{"id":1010,"date":"2025-06-20T17:28:35","date_gmt":"2025-06-20T17:28:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1010"},"modified":"2025-08-28T13:48:42","modified_gmt":"2025-08-28T13:48:42","slug":"polar-coordinates-and-conic-sections-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/polar-coordinates-and-conic-sections-background-youll-need-2\/","title":{"raw":"Polar Coordinates and Conic Sections: Background You'll Need 2","rendered":"Polar Coordinates and Conic Sections: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the distance formula and the Pythagorean theorem<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Distance Formula<\/h2>\r\n[caption id=\"\" align=\"alignright\" width=\"400\"]<img 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=\"400\" height=\"272\" \/> Triangle with sides labeled on an x,y coordinate plane[\/caption]\r\n\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 [latex]a[\/latex]<i> <\/i>and [latex]b[\/latex] are the lengths of the legs adjacent to the right angle, and [latex]c[\/latex]\u00a0is the length of the hypotenuse.\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\n<section class=\"textbox keyTakeaway\">\r\n<h3>distance formula<\/h3>\r\nThe <strong>distance formula<\/strong> is a mathematical equation used to determine the exact distance between two points ([latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]) on a coordinate plane.\r\n<p style=\"text-align: center;\">[latex]\\text{Distance}: d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\">Find the distance between the points [latex]\\left(-3,-1\\right)[\/latex] and [latex]\\left(2,3\\right)[\/latex].[reveal-answer q=\"891596\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"891596\"]Let us first look at the graph of the two points. Connect the points to form a right triangle.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"324\"]<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=\"324\" height=\"192\" \/> Triangle with points labeled on an x,y, coordinate plane[\/caption]\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<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18922[\/ohm2_question]<\/section><section>\r\n<h2>Using the Pythagorean Theorem<\/h2>\r\n<strong>The Pythagorean Theorem <\/strong>is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[\/latex] BCE.\r\n\r\nRemember that a right triangle has a [latex]90^\\circ [\/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\\circ [\/latex] angle is called the <strong>hypotenuse<\/strong>, and the other two sides are called the legs.\r\n\r\n<center>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"561\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223735\/CNX_BMath_Figure_09_03_024.png\" alt=\"Three right triangles, each with a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled \" width=\"561\" height=\"121\" \/> Figure 1. These right triangles have two legs and a hypotenuse[\/caption]\r\n\r\n<\/center>&nbsp;\r\n\r\nThe Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>the Pythagorean Theorem<\/h3>\r\nIn any right triangle [latex]\\Delta ABC[\/latex],\r\n<p style=\"text-align: center;\">[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nwhere [latex]c[\/latex] is the length of the hypotenuse [latex]a[\/latex] and [latex]b[\/latex] are the lengths of the legs.\r\n\r\n&nbsp;\r\n\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223736\/CNX_BMath_Figure_09_03_025.png\" alt=\"A right triangle, with the right angle marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" width=\"180\" height=\"156\" \/><\/center>&nbsp;\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox recall\">\r\n<div>\r\n\r\nTo solve problems that use the Pythagorean Theorem, we will need to find square roots. Recall the notation [latex]\\sqrt{m}[\/latex] and that it is defined in this way:\r\n<p style=\"text-align: center;\">[latex]\\text{If }m={n}^{2},\\text{ then }\\sqrt{m}=n\\text{ for }n\\ge 0[\/latex]<\/p>\r\nFor example, [latex]\\sqrt{25}[\/latex] is [latex]5[\/latex] because [latex]{5}^{2}=25[\/latex].\r\n\r\nWe will use this definition of square roots to solve for the length of a side in a right triangle.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\">Use the Pythagorean Theorem to find the length of the hypotenuse.\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223737\/CNX_BMath_Figure_09_03_026_img.png\" alt=\"Right triangle with legs labeled as 3 and 4.\" \/><\/center>\r\n[reveal-answer q=\"57881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"57881\"]\r\n<table id=\"eip-id1168469450887\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the length of the hypotenuse of the triangle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]c=\\text{the length of the hypotenuse}[\/latex]\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223738\/CNX_BMath_Figure_09_03_053_img-01.png\" alt=\"Right triangle with legs labeled 3 and 4 and the hypotenuse labeled c\" width=\"188\" height=\"160\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula. Substitute.<\/td>\r\n<td>[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]\r\n\r\n[latex]{3}^{2}+{4}^{2}={c}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]9+16={c}^{2}[\/latex]\r\n\r\n[latex]25={c}^{2}[\/latex]\r\n\r\n[latex]\\sqrt{25}={c}^{2}[\/latex]\r\n\r\n[latex]5=c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong><\/td>\r\n<td>[latex]{3}^{2}+{4}^{2}={\\color{red}{5}}^{2}[\/latex]\r\n\r\n[latex]9+16\\stackrel{?}{=}25[\/latex]\r\n\r\n[latex]25+25\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The length of the hypotenuse is [latex]5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7037[\/ohm2_question]<\/section><section class=\"textbox example\">Use the Pythagorean Theorem to find the length of the longer leg.\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223744\/CNX_BMath_Figure_09_03_031_img.png\" alt=\"A right triangle with one leg labeled as 5 and hypotenuse labeled as 13.\" width=\"176\" height=\"88\" \/><\/center>&nbsp;\r\n\r\n[reveal-answer q=\"477507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"477507\"]\r\n<table id=\"eip-id1168467480162\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>The length of the leg of the triangle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]b=\\text{the leg of the triangle}[\/latex]\r\n\r\nLabel side <em>b<\/em>\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223745\/CNX_BMath_Figure_09_03_054_img-01.png\" alt=\"A right triangle with one leg labeled as 5, the other leg labeled as b, and hypotenuse labeled as 13.\" width=\"246\" height=\"146\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula. Substitute.<\/td>\r\n<td>[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]\r\n\r\n[latex]{5}^{2}+{b}^{2}={13}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation. Isolate the variable term. Use the definition of the square root.\r\n\r\nSimplify.<\/td>\r\n<td>[latex]25+{b}^{2}=169[\/latex]\r\n\r\n[latex]{b}^{2}=144[\/latex]\r\n\r\n[latex]{b}^{2}=\\sqrt{144}[\/latex]\r\n\r\n[latex]b=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong><\/td>\r\n<td>[latex]{5}^{2}+{\\color{red}{12}}^{2}\\stackrel{?}{=}{13}^{2}[\/latex]\r\n\r\n[latex]25+144\\stackrel{?}{=}169[\/latex]\r\n\r\n[latex]169=169\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The length of the leg is [latex]12[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the distance formula and the Pythagorean theorem<\/li>\n<\/ul>\n<\/section>\n<h2>Distance Formula<\/h2>\n<figure style=\"width: 400px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" 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=\"400\" height=\"272\" \/><figcaption class=\"wp-caption-text\">Triangle with sides labeled on an x,y coordinate plane<\/figcaption><\/figure>\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 [latex]a[\/latex]<i> <\/i>and [latex]b[\/latex] are the lengths of the legs adjacent to the right angle, and [latex]c[\/latex]\u00a0is the length of the hypotenuse.<\/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<section class=\"textbox keyTakeaway\">\n<h3>distance formula<\/h3>\n<p>The <strong>distance formula<\/strong> is a mathematical equation used to determine the exact distance between two points ([latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]) on a coordinate plane.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Distance}: d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q891596\">Show Answer<\/button><\/p>\n<div id=\"q891596\" class=\"hidden-answer\" style=\"display: none\">Let us first look at the graph of the two points. Connect the points to form a right triangle.<\/p>\n<figure style=\"width: 324px\" 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\/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=\"324\" height=\"192\" \/><figcaption class=\"wp-caption-text\">Triangle with points labeled on an x,y, coordinate plane<\/figcaption><\/figure>\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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18922\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18922&theme=lumen&iframe_resize_id=ohm18922&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Using the Pythagorean Theorem<\/h2>\n<p><strong>The Pythagorean Theorem <\/strong>is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[\/latex] BCE.<\/p>\n<p>Remember that a right triangle has a [latex]90^\\circ[\/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\\circ[\/latex] angle is called the <strong>hypotenuse<\/strong>, and the other two sides are called the legs.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 561px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223735\/CNX_BMath_Figure_09_03_024.png\" alt=\"Three right triangles, each with a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled\" width=\"561\" height=\"121\" \/><figcaption class=\"wp-caption-text\">Figure 1. These right triangles have two legs and a hypotenuse<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the Pythagorean Theorem<\/h3>\n<p>In any right triangle [latex]\\Delta ABC[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>where [latex]c[\/latex] is the length of the hypotenuse [latex]a[\/latex] and [latex]b[\/latex] are the lengths of the legs.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223736\/CNX_BMath_Figure_09_03_025.png\" alt=\"A right triangle, with the right angle marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" width=\"180\" height=\"156\" \/><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox recall\">\n<div>\n<p>To solve problems that use the Pythagorean Theorem, we will need to find square roots. Recall the notation [latex]\\sqrt{m}[\/latex] and that it is defined in this way:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{If }m={n}^{2},\\text{ then }\\sqrt{m}=n\\text{ for }n\\ge 0[\/latex]<\/p>\n<p>For example, [latex]\\sqrt{25}[\/latex] is [latex]5[\/latex] because [latex]{5}^{2}=25[\/latex].<\/p>\n<p>We will use this definition of square roots to solve for the length of a side in a right triangle.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<div style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223737\/CNX_BMath_Figure_09_03_026_img.png\" alt=\"Right triangle with legs labeled as 3 and 4.\" \/><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57881\">Show Solution<\/button><\/p>\n<div id=\"q57881\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168469450887\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the length of the hypotenuse of the triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]c=\\text{the length of the hypotenuse}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223738\/CNX_BMath_Figure_09_03_053_img-01.png\" alt=\"Right triangle with legs labeled 3 and 4 and the hypotenuse labeled c\" width=\"188\" height=\"160\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula. Substitute.<\/td>\n<td>[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\n<p>[latex]{3}^{2}+{4}^{2}={c}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]9+16={c}^{2}[\/latex]<\/p>\n<p>[latex]25={c}^{2}[\/latex]<\/p>\n<p>[latex]\\sqrt{25}={c}^{2}[\/latex]<\/p>\n<p>[latex]5=c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>[latex]{3}^{2}+{4}^{2}={\\color{red}{5}}^{2}[\/latex]<\/p>\n<p>[latex]9+16\\stackrel{?}{=}25[\/latex]<\/p>\n<p>[latex]25+25\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The length of the hypotenuse is [latex]5[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7037\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7037&theme=lumen&iframe_resize_id=ohm7037&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Use the Pythagorean Theorem to find the length of the longer leg.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223744\/CNX_BMath_Figure_09_03_031_img.png\" alt=\"A right triangle with one leg labeled as 5 and hypotenuse labeled as 13.\" width=\"176\" height=\"88\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q477507\">Show Solution<\/button><\/p>\n<div id=\"q477507\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467480162\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>The length of the leg of the triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]b=\\text{the leg of the triangle}[\/latex]<\/p>\n<p>Label side <em>b<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223745\/CNX_BMath_Figure_09_03_054_img-01.png\" alt=\"A right triangle with one leg labeled as 5, the other leg labeled as b, and hypotenuse labeled as 13.\" width=\"246\" height=\"146\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula. Substitute.<\/td>\n<td>[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\n<p>[latex]{5}^{2}+{b}^{2}={13}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation. Isolate the variable term. Use the definition of the square root.<\/p>\n<p>Simplify.<\/td>\n<td>[latex]25+{b}^{2}=169[\/latex]<\/p>\n<p>[latex]{b}^{2}=144[\/latex]<\/p>\n<p>[latex]{b}^{2}=\\sqrt{144}[\/latex]<\/p>\n<p>[latex]b=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>[latex]{5}^{2}+{\\color{red}{12}}^{2}\\stackrel{?}{=}{13}^{2}[\/latex]<\/p>\n<p>[latex]25+144\\stackrel{?}{=}169[\/latex]<\/p>\n<p>[latex]169=169\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The length of the leg is [latex]12[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":677,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1010"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1010\/revisions"}],"predecessor-version":[{"id":2064,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1010\/revisions\/2064"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/677"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1010\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1010"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1010"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1010"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1010"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}