{"id":10763,"date":"2017-06-05T15:55:04","date_gmt":"2017-06-05T15:55:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10763"},"modified":"2024-04-30T16:47:47","modified_gmt":"2024-04-30T16:47:47","slug":"using-the-properties-of-trapezoids-to-solve-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/using-the-properties-of-trapezoids-to-solve-problems\/","title":{"raw":"Using the Properties of Trapezoids to Solve Problems","rendered":"Using the Properties of Trapezoids 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 trapezoids<\/li>\r\n<\/ul>\r\n<\/div>\r\nA trapezoid is a four-sided figure, a <em>quadrilateral<\/em>, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[\/latex], and the length of the bigger base [latex]B[\/latex]. The height, [latex]h[\/latex], of a trapezoid is the distance between the two bases as shown in the image below.\r\n\r\nA trapezoid has a larger base, [latex]B[\/latex], and a smaller base, [latex]b[\/latex]. The height [latex]h[\/latex] is the distance between the bases.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223946\/CNX_BMath_Figure_09_04_052.png\" alt=\"A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.\" \/>\r\n<div class=\"textbox shaded\">\r\n\r\nThe formula for the area of a trapezoid is:\r\n\r\n[latex]{\\text{Area}}_{\\text{trapezoid}}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex]\r\n\r\n<\/div>\r\nSplitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See the image below.\r\n\r\nSplitting a trapezoid into two triangles may help you understand the formula for its area.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223948\/CNX_BMath_Figure_09_04_053.png\" alt=\"An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.\" \/>\r\nThe height of the trapezoid is also the height of each of the two triangles. See the image below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223949\/CNX_BMath_Figure_09_04_078.png\" alt=\"An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. There is an arrow pointing to a second trapezoid. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.\" \/>\r\nThe formula for the area of a trapezoid is\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223950\/CNX_BMath_Figure_09_04_056_img.png\" alt=\"This image shows the formula for the area of a trapezoid and says \u201carea of trapezoid equals one-half h times smaller base b plus larger base B).\" width=\"185\" height=\"40\" \/>\r\nIf we distribute, we get,\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223951\/CNX_BMath_Figure_09_04_079_img.png\" alt=\"The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Trapezoids<\/h3>\r\n<ul id=\"fs-id1429217\">\r\n \t<li>A trapezoid has four sides.<\/li>\r\n \t<li>Two of its sides are parallel and two sides are not.<\/li>\r\n \t<li>The area, [latex]A[\/latex], of a trapezoid is [latex]\\text{A}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex] .<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the area of a trapezoid whose height is [latex]6[\/latex] inches and whose bases are [latex]14[\/latex] and [latex]11[\/latex] inches.\r\n\r\nSolution\r\n<table id=\"eip-id1168468452905\" 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\/24223952\/CNX_BMath_Figure_09_04_080_img-01.png\" alt=\"A trapezoid of base 11 and 14 inches and height of 6 inches.\" width=\"243\" height=\"148\" \/><\/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 trapezoid<\/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]A=\\text{the area}[\/latex]<\/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\/24223953\/CNX_BMath_Figure_09_04_080_img-02.png\" alt=\"Plugging in 11 and 14 for bases and 6 for height the equation area A equals one half times height times base one plus base two becomes area equals one half times 6 times 11 plus 14.\" width=\"392\" height=\"92\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]A={\\Large\\frac{1}{2}}\\normalsize\\cdot 6(25)[\/latex]\r\n\r\n[latex]A=3(25)[\/latex]\r\n\r\n[latex]A=75[\/latex] square inches<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Is this answer reasonable?<\/td>\r\n<td>\u00a0[latex]\\checkmark[\/latex]\u00a0 see reasoning below<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf we draw a rectangle around the trapezoid that has the same big base [latex]B[\/latex] and a height [latex]h[\/latex], its area should be greater than that of the trapezoid.\r\nIf we draw a rectangle inside the trapezoid that has the same little base [latex]b[\/latex] and a height [latex]h[\/latex], its area should be smaller than that of the trapezoid.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223956\/CNX_BMath_Figure_09_04_060.png\" alt=\"A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.\" \/>\r\nThe area of the larger rectangle is [latex]84[\/latex] square inches and the area of the smaller rectangle is [latex]66[\/latex] square inches. So it makes sense that the area of the trapezoid is between [latex]84[\/latex] and [latex]66[\/latex] square inches\r\n\r\nStep 7. <strong>Answer<\/strong> the question. The area of the trapezoid is [latex]75[\/latex] square inches.\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]146533[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show another example of how to use the formula to find the area of a trapezoid given the lengths of it's height and bases.\r\n\r\nhttps:\/\/youtu.be\/WNo7s-XoI4w\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the area of a trapezoid whose height is [latex]5[\/latex] feet and whose bases are [latex]10.3[\/latex] and [latex]13.7[\/latex] feet.\r\n[reveal-answer q=\"158293\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"158293\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469577474\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr style=\"height: 178.867px;\">\r\n<td style=\"width: 417.267px; height: 178.867px;\">Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td style=\"width: 414.733px; height: 178.867px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223958\/CNX_BMath_Figure_09_04_081_img-01.png\" alt=\"A trapezoid of base 10.3 and 13.7 feet and height of 5 feet.\" width=\"421\" height=\"141\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 417.267px; height: 15px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td style=\"width: 414.733px; height: 15px;\">the area of the trapezoid<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 417.267px; height: 15px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td style=\"width: 414.733px; height: 15px;\">Let <em>A<\/em> = the area<\/td>\r\n<\/tr>\r\n<tr style=\"height: 102px;\">\r\n<td style=\"width: 417.267px; height: 102px;\">Step 4.<strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td style=\"width: 414.733px; height: 102px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223959\/CNX_BMath_Figure_09_04_081_img-02.png\" alt=\"Plugging in 10.3 and 13.7 for bases and 5 for height the equation area A equals one half times height times base one plus base two becomes area equals one half times 5 times 13.7 plus 10.3.\" width=\"421\" height=\"99\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 90px;\">\r\n<td style=\"width: 417.267px; height: 90px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td style=\"width: 414.733px; height: 90px;\">[latex]A={\\Large\\frac{1}{2}}\\normalsize\\cdot 5(24)[\/latex]\r\n\r\n[latex]A=12(5)[\/latex]\r\n\r\n[latex]A=60[\/latex] square feet<\/td>\r\n<\/tr>\r\n<tr style=\"height: 305px;\">\r\n<td style=\"width: 417.267px; height: 305px;\">Step 6. <strong>Check:<\/strong> Is this answer reasonable?\r\n\r\nThe area of the trapezoid should be less than the area of a rectangle with base [latex]13.7[\/latex] and height [latex]5[\/latex], but more than the area of a rectangle with base [latex]10.3[\/latex] and height [latex]5[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224003\/CNX_BMath_Figure_09_04_063.png\" alt=\"An image of a trapezoid is shown with a red rectangle drawn around it. The larger base of the trapezoid is labeled 13.7 ft. and is the same as the base of the rectangle. The height of both the trapezoid and the rectangle is 5 ft. Next to this is an image of a trapezoid with a black rectangle drawn inside it. The smaller base of the trapezoid is labeled 10.3 ft. and is the same as the base of the rectangle. Below the images is A sub red rectangle is greater than A sub trapezoid is greater than A sub rectangle. Below this is 68.5, 60, and 51.5.\" \/><\/td>\r\n<td style=\"width: 414.733px; height: 305px;\">\u00a0[latex]\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 417.267px; height: 15px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td style=\"width: 414.733px; height: 15px;\">The area of the trapezoid is [latex]60[\/latex] square feet.<\/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]146534[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nVinny has a garden that is shaped like a trapezoid. The trapezoid has a height of [latex]3.4[\/latex] yards and the bases are [latex]8.2[\/latex] and [latex]5.6[\/latex] yards. How many square yards will be available to plant?\r\n[reveal-answer q=\"676574\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"676574\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469766721\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td style=\"width: 424.817px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224005\/CNX_BMath_Figure_09_04_082_img-01.png\" alt=\"A trapezoid of bases 5.6 and 8.2 yards and height of 3.4 yards.\" width=\"418\" height=\"129\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td style=\"width: 424.817px;\">the area of a trapezoid<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td style=\"width: 424.817px;\">Let [latex]A[\/latex] = the area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 4.<strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td style=\"width: 424.817px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224007\/CNX_BMath_Figure_09_04_082_img-02.png\" alt=\"Plugging in 5.6 and 8.2 for bases and 3.4 for height the equation area A equals one half times height times base one plus base two becomes area equals one half times 3.4 times 5.6 plus 8.2.\" width=\"418\" height=\"101\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td style=\"width: 424.817px;\">[latex]A={\\Large\\frac{1}{2}}\\normalsize(3.4)(13.8)[\/latex]\r\n\r\n[latex]A=23.46[\/latex] square yards.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 834px;\" colspan=\"2\">Step 6. <strong>Check:<\/strong> Is this answer reasonable?\r\n\r\nYes. The area of the trapezoid is less than the area of a rectangle with a base of [latex]8.2[\/latex] yd and height [latex]3.4[\/latex] yd, but more than the area of a rectangle with base [latex]5.6[\/latex] yd and height [latex]3.4[\/latex] yd.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224012\/CNX_BMath_Figure_09_04_066.png\" alt=\"This image is a table with two rows. the first row is split into three columns. The first column is the formula Area of a rectangle equals base times height. On the next line under this it has numbers plugged into the formula; the base, 8.2 in parentheses times the height 3.4 in parentheses. Under this is it has \u201cequals 27.88 yards squared\u201d. The center column includes the formula of a trapezoid and says Area of a trapezoid equals one half times 3.5 yards in parentheses times 5.8 plus 8.2 in parentheses. Under this it has \u201cequals 23.46 yards squared\u201d. In the third column it it has the formula the area of a rectangle equals base times height. Under this it has equals 5.6 in parentheses times 3.4 in parentheses. Under this it has \u201cequals 19.04 yards squared.\u201d In the second row, centered from left to right it has \u201cArea of a rectangle\u201d and a \u201cgreater than\u201d sign, \u201cArea of a trapezoid\u201d and a greater than sign and \u201carea of a rectangle\u201d. Under Area of a rectangle it has 27.88, then 23.46 under \u201carea of a trapezoid\u201d, then 19.04 under \u201carea of a rectangle\u201d.\" width=\"850\" height=\"200\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td style=\"width: 424.817px;\">Vinny has [latex]23.46[\/latex] square yards in which he can plant.<\/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]146535[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use properties of trapezoids<\/li>\n<\/ul>\n<\/div>\n<p>A trapezoid is a four-sided figure, a <em>quadrilateral<\/em>, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[\/latex], and the length of the bigger base [latex]B[\/latex]. The height, [latex]h[\/latex], of a trapezoid is the distance between the two bases as shown in the image below.<\/p>\n<p>A trapezoid has a larger base, [latex]B[\/latex], and a smaller base, [latex]b[\/latex]. The height [latex]h[\/latex] is the distance between the bases.<\/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\/24223946\/CNX_BMath_Figure_09_04_052.png\" alt=\"A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.\" \/><\/p>\n<div class=\"textbox shaded\">\n<p>The formula for the area of a trapezoid is:<\/p>\n<p>[latex]{\\text{Area}}_{\\text{trapezoid}}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex]<\/p>\n<\/div>\n<p>Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See the image below.<\/p>\n<p>Splitting a trapezoid into two triangles may help you understand the formula for its area.<\/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\/24223948\/CNX_BMath_Figure_09_04_053.png\" alt=\"An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.\" \/><br \/>\nThe height of the trapezoid is also the height of each of the two triangles. See the image below.<\/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\/24223949\/CNX_BMath_Figure_09_04_078.png\" alt=\"An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. There is an arrow pointing to a second trapezoid. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.\" \/><br \/>\nThe formula for the area of a trapezoid is<\/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\/24223950\/CNX_BMath_Figure_09_04_056_img.png\" alt=\"This image shows the formula for the area of a trapezoid and says \u201carea of trapezoid equals one-half h times smaller base b plus larger base B).\" width=\"185\" height=\"40\" \/><br \/>\nIf we distribute, we get,<\/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\/24223951\/CNX_BMath_Figure_09_04_079_img.png\" alt=\"The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Trapezoids<\/h3>\n<ul id=\"fs-id1429217\">\n<li>A trapezoid has four sides.<\/li>\n<li>Two of its sides are parallel and two sides are not.<\/li>\n<li>The area, [latex]A[\/latex], of a trapezoid is [latex]\\text{A}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex] .<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the area of a trapezoid whose height is [latex]6[\/latex] inches and whose bases are [latex]14[\/latex] and [latex]11[\/latex] inches.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468452905\" 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\/24223952\/CNX_BMath_Figure_09_04_080_img-01.png\" alt=\"A trapezoid of base 11 and 14 inches and height of 6 inches.\" width=\"243\" height=\"148\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of the trapezoid<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]A=\\text{the area}[\/latex]<\/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\/24223953\/CNX_BMath_Figure_09_04_080_img-02.png\" alt=\"Plugging in 11 and 14 for bases and 6 for height the equation area A equals one half times height times base one plus base two becomes area equals one half times 6 times 11 plus 14.\" width=\"392\" height=\"92\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A={\\Large\\frac{1}{2}}\\normalsize\\cdot 6(25)[\/latex]<\/p>\n<p>[latex]A=3(25)[\/latex]<\/p>\n<p>[latex]A=75[\/latex] square inches<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Is this answer reasonable?<\/td>\n<td>\u00a0[latex]\\checkmark[\/latex]\u00a0 see reasoning below<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If we draw a rectangle around the trapezoid that has the same big base [latex]B[\/latex] and a height [latex]h[\/latex], its area should be greater than that of the trapezoid.<br \/>\nIf we draw a rectangle inside the trapezoid that has the same little base [latex]b[\/latex] and a height [latex]h[\/latex], its area should be smaller than that of the trapezoid.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223956\/CNX_BMath_Figure_09_04_060.png\" alt=\"A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.\" \/><br \/>\nThe area of the larger rectangle is [latex]84[\/latex] square inches and the area of the smaller rectangle is [latex]66[\/latex] square inches. So it makes sense that the area of the trapezoid is between [latex]84[\/latex] and [latex]66[\/latex] square inches<\/p>\n<p>Step 7. <strong>Answer<\/strong> the question. The area of the trapezoid is [latex]75[\/latex] square inches.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146533\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146533&theme=oea&iframe_resize_id=ohm146533&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show another example of how to use the formula to find the area of a trapezoid given the lengths of it&#8217;s height and bases.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Find the Area of a Trapezoid\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/WNo7s-XoI4w?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the area of a trapezoid whose height is [latex]5[\/latex] feet and whose bases are [latex]10.3[\/latex] and [latex]13.7[\/latex] feet.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q158293\">Show Solution<\/span><\/p>\n<div id=\"q158293\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469577474\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says,\">\n<tbody>\n<tr style=\"height: 178.867px;\">\n<td style=\"width: 417.267px; height: 178.867px;\">Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td style=\"width: 414.733px; height: 178.867px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223958\/CNX_BMath_Figure_09_04_081_img-01.png\" alt=\"A trapezoid of base 10.3 and 13.7 feet and height of 5 feet.\" width=\"421\" height=\"141\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 417.267px; height: 15px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td style=\"width: 414.733px; height: 15px;\">the area of the trapezoid<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 417.267px; height: 15px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td style=\"width: 414.733px; height: 15px;\">Let <em>A<\/em> = the area<\/td>\n<\/tr>\n<tr style=\"height: 102px;\">\n<td style=\"width: 417.267px; height: 102px;\">Step 4.<strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td style=\"width: 414.733px; height: 102px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223959\/CNX_BMath_Figure_09_04_081_img-02.png\" alt=\"Plugging in 10.3 and 13.7 for bases and 5 for height the equation area A equals one half times height times base one plus base two becomes area equals one half times 5 times 13.7 plus 10.3.\" width=\"421\" height=\"99\" \/><\/td>\n<\/tr>\n<tr style=\"height: 90px;\">\n<td style=\"width: 417.267px; height: 90px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td style=\"width: 414.733px; height: 90px;\">[latex]A={\\Large\\frac{1}{2}}\\normalsize\\cdot 5(24)[\/latex]<\/p>\n<p>[latex]A=12(5)[\/latex]<\/p>\n<p>[latex]A=60[\/latex] square feet<\/td>\n<\/tr>\n<tr style=\"height: 305px;\">\n<td style=\"width: 417.267px; height: 305px;\">Step 6. <strong>Check:<\/strong> Is this answer reasonable?<\/p>\n<p>The area of the trapezoid should be less than the area of a rectangle with base [latex]13.7[\/latex] and height [latex]5[\/latex], but more than the area of a rectangle with base [latex]10.3[\/latex] and height [latex]5[\/latex].<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224003\/CNX_BMath_Figure_09_04_063.png\" alt=\"An image of a trapezoid is shown with a red rectangle drawn around it. The larger base of the trapezoid is labeled 13.7 ft. and is the same as the base of the rectangle. The height of both the trapezoid and the rectangle is 5 ft. Next to this is an image of a trapezoid with a black rectangle drawn inside it. The smaller base of the trapezoid is labeled 10.3 ft. and is the same as the base of the rectangle. Below the images is A sub red rectangle is greater than A sub trapezoid is greater than A sub rectangle. Below this is 68.5, 60, and 51.5.\" \/><\/td>\n<td style=\"width: 414.733px; height: 305px;\">\u00a0[latex]\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 417.267px; height: 15px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td style=\"width: 414.733px; height: 15px;\">The area of the trapezoid is [latex]60[\/latex] square feet.<\/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=\"ohm146534\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146534&theme=oea&iframe_resize_id=ohm146534&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>Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of [latex]3.4[\/latex] yards and the bases are [latex]8.2[\/latex] and [latex]5.6[\/latex] yards. How many square yards will be available to plant?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q676574\">Show Solution<\/span><\/p>\n<div id=\"q676574\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469766721\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td style=\"width: 409.183px;\">Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td style=\"width: 424.817px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224005\/CNX_BMath_Figure_09_04_082_img-01.png\" alt=\"A trapezoid of bases 5.6 and 8.2 yards and height of 3.4 yards.\" width=\"418\" height=\"129\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td style=\"width: 424.817px;\">the area of a trapezoid<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td style=\"width: 424.817px;\">Let [latex]A[\/latex] = the area<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 4.<strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td style=\"width: 424.817px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224007\/CNX_BMath_Figure_09_04_082_img-02.png\" alt=\"Plugging in 5.6 and 8.2 for bases and 3.4 for height the equation area A equals one half times height times base one plus base two becomes area equals one half times 3.4 times 5.6 plus 8.2.\" width=\"418\" height=\"101\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td style=\"width: 424.817px;\">[latex]A={\\Large\\frac{1}{2}}\\normalsize(3.4)(13.8)[\/latex]<\/p>\n<p>[latex]A=23.46[\/latex] square yards.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 834px;\" colspan=\"2\">Step 6. <strong>Check:<\/strong> Is this answer reasonable?<\/p>\n<p>Yes. The area of the trapezoid is less than the area of a rectangle with a base of [latex]8.2[\/latex] yd and height [latex]3.4[\/latex] yd, but more than the area of a rectangle with base [latex]5.6[\/latex] yd and height [latex]3.4[\/latex] yd.<\/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\/24224012\/CNX_BMath_Figure_09_04_066.png\" alt=\"This image is a table with two rows. the first row is split into three columns. The first column is the formula Area of a rectangle equals base times height. On the next line under this it has numbers plugged into the formula; the base, 8.2 in parentheses times the height 3.4 in parentheses. Under this is it has \u201cequals 27.88 yards squared\u201d. The center column includes the formula of a trapezoid and says Area of a trapezoid equals one half times 3.5 yards in parentheses times 5.8 plus 8.2 in parentheses. Under this it has \u201cequals 23.46 yards squared\u201d. In the third column it it has the formula the area of a rectangle equals base times height. Under this it has equals 5.6 in parentheses times 3.4 in parentheses. Under this it has \u201cequals 19.04 yards squared.\u201d In the second row, centered from left to right it has \u201cArea of a rectangle\u201d and a \u201cgreater than\u201d sign, \u201cArea of a trapezoid\u201d and a greater than sign and \u201carea of a rectangle\u201d. Under Area of a rectangle it has 27.88, then 23.46 under \u201carea of a trapezoid\u201d, then 19.04 under \u201carea of a rectangle\u201d.\" width=\"850\" height=\"200\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td style=\"width: 424.817px;\">Vinny has [latex]23.46[\/latex] square yards in which he can plant.<\/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=\"ohm146535\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146535&theme=oea&iframe_resize_id=ohm146535&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-10763\">\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 146533, 146534, 146535. <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 Trapezoid. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/WNo7s-XoI4w\">https:\/\/youtu.be\/WNo7s-XoI4w<\/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":11,"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 146533, 146534, 146535\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Find the Area of a Trapezoid\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/WNo7s-XoI4w\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"096dc39df9044b6cb7595a8b3f1f9cd8, 45a631590b4e49b595c469833f7c312e","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10763","chapter","type-chapter","status-publish","hentry"],"part":7831,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10763","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":28,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10763\/revisions"}],"predecessor-version":[{"id":20413,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10763\/revisions\/20413"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/7831"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10763\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=10763"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=10763"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=10763"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=10763"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}