{"id":10752,"date":"2017-06-05T15:46:21","date_gmt":"2017-06-05T15:46:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10752"},"modified":"2024-04-30T16:46:35","modified_gmt":"2024-04-30T16:46:35","slug":"using-the-pythagorean-theorem-to-solve-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/using-the-pythagorean-theorem-to-solve-problems\/","title":{"raw":"Using the Pythagorean Theorem to Solve Problems","rendered":"Using the Pythagorean Theorem to Solve Problems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the Pythagorean theorem to find the unknown side length of a right triangle<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe Pythagorean theorem 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 hypotenuse, and the other two sides are called the legs. See the triangles below.\r\n\r\nIn a right triangle, the side opposite the [latex]90^\\circ [\/latex] angle is called the hypotenuse and each of the other sides is called a leg.\r\n\r\n<img class=\"aligncenter\" 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 are shown. Each has 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 \u201cleg\u201d in each triangle. The sides across from the right angles are labeled \u201chypotenuse.\u201d\" width=\"561\" height=\"121\" \/>\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<div class=\"textbox shaded\">\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\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<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 is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" \/>\r\n\r\n<\/div>\r\nTo solve problems that use the Pythagorean theorem, we will need to find square roots. <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/modeling-squares-and-finding-the-square-root-of-a-number\/\">Previously, we introduced the notation [latex]\\sqrt{m}[\/latex]<\/a> and defined it 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, we found that [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<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the Pythagorean theorem to find 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\/277\/2017\/04\/24223737\/CNX_BMath_Figure_09_03_026_img.png\" alt=\"Right triangle with legs labeled as 3 and 4.\" \/>\r\n\r\nSolution\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 side lengths 3 and 4 and hypotenuse 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.\r\n\r\nSubstitute.<\/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}[\/latex]\r\n\r\n[latex]5=c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n&nbsp;<\/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<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146913[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the Pythagorean theorem to find the length of the longer leg.\r\n\r\n<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=\"Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.\" \/>\r\n[reveal-answer q=\"477507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"477507\"]\r\n\r\nSolution\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=\"Right triangle with side lengths 5 and b and hypotenuse 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}=\\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>\r\n\r\n[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<td><\/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<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146914[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nKelvin is building a gazebo and wants to brace each corner by placing a [latex]\\text{10-inch}[\/latex] wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223751\/CNX_BMath_Figure_09_03_036.png\" alt=\"A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.\" \/>\r\n[reveal-answer q=\"730318\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"730318\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468762748\" 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 distance from the corner that the bracket should be attached<\/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 <em>x<\/em> = the distance from the corner\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223753\/CNX_BMath_Figure_09_03_055_img-01.png\" alt=\"Right triangle with side lengths x and x and hypotenuse 10 inches.\" width=\"97\" height=\"86\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td>[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]\r\n\r\n[latex]{x}^{2}+{x}^{2}={10}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.\r\n\r\nIsolate the variable.\r\n\r\nUse the definition of the square root.\r\n\r\nSimplify. Approximate to the nearest tenth.<\/td>\r\n<td>[latex]2x^2=100[\/latex]\r\n\r\n[latex]x^2=50[\/latex]\r\n\r\n[latex]x=\\sqrt{50}[\/latex]\r\n\r\n[latex]b\\approx{7.1}[\/latex]\r\n\r\n&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]\r\n\r\n[latex](\\color{red}{7.1})^2+(\\color{red}{7.1})^{2}\\stackrel{\\text{?}}{\\approx}{10}^{2}[\/latex]\r\n\r\n[latex]50.41+50.41=100.82\\approx{100}\\quad\\checkmark[\/latex]\r\n\r\nYes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>Kelvin should fasten each piece of wood approximately [latex]7.1in[\/latex] from the corner.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146916[\/ohm_question]\r\n\r\n[ohm_question]146918[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show two more examples of how to use the Pythagorean theorem to solve application problems.\r\n\r\nhttps:\/\/youtu.be\/2P0dJxpwFMY","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the Pythagorean theorem to find the unknown side length of a right triangle<\/li>\n<\/ul>\n<\/div>\n<p>The Pythagorean theorem 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 hypotenuse, and the other two sides are called the legs. See the triangles below.<\/p>\n<p>In a right triangle, the side opposite the [latex]90^\\circ[\/latex] angle is called the hypotenuse and each of the other sides is called a leg.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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 are shown. Each has 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 \u201cleg\u201d in each triangle. The sides across from the right angles are labeled \u201chypotenuse.\u201d\" width=\"561\" height=\"121\" \/><br \/>\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.<\/p>\n<div class=\"textbox shaded\">\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>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><img 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 is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" \/><\/p>\n<\/div>\n<p>To solve problems that use the Pythagorean theorem, we will need to find square roots. <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/modeling-squares-and-finding-the-square-root-of-a-number\/\">Previously, we introduced the notation [latex]\\sqrt{m}[\/latex]<\/a> and defined it 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, we found that [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 class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the Pythagorean theorem to find the length of the hypotenuse.<\/p>\n<p><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.\" \/><\/p>\n<p>Solution<\/p>\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 side lengths 3 and 4 and hypotenuse c.\" width=\"188\" height=\"160\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>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}[\/latex]<\/p>\n<p>[latex]5=c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>&nbsp;<\/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<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146913\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146913&theme=oea&iframe_resize_id=ohm146913&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the Pythagorean theorem to find the length of the longer leg.<\/p>\n<p><img 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=\"Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q477507\">Show Solution<\/span><\/p>\n<div id=\"q477507\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\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=\"Right triangle with side lengths 5 and b and hypotenuse 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}=\\sqrt{144}[\/latex]<\/p>\n<p>[latex]b=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[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<td><\/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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146914\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146914&theme=oea&iframe_resize_id=ohm146914&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Kelvin is building a gazebo and wants to brace each corner by placing a [latex]\\text{10-inch}[\/latex] wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223751\/CNX_BMath_Figure_09_03_036.png\" alt=\"A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q730318\">Show Solution<\/span><\/p>\n<div id=\"q730318\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468762748\" 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 distance from the corner that the bracket should be attached<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>x<\/em> = the distance from the corner<\/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\/24223753\/CNX_BMath_Figure_09_03_055_img-01.png\" alt=\"Right triangle with side lengths x and x and hypotenuse 10 inches.\" width=\"97\" height=\"86\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td>[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\n<p>[latex]{x}^{2}+{x}^{2}={10}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/p>\n<p>Isolate the variable.<\/p>\n<p>Use the definition of the square root.<\/p>\n<p>Simplify. Approximate to the nearest tenth.<\/td>\n<td>[latex]2x^2=100[\/latex]<\/p>\n<p>[latex]x^2=50[\/latex]<\/p>\n<p>[latex]x=\\sqrt{50}[\/latex]<\/p>\n<p>[latex]b\\approx{7.1}[\/latex]<\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\n<p>[latex](\\color{red}{7.1})^2+(\\color{red}{7.1})^{2}\\stackrel{\\text{?}}{\\approx}{10}^{2}[\/latex]<\/p>\n<p>[latex]50.41+50.41=100.82\\approx{100}\\quad\\checkmark[\/latex]<\/p>\n<p>Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>Kelvin should fasten each piece of wood approximately [latex]7.1in[\/latex] from the corner.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146916\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146916&theme=oea&iframe_resize_id=ohm146916&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146918\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146918&theme=oea&iframe_resize_id=ohm146918&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show two more examples of how to use the Pythagorean theorem to solve application problems.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solve Applications Using the Pythagorean Theorem  (c only)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2P0dJxpwFMY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10752\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146918, 146916, 146914, 146913. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Solve Applications Using the Pythagorean Theorem (c only). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2P0dJxpwFMY\">https:\/\/youtu.be\/2P0dJxpwFMY<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146918, 146916, 146914, 146913\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Solve Applications Using the Pythagorean Theorem (c only)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/2P0dJxpwFMY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"0abe40ae7c7b4d9bb490a70e0ab41de2, 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