0[latex](23,-34)[\/latex]<\/p>\n<p>For each of the following exercises, identify the information requested.<\/p>\n<p>11. What are the coordinates of the origin?<\/p>\n<p>12. If a point is located on the <em>y<\/em>-axis, what is the <em>x<\/em>-coordinate?<\/p>\n<p>13. If a point is located on the <em>x<\/em>-axis, what is the <em>y<\/em>-coordinate?<\/p>\n<p>For each of the following exercises, plot the three points on the same coordinate plane.\u00a0 State whether the three points you plotted appear to be collinear (on the same line).<\/p>\n<p>14. [latex](4,1),(-2,3),(5,0)[\/latex]<\/p>\n<p>15. [latex](-1,2),(0,4),(2,1)[\/latex]<\/p>\n<p>16. [latex](-3,0),(-3,4),(-3,-3)[\/latex]<\/p>\n<p>17. Name the coordinates of the points graphed below.<\/p>\n<div style=\"text-align: left;\">\n<p><img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/PlotGraph.jpg\" alt=\"Graph with three points A, B, C, plotted\" \/><\/p>\n<p>18. Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.<\/p>\n<p>a.\u00a0 [latex](-3,4)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 b.\u00a0 [latex](-5,0)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0c.\u00a0 [latex](1,-4)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 d.\u00a0 [latex](-2,7)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0e.\u00a0\u00a0[latex](0,-3)[\/latex]<\/p>\n<p>For questions 19 &#8211; 23, use the graph in the figure below.<br \/>\n<img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/PlotLine.jpg\" alt=\"Graph of line segment connecting (-3,4) and (5,2)\" \/><\/p>\n<p>19. Find the distance between the two endpoints using the distance formula. Round to three decimal places.<\/p>\n<p>20. Find the coordinates of the midpoint of the line segment connecting the two points.<\/p>\n<p>21. Find the distance that [latex](-3,4)[\/latex] is from the origin.<\/p>\n<p>22. Find the distance that [latex](5,2)[\/latex] is from the origin.\u00a0 Round to three decimal places.<\/p>\n<p>23. Which point is closer to the origin?<\/p>\n<p>24. A man drove 10 mi directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?<\/p>\n<p>25. If the road was made in the previous exercise, how much shorter would the man&#8217;s one way trip be everyday?<\/p>\n<p>26 Given these four points [latex]A=(1,3),\\text{ }B=(-3,5),\\text{ }C=(4,7),\\text{ }D=(5,-4)[\/latex], find the coordinates of the midpoint of line segments [latex]\\overline{AB}[\/latex] and [latex]\\overline{CD}[\/latex].<\/p>\n<p>27. After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.<\/p>\n<p>28. The coordinates on a map for San Francisco are [latex](53,17)[\/latex] and those for Sacramento are [latex](123,78)[\/latex]. Note that the coordinates represent miles. Find the distance between the cities to the nearest mile.<\/p>\n<p>29. If San Jose&#8217;s coordinates are [latex](76,-12)[\/latex], where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.<\/p>\n<p>30. A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were [latex](49,64)[\/latex]. One rescue boat is at the coordinates [latex](60,82)[\/latex] and a second Coast Guard craft is at coordinates [latex](58,47)[\/latex]. Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?<\/p>\n<\/div>\n","protected":false},"author":264444,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17655","chapter","type-chapter","status-publish","hentry"],"part":10705,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17655","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17655\/revisions"}],"predecessor-version":[{"id":17777,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17655\/revisions\/17777"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/10705"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17655\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=17655"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=17655"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=17655"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=17655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}