document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209009>Question 1209009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Farmer Jessie has a field shaped as quadrilateral ABCD. She measures three of the sides AB = 50 meters, BC = 65 meters, and CD = 80 meters. She also determines that angle ABC = 130 degrees and BCD = 52 degrees. <BR>\n" );
document.write( "(a) What is the distance between points A and C?<BR>\n" );
document.write( "(b) Show one way to find the area of triangle ABC.<BR>\n" );
document.write( "(c) Show a different way to find the area of triangle ABC.<BR>\n" );
document.write( "(d) What is the area of quadrilateral ABCD?<BR>\n" );
document.write( "Round each result to 3 decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847575 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=847575\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)<BR>\n" );
document.write( "Here is a diagram if your teacher/textbook has not provided it.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-3.1%2C2.56%2C-4.88%2C4.04%2C%0D%0Aline%28-0.62%2C3.04%2C1.56%2C0%29%2Cline%281.56%2C0%2C0.74%2C-3.88%29%2Cline%280.74%2C-3.88%2C-2.1%2C-0.56%29%2Cline%28-2.1%2C-0.56%2C-0.62%2C3.04%29%2C%0D%0Alocate%28-0.62%2C3.44%2C%22A%22%29%2Clocate%281.62%2C0%2C%22B%22%29%2Clocate%280.74%2C-3.88%2C%22C%22%29%2Clocate%28-2.2%2C-0.56%2C%22D%22%29%2Clocate%280.66%2C1.7%2C%2250%22%29%2Clocate%281.48-0.2%2C-1.8%2C%2265%22%29%2Clocate%28-1.1%2C-2.22%2C%2280%22%29%2Clocate%280.82%2C0.12%2C130%5Eo%29%2Clocate%280.43%2C-2.94%2C52%5Eo%29%2C%0D%0Alocate%28-2.8%2C-4.2%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's draw in diagonal AC so we split the quadrilateral into triangles ABC and ACD.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-3.1%2C2.56%2C-4.88%2C4.04%2C%0D%0Aline%28-0.62%2C3.04%2C1.56%2C0%29%2Cline%281.56%2C0%2C0.74%2C-3.88%29%2Cline%280.74%2C-3.88%2C-2.1%2C-0.56%29%2Cline%28-2.1%2C-0.56%2C-0.62%2C3.04%29%2C%0D%0Aline%28-0.62%2C3.04%2C0.74%2C-3.88%29%2C%0D%0Alocate%28-0.62%2C3.44%2C%22A%22%29%2Clocate%281.62%2C0%2C%22B%22%29%2Clocate%280.74%2C-3.88%2C%22C%22%29%2Clocate%28-2.2%2C-0.56%2C%22D%22%29%2Clocate%280.66%2C1.7%2C%2250%22%29%2Clocate%281.48-0.2%2C-1.8%2C%2265%22%29%2Clocate%28-1.1%2C-2.22%2C%2280%22%29%2Clocate%280.82%2C0.12%2C130%5Eo%29%2C%0D%0Alocate%28-0.2%2C0%2C%22x%22%29%2C%0D%0Alocate%28-2.8%2C-4.2%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = length of segment AC\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focus on triangle ABC only. <BR>\n" );
document.write( "Either cover up the other triangle, or draw triangle ABC off to the side.<BR>\n" );
document.write( "We use the Law of Cosines to find x.<BR>\n" );
document.write( "b^2 = a^2 + c^2 - 2*a*c*cos(B)<BR>\n" );
document.write( "x^2 = 65^2 + 50^2 - 2*65*50*cos(130)<BR>\n" );
document.write( "x^2 = 10903.11946296<BR>\n" );
document.write( "x = sqrt(10903.11946296)<BR>\n" );
document.write( "x = 104.41800354<BR>\n" );
document.write( "This value is approximate. <BR>\n" );
document.write( "Unless otherwise stated, each decimal value mentioned from here on out will also be approximate. <BR>\n" );
document.write( "The distance from A to C is <font color=red>roughly 104.418 meters</font> when rounding to 3 decimal places.<BR>\n" );
document.write( "Please make sure that your calculator is set to degrees mode.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "------------------------<BR>\n" );
document.write( "Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Triangle ABC has these side lengths<BR>\n" );
document.write( "AB = 50 (given)<BR>\n" );
document.write( "BC = 65 (given)<BR>\n" );
document.write( "AC = 104.41800354 (approximate; see part (a) )\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can use Heron's triangle area formula to get the following<BR>\n" );
document.write( "s = semi perimeter = (a+b+c)/2 = (65+104.41800354+50)/2 = 109.70900177<BR>\n" );
document.write( "area = sqrt(s*(s-a)*(s-b)*(s-c))<BR>\n" );
document.write( "area = sqrt(109.70900177*(109.70900177-65)*(109.70900177-104.41800354)*(109.70900177-50))<BR>\n" );
document.write( "area = 1244.822220001852<BR>\n" );
document.write( "Triangle ABC has area of <font color=red>approximately 1244.822 square meters</font> when rounding to 3 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "------------------------<BR>\n" );
document.write( "Part (c)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The previous part relies on calculating the length of AC. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Luckily there's a much faster way to find the area of triangle ABC without needing AC at all. <BR>\n" );
document.write( "I'll use the SAS triangle area formula. SAS stands for \"side angle side\".<BR>\n" );
document.write( "area = 0.5*side1*side2*sin( included angle )<BR>\n" );
document.write( "area = 0.5*AB*BC*sin(angle ABC )<BR>\n" );
document.write( "area = 0.5*50*65*sin(130)<BR>\n" );
document.write( "area = 1244.822220068339<BR>\n" );
document.write( "When rounding to 3 decimal places, we get <font color=red>1244.822 square meters</font> once again. \r<BR>\n" );
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document.write( "-----------------------<BR>\n" );
document.write( "Part (d)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let y = measure of angle ACB<BR>\n" );
document.write( "The adjacent angle ACD is 52-y. Angles ACB and ACD add to 52 degrees (measure of angle BCD).\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Since we found x = AC = 104.41800354 approximately, we can update the diagram to this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-3.1%2C2.56%2C-4.88%2C4.04%2C%0D%0Aline%28-0.62%2C3.04%2C1.56%2C0%29%2Cline%281.56%2C0%2C0.74%2C-3.88%29%2Cline%280.74%2C-3.88%2C-2.1%2C-0.56%29%2Cline%28-2.1%2C-0.56%2C-0.62%2C3.04%29%2C%0D%0Aline%28-0.62%2C3.04%2C0.74%2C-3.88%29%2C%0D%0Alocate%28-0.62%2C3.44%2C%22A%22%29%2Clocate%281.62%2C0%2C%22B%22%29%2Clocate%280.74%2C-3.88%2C%22C%22%29%2Clocate%28-2.2%2C-0.56%2C%22D%22%29%2Clocate%280.66%2C1.7%2C%2250%22%29%2Clocate%281.48-0.2%2C-1.8%2C%2265%22%29%2Clocate%28-1.1%2C-2.22%2C%2280%22%29%2Clocate%280.82%2C0.12%2C130%5Eo%29%2C%0D%0Alocate%28-0.2-1.3%2C0%2C104.41800354%29%2C%0D%0Alocate%280.43%2B0.25%2C-2.94%2B0.5%2Cy%5Eo%29%2C%0D%0Alocate%280.43-0.9%2C-2.94%2B1.1%2C%2852-y%29%5Eo%29%2C%0D%0Alocate%28-2.8%2C-4.2%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Focus on triangle ABC.<BR>\n" );
document.write( "Use the Law of Cosines to determine angle y.<BR>\n" );
document.write( "c^2 = a^2+b^2 - 2*a*b*cos(C)<BR>\n" );
document.write( "50^2 = 65^2+104.41800354^2 - 2*65*104.41800354*cos(y)<BR>\n" );
document.write( "I'll skip a few steps and leave the arithmetic for the student to do. <BR>\n" );
document.write( "You should arrive at y = 21.51940163 degrees approximately.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Alternatively, you can use the Law of Sines<BR>\n" );
document.write( "sin(B)/b = sin(C)/c<BR>\n" );
document.write( "sin(130)/104.41800354 = sin(y)/50<BR>\n" );
document.write( "I'll let the student handle the scratch work to solve for angle y. <BR>\n" );
document.write( "You should arrive at the previously mentioned y value. Or very close to it.<BR>\n" );
document.write( "This is the approximate measure of angle ACB.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Adjacent to this is angle ACD, which has measure of 52-y = 52-21.51940163 = 30.48059837 degrees approximately.<BR>\n" );
document.write( "Now we can use the SAS area formula to find the area of triangle ACD.<BR>\n" );
document.write( "area = 0.5*side1*side2*sin( included angle )<BR>\n" );
document.write( "area = 0.5*AC*CD*sin(angle ACD )<BR>\n" );
document.write( "area = 0.5*104.41800354*80*sin(30.48059837)<BR>\n" );
document.write( "area = 2118.62695314<BR>\n" );
document.write( "Like with all of the other decimal values mentioned, this value is approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then we add the two triangle areas to get the area of the quadrilateral. <BR>\n" );
document.write( "Refer to either part (b) or part (c) to get the area of triangle ABC.<BR>\n" );
document.write( "area(ABCD) = area(ABC) + area(ACD)<BR>\n" );
document.write( "area(ABCD) = 1244.82222000 + 2118.62695314<BR>\n" );
document.write( "area(ABCD) = 3363.44917314<BR>\n" );
document.write( "The area of quadrilateral ABCD is <font color=red>approximately 3363.449 square meters</font>.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "--------------------------------------------------------------------------<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answers Summary<BR>\n" );
document.write( "(a) <font color=red>104.418 meters</font><BR>\n" );
document.write( "(b) <font color=red>1244.822 square meters</font><BR>\n" );
document.write( "(c) <font color=red>1244.822 square meters</font><BR>\n" );
document.write( "(d) <font color=red>3363.449 square meters</font><BR>\n" );
document.write( "Each decimal value is approximate.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );