document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102095>Question 1102095</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The longer base of an isosceles trapezoid measures 18 ft. The nonparallel sides measure 11 &#8203;ft, and the base angles measure 80°.\r<BR>\n" );
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document.write( "I was able to find the length of a diagonal which is 19 ft. I need to know how to find the area of the trapezoid.\r<BR>\n" );
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document.write( "Thanks friends </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #716738 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=716738\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "First let's draw a picture of what this could look like<BR>\n" );
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document.write( "The longer base is horizontally placed along the bottom, which is the distance from point A to point B. So AB = 18 feet.\r<BR>\n" );
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document.write( "The nonparallel sides, AD and BC are both 11 feet long. So AD = 11 and BC = 11. <BR>\n" );
document.write( "We don't know the length of CD, which is the length of the other parallel side. <BR>\n" );
document.write( "We'll need this to find the area of the trapezoid. For now, let's call it x. So CD = x.\r<BR>\n" );
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document.write( "To help find CD, let's drop a perpendicular line straight down from points C and D like shown below<BR>\n" );
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document.write( "Those perpendicular lines, the dashed vertical ones, drop to points E and F as shown in the figure. <BR>\n" );
document.write( "The distance from D to E is some unknown height h.\r<BR>\n" );
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document.write( "The distance from A to E is unknown. We'll call it y. So AE = y. <BR>\n" );
document.write( "Because we have an isosceles trapezoid, we have symmetry along the vertical center line. <BR>\n" );
document.write( "This means FB = y as well. The length of EF is equal to the length of CD. This means EF = x. \r<BR>\n" );
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document.write( "Overall, AE+EF+FB = AB translating to y+x+y = 18 or x+2y = 18. Solve for x to get x = -2y+18. <BR>\n" );
document.write( "If we knew the value of y, then we can find x.\r<BR>\n" );
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document.write( "To find the value of y, we'll use trigonometry. We'll use the cosine rule\r<BR>\n" );
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document.write( "cos(angle) = adjacent/hypotenuse<BR>\n" );
document.write( "cos(80) = AE/AD<BR>\n" );
document.write( "cos(80) = y/11<BR>\n" );
document.write( "11*cos(80) = y<BR>\n" );
document.write( "y = 11*cos(80) <font color=blue>this is the exact length of AE and FB</font><BR>\n" );
document.write( "y = 1.910130 <font color=blue>this is the approximate length of AE and FB (rounded to 6 decimal places)</font>\r<BR>\n" );
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document.write( "We now know the approximate value of y so that makes x to be roughly...<BR>\n" );
document.write( "x = -2*y+18<BR>\n" );
document.write( "x = -2*1.910130+18<BR>\n" );
document.write( "x = 14.17974\r<BR>\n" );
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document.write( "So the other parallel side, CD, is roughly 14.17974 feet long\r<BR>\n" );
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document.write( "We're almost at the finish line. We need to find the height h. <BR>\n" );
document.write( "We'll use trig again, but this time the sine rule\r<BR>\n" );
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document.write( "sin(angle) = opposite/hypotenuse<BR>\n" );
document.write( "sin(80) = h/11<BR>\n" );
document.write( "h = 11*sin(80)<BR>\n" );
document.write( "h = 10.832885\r<BR>\n" );
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document.write( "The height of this trapezoid is roughly 10.832885 feet.\r<BR>\n" );
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document.write( "In summary so far, we found\r<BR>\n" );
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document.write( "x = 14.17974<BR>\n" );
document.write( "h = 10.832885\r<BR>\n" );
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document.write( "Which allows us to finally find the area of this trapezoid\r<BR>\n" );
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document.write( "Use this area formula<BR>\n" );
document.write( "Area of Trapezoid = h*(b1+b2)/2<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "h = height<BR>\n" );
document.write( "b1 & b2 = the parallel bases\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "Area of Trapezoid = h*(b1+b2)/2<BR>\n" );
document.write( "Area of Trapezoid = 10.832885*(14.17974+18)/2<BR>\n" );
document.write( "Area of Trapezoid = 174.29971137495\r<BR>\n" );
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document.write( "Which is approximate. Due to the approximate nature of the answer, there is likely to be rounding error. <BR>\n" );
document.write( "Use more decimal digits in the values of x and h to reduce the rounding error. \r<BR>\n" );
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document.write( "Side Note: the units for the answer are \"square feet\".<BR>\n" );
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