{"id":5384,"date":"2021-03-17T23:41:43","date_gmt":"2021-03-17T23:41:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nwfsc-mathforliberalartscorequisite\/chapter\/using-the-properties-of-triangles-to-solve-problems-2-2\/"},"modified":"2022-03-01T01:06:25","modified_gmt":"2022-03-01T01:06:25","slug":"using-the-properties-of-triangles-to-solve-problems-2-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-mathforliberalartscorequisite\/chapter\/using-the-properties-of-triangles-to-solve-problems-2-2\/","title":{"raw":"Using the Properties of Triangles to Solve Problems","rendered":"Using the Properties of Triangles to Solve Problems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use properties of triangles<\/li>\r\n<\/ul>\r\n<\/div>\r\nTo find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a [latex]\\text{90}^ \\circ[\/latex] angle with the base. The image below\u00a0shows three triangles with the base and height of each marked.\r\n\r\nThe height [latex]h[\/latex] of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a [latex]\\text{90}^ \\circ[\/latex] angle with the base.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223914\/CNX_BMath_Figure_09_04_037.png\" alt=\"Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Triangle Properties<\/h3>\r\nFor any triangle [latex]\\Delta ABC[\/latex], the sum of the measures of the angles is [latex]\\text{180}^ \\circ[\/latex].\r\n<p style=\"text-align: center;\">[latex]m\\angle{A}+m\\angle{B}+m\\angle{C}=180^\\circ [\/latex]<\/p>\r\nThe perimeter of a triangle is the sum of the lengths of the sides.\r\n<p style=\"text-align: center;\">[latex]P=a+b+c[\/latex]<\/p>\r\nThe area of a triangle is one-half the base, [latex]b[\/latex], times the height, [latex]h[\/latex].\r\n<p style=\"text-align: center;\">[latex]A={\\Large\\frac{1}{2}}bh[\/latex]<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223917\/CNX_BMath_Figure_09_04_038_img.png\" alt=\"A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the area of a triangle whose base is [latex]11[\/latex] inches and whose height is [latex]8[\/latex] inches.\r\n\r\nSolution\r\n<table id=\"eip-id1168468457178\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223918\/CNX_BMath_Figure_09_04_073_img-01.png\" alt=\"A triangle with base 11 inches and height 8 inches\" width=\"318\" height=\"202\" \/><\/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 area 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 <em>A<\/em> = area of the triangle<\/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><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223920\/CNX_BMath_Figure_09_04_073_img-02.png\" alt=\" A = 1\/2 * b * h.  A = 1\/2 * 11 * 8\" width=\"318\" height=\"110\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]A=44[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]A={\r\n\r\n\\Large\\frac{1}{2}}bh[\/latex]\r\n\r\n[latex]44\\stackrel{?}{=}{\r\n\r\n\\Large\\frac{1}{2}}(11)8[\/latex]\r\n\r\n[latex]44=44\\quad\\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 area is [latex]44[\/latex] square inches.<\/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]146525[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video provides another example of how to use the area formula for triangles.\r\n\r\nhttps:\/\/youtu.be\/jXbPAk2jorM\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe perimeter of a triangular garden is [latex]24[\/latex] feet. The lengths of two sides are [latex]4[\/latex] feet and [latex]9[\/latex] feet. How long is the third side?\r\n[reveal-answer q=\"371512\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371512\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466081900\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223923\/CNX_BMath_Figure_09_04_074_img-01.png\" alt=\"A triangle with sides labelled 4 ft, c, and 9 ft.  P = 24 ft is also given.\" width=\"317\" height=\"188\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>length of the third side of a 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 <em>c<\/em> = the third side<\/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 in the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223925\/CNX_BMath_Figure_09_04_074_img-02.png\" alt=\"P = a + b + c.  24 = 4 + 9 + c\" width=\"317\" height=\"67\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]24=13+c[\/latex]\r\n\r\n[latex]11=c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]P=a+b+c[\/latex]\r\n\r\n[latex]24\\stackrel{?}{=}4+9+11[\/latex]\r\n\r\n[latex]24=24\\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 third side is [latex]11[\/latex] feet long.<\/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]146526[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe area of a triangular church window is [latex]90[\/latex] square meters. The base of the window is [latex]15[\/latex] meters. What is the window\u2019s height?\r\n[reveal-answer q=\"632571\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632571\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467155173\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223929\/CNX_BMath_Figure_09_04_075_img-01.png\" alt=\"Triangle with base 15 meters and height labelled h.\" width=\"225\" height=\"222\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>height of a 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 <em>h<\/em> = the height<\/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 in the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223931\/CNX_BMath_Figure_09_04_075_img-02.png\" alt=\"A = 1\/2 * b * h. 90 = 1\/2 * 15 * h.\" width=\"303\" height=\"99\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]90={\\Large\\frac{1}{2}}\\normalsize(15)h[\/latex]\r\n\r\n[latex]12=h[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]A={\\Large\\frac{1}{2}}\\normalsize bh[\/latex]\r\n\r\n[latex]90\\stackrel{?}{=}{\\Large\\frac{1}{2}}\\normalsize\\cdot 15\\cdot 12[\/latex]\r\n\r\n[latex]90=90\\quad\\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 height of the triangle is [latex]12[\/latex] meters.<\/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]146528[\/ohm_question]\r\n\r\n[ohm_question]146527[\/ohm_question]\r\n\r\n<\/div>\r\nIn our next video, we show another example of how to find the height of a triangle given it's area.\r\n\r\nhttps:\/\/youtu.be\/C0vUVK_o5r0\r\n<h3>Isosceles and Equilateral Triangles<\/h3>\r\nBesides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. The image below\u00a0shows both types of triangles.\r\n\r\nIn an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223935\/CNX_BMath_Figure_09_04_045.png\" alt=\"Two triangles are shown.  One is an isosceles triangle with two sides labelled s.  The other is an equilateral triangle with all sides labelled s.\" width=\"367\" height=\"231\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Isosceles and Equilateral Triangles<\/h3>\r\nAn <strong>isosceles<\/strong> triangle has two sides the same length.\r\nAn <strong>equilateral<\/strong> triangle has three sides of equal length.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe perimeter of an equilateral triangle is [latex]93[\/latex] inches. Find the length of each side.\r\n[reveal-answer q=\"157458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157458\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468574026\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223937\/CNX_BMath_Figure_09_04_076_img-01.png\" alt=\"A triangle with all sides labelled s.\" width=\"185\" height=\"174\" \/>\r\n\r\nPerimeter = [latex]93[\/latex] in.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>length of the sides of an equilateral 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 <em>s<\/em> = length of each side<\/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><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223938\/CNX_BMath_Figure_09_04_076_img-02.png\" alt=\"P = a + b + c. 93 = s + s + s\" width=\"298\" height=\"60\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]93=3s[\/latex]\r\n\r\n[latex]31=s[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223940\/CNX_BMath_Figure_09_04_076_img-04.png\" alt=\".\" \/>\r\n\r\n[latex]93\\stackrel{?}{=}31+31+31[\/latex]\r\n\r\n[latex]93=93\\quad\\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>Each side is [latex]31[\/latex] inches.<\/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]146529[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nArianna has [latex]156[\/latex] inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of [latex]60[\/latex] inches. How long can she make the two equal sides?\r\n[reveal-answer q=\"327649\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"327649\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466073183\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223942\/CNX_BMath_Figure_09_04_077_img-01.png\" alt=\"A triangle with the base labelled 60 in.  The other two sides are labelled s.\" width=\"189\" height=\"115\" \/>\r\n\r\n<em>P<\/em> = [latex]156[\/latex] in.<\/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 lengths of the two equal sides<\/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>s<\/em> = the length of each side<\/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 in the given information.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223943\/CNX_BMath_Figure_09_04_077_img-02.png\" alt=\"P = a + b + c. 156 = s + 60 + s\" width=\"300\" height=\"51\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]156=2s=60[\/latex]\r\n\r\n[latex]96=2s[\/latex]\r\n\r\n[latex]48=s[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n[latex]P=a+b+c[\/latex]\r\n\r\n[latex]156\\stackrel{?}{=}48+60+48[\/latex]\r\n\r\n[latex]156=156\\quad\\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>Arianna can make each of the two equal sides [latex]48[\/latex] inches long.<\/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]146531[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use properties of triangles<\/li>\n<\/ul>\n<\/div>\n<p>To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a [latex]\\text{90}^ \\circ[\/latex] angle with the base. The image below\u00a0shows three triangles with the base and height of each marked.<\/p>\n<p>The height [latex]h[\/latex] of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a [latex]\\text{90}^ \\circ[\/latex] angle with the base.<\/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\/24223914\/CNX_BMath_Figure_09_04_037.png\" alt=\"Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Triangle Properties<\/h3>\n<p>For any triangle [latex]\\Delta ABC[\/latex], the sum of the measures of the angles is [latex]\\text{180}^ \\circ[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]m\\angle{A}+m\\angle{B}+m\\angle{C}=180^\\circ[\/latex]<\/p>\n<p>The perimeter of a triangle is the sum of the lengths of the sides.<\/p>\n<p style=\"text-align: center;\">[latex]P=a+b+c[\/latex]<\/p>\n<p>The area of a triangle is one-half the base, [latex]b[\/latex], times the height, [latex]h[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]A={\\Large\\frac{1}{2}}bh[\/latex]<\/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\/24223917\/CNX_BMath_Figure_09_04_038_img.png\" alt=\"A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the area of a triangle whose base is [latex]11[\/latex] inches and whose height is [latex]8[\/latex] inches.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468457178\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223918\/CNX_BMath_Figure_09_04_073_img-01.png\" alt=\"A triangle with base 11 inches and height 8 inches\" width=\"318\" height=\"202\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area 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 <em>A<\/em> = area of the triangle<\/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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223920\/CNX_BMath_Figure_09_04_073_img-02.png\" alt=\"A = 1\/2 * b * h.  A = 1\/2 * 11 * 8\" width=\"318\" height=\"110\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A=44[\/latex] square inches.<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]A={    \\Large\\frac{1}{2}}bh[\/latex]<\/p>\n<p>[latex]44\\stackrel{?}{=}{    \\Large\\frac{1}{2}}(11)8[\/latex]<\/p>\n<p>[latex]44=44\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The area is [latex]44[\/latex] square inches.<\/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=\"ohm146525\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146525&theme=oea&iframe_resize_id=ohm146525&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video provides another example of how to use the area formula for triangles.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Find the Area of a Triangle (Whole Number)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jXbPAk2jorM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The perimeter of a triangular garden is [latex]24[\/latex] feet. The lengths of two sides are [latex]4[\/latex] feet and [latex]9[\/latex] feet. How long is the third side?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q371512\">Show Solution<\/span><\/p>\n<div id=\"q371512\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466081900\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223923\/CNX_BMath_Figure_09_04_074_img-01.png\" alt=\"A triangle with sides labelled 4 ft, c, and 9 ft.  P = 24 ft is also given.\" width=\"317\" height=\"188\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>length of the third side of a triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>c<\/em> = the third side<\/td>\n<\/tr>\n<tr>\n<td>Step 4.<strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute in the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223925\/CNX_BMath_Figure_09_04_074_img-02.png\" alt=\"P = a + b + c.  24 = 4 + 9 + c\" width=\"317\" height=\"67\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]24=13+c[\/latex]<\/p>\n<p>[latex]11=c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]P=a+b+c[\/latex]<\/p>\n<p>[latex]24\\stackrel{?}{=}4+9+11[\/latex]<\/p>\n<p>[latex]24=24\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The third side is [latex]11[\/latex] feet long.<\/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=\"ohm146526\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146526&theme=oea&iframe_resize_id=ohm146526&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>The area of a triangular church window is [latex]90[\/latex] square meters. The base of the window is [latex]15[\/latex] meters. What is the window\u2019s height?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q632571\">Show Solution<\/span><\/p>\n<div id=\"q632571\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467155173\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223929\/CNX_BMath_Figure_09_04_075_img-01.png\" alt=\"Triangle with base 15 meters and height labelled h.\" width=\"225\" height=\"222\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>height of a triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>h<\/em> = the height<\/td>\n<\/tr>\n<tr>\n<td>Step 4.<strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute in the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223931\/CNX_BMath_Figure_09_04_075_img-02.png\" alt=\"A = 1\/2 * b * h. 90 = 1\/2 * 15 * h.\" width=\"303\" height=\"99\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]90={\\Large\\frac{1}{2}}\\normalsize(15)h[\/latex]<\/p>\n<p>[latex]12=h[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]A={\\Large\\frac{1}{2}}\\normalsize bh[\/latex]<\/p>\n<p>[latex]90\\stackrel{?}{=}{\\Large\\frac{1}{2}}\\normalsize\\cdot 15\\cdot 12[\/latex]<\/p>\n<p>[latex]90=90\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The height of the triangle is [latex]12[\/latex] meters.<\/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=\"ohm146528\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146528&theme=oea&iframe_resize_id=ohm146528&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146527\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146527&theme=oea&iframe_resize_id=ohm146527&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In our next video, we show another example of how to find the height of a triangle given it&#8217;s area.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Find the Height of a Triangle Given Area (Even Base)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/C0vUVK_o5r0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Isosceles and Equilateral Triangles<\/h3>\n<p>Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. The image below\u00a0shows both types of triangles.<\/p>\n<p>In an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.<\/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\/24223935\/CNX_BMath_Figure_09_04_045.png\" alt=\"Two triangles are shown.  One is an isosceles triangle with two sides labelled s.  The other is an equilateral triangle with all sides labelled s.\" width=\"367\" height=\"231\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Isosceles and Equilateral Triangles<\/h3>\n<p>An <strong>isosceles<\/strong> triangle has two sides the same length.<br \/>\nAn <strong>equilateral<\/strong> triangle has three sides of equal length.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The perimeter of an equilateral triangle is [latex]93[\/latex] inches. Find the length of each side.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157458\">Show Solution<\/span><\/p>\n<div id=\"q157458\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468574026\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223937\/CNX_BMath_Figure_09_04_076_img-01.png\" alt=\"A triangle with all sides labelled s.\" width=\"185\" height=\"174\" \/><\/p>\n<p>Perimeter = [latex]93[\/latex] in.<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>length of the sides of an equilateral triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>s<\/em> = length of each side<\/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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223938\/CNX_BMath_Figure_09_04_076_img-02.png\" alt=\"P = a + b + c. 93 = s + s + s\" width=\"298\" height=\"60\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]93=3s[\/latex]<\/p>\n<p>[latex]31=s[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223940\/CNX_BMath_Figure_09_04_076_img-04.png\" alt=\".\" \/><\/p>\n<p>[latex]93\\stackrel{?}{=}31+31+31[\/latex]<\/p>\n<p>[latex]93=93\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>Each side is [latex]31[\/latex] inches.<\/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=\"ohm146529\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146529&theme=oea&iframe_resize_id=ohm146529&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>Arianna has [latex]156[\/latex] inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of [latex]60[\/latex] inches. How long can she make the two equal sides?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q327649\">Show Solution<\/span><\/p>\n<div id=\"q327649\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466073183\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223942\/CNX_BMath_Figure_09_04_077_img-01.png\" alt=\"A triangle with the base labelled 60 in.  The other two sides are labelled s.\" width=\"189\" height=\"115\" \/><\/p>\n<p><em>P<\/em> = [latex]156[\/latex] in.<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the lengths of the two equal sides<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>s<\/em> = the length of each side<\/td>\n<\/tr>\n<tr>\n<td>Step 4.<strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute in the given information.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223943\/CNX_BMath_Figure_09_04_077_img-02.png\" alt=\"P = a + b + c. 156 = s + 60 + s\" width=\"300\" height=\"51\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]156=2s=60[\/latex]<\/p>\n<p>[latex]96=2s[\/latex]<\/p>\n<p>[latex]48=s[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>[latex]P=a+b+c[\/latex]<\/p>\n<p>[latex]156\\stackrel{?}{=}48+60+48[\/latex]<\/p>\n<p>[latex]156=156\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>Arianna can make each of the two equal sides [latex]48[\/latex] inches long.<\/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=\"ohm146531\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146531&theme=oea&iframe_resize_id=ohm146531&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/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-5384\">\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 146531, 146528, 146526, 146525. <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>Ex: Find the Area of a Triangle (Whole Number). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jXbPAk2jorM\">https:\/\/youtu.be\/jXbPAk2jorM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Find the Height of a Triangle Given Area (Even Base). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/C0vUVK_o5r0\">https:\/\/youtu.be\/C0vUVK_o5r0<\/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":167848,"menu_order":4,"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 146531, 146528, 146526, 146525\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Find the Area of a Triangle (Whole Number)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/jXbPAk2jorM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Find the Height of a Triangle Given Area (Even Base)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/C0vUVK_o5r0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"096dc39df9044b6cb7595a8b3f1f9cd8, 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