document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158473>Question 1158473</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An isosceles trapezoid has a height of 4cm and bases 3cm and 7cm long. How long are its diagonals? Use the law of cosines. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854607 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5396)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854607\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In the USA, a trapezoid is a quadrilateral with two parallel sides, and an isosceles trapezoid is one whose other two sides have the same length.<BR>\n" );
document.write( "The names of trapezium and trapezoid are not the same in all English-speaking countries.<BR>\n" );
document.write( "Here is the American isosceles trapezoid ABCD described in the question, with diagonal AC and BD:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C+-4%2C4%2C-1%2C5%2C%0D%0Atriangle%28-3.5%2C0%2C3.5%2C0%2C1.5%2C4%29%2C%0D%0Atriangle%28-3.5%2C0%2C-1.5%2C4%2C1.5%2C4%29%2C%0D%0Atriangle%283.5%2C0%2C-1.5%2C4%2C1.5%2C4%29%2C%0D%0Alocate%28-0.2%2C-0.03%2C7cm%29%2C%0D%0Alocate%28-0.2%2C4.3%2C3cm%29%2C%0D%0Alocate%283.6%2C0.1%2CA%29%2Clocate%28-3.7%2C0.1%2CD%29%0D%0Alocate%281.55%2C4.15%2CB%29%2Clocate%28-1.75%2C4.15%2CC%29%2C%0D%0Ared%28arc%28-1.5%2C4%2C1%2C1%2C0%2C120%29%29%2Cred%28arc%281.5%2C4%2C1%2C1%2C60%2C180%29%29%2C%0D%0Ared%28arc%28-3.5%2C0%2C1%2C1%2C-60%2C0%29%29%2Cred%28arc%28-3.5%2C0%2C0.9%2C0.9%2C-65%2C0%29%29%2C%0D%0Ared%28arc%283.5%2C0%2C1%2C1%2C180%2C240%29%29%2Cred%28arc%283.5%2C0%2C0.9%2C0.9%2C180%2C240%29%29%2C%0D%0Agreen%28arrow%281.5%2C2%2C1.5%2C0%29%29%2Cgreen%28arrow%281.5%2C2%2C1.5%2C4%29%29%2C%0D%0Alocate%281.52%2C2.5%2Cgreen%284cm%29%29%2C%0D%0Acircle%281.5%2C0%2C0.02%29%2Clocate%281.49%2C-0.03%2CP%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Diagonal AC is a side in triangles ACD and ACB. Diagonal BD is a side of BDA and BDC.<BR>\n" );
document.write( "All we know about each of the triangles mentioned above  is the length of one side. No angle measures.<BR>\n" );
document.write( "The law of cosines allows us to find the length of one side of a triangle when we know the measure of the opposite angle and the length of the other two sides.<BR>\n" );
document.write( "Had the lengths of sides AB and CD been given, using the law of cosines would make sense.<BR>\n" );
document.write( "Height BP cuts right triangle ABP from the rest of the trapezoid,<BR>\n" );
document.write( "and it is easy to figure out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP=%287cm-3cm%29 ALIGN=MIDDLE  ALT=\"AP=%287cm-3cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "From that we can easily calculate measure of the angle PAB (angle A, for short).<BR>\n" );
document.write( "We can also calculate the length of AB<BR>\n" );
document.write( "However, as soon as we figure out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP=2cm ALIGN=MIDDLE  ALT=\"AP=2cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we can calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DP=7cm-2cm=5cm ALIGN=MIDDLE  ALT=\"DP=7cm-2cm=5cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then it would be easy to calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD ALIGN=MIDDLE  ALT=\"BD\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the hypotenuse of right triangle PBD using the Pythagorean theorem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD=sqrt%28%284cm%29%5E2%2B%285cm%29%5E2%29=sqrt%2841cm%5E2%29=6.403cm ALIGN=MIDDLE  ALT=\"BD=sqrt%28%284cm%29%5E2%2B%285cm%29%5E2%29=sqrt%2841cm%5E2%29=6.403cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded)<BR>\n" );
document.write( " <BR>\n" );
document.write( "To use the law of cosines on triangle ABD to calculate the length of BD, it appears like I would need the measure of angle A, and the length of AB.<BR>\n" );
document.write( "Based on right triangle ABP, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%29=BP%2FAp=4cm%2F2cm=2 ALIGN=MIDDLE  ALT=\"tan%28A%29=BP%2FAp=4cm%2F2cm=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=tan%5E-1%282%29=63.435%5Eo ALIGN=MIDDLE  ALT=\"A=tan%5E-1%282%29=63.435%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded)<BR>\n" );
document.write( "Using the Pythagorean theorem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=sqrt%28%284cm%29%5E2%2B%282cm%29%5E2%29=sqrt%2820cm%5E2%29=4.472cm ALIGN=MIDDLE  ALT=\"AB=sqrt%28%284cm%29%5E2%2B%282cm%29%5E2%29=sqrt%2820cm%5E2%29=4.472cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded)<BR>\n" );
document.write( "The law of cosines for triangle ABD would give us BD from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD%5E2=AB%5E2%2BAD%5E2-2%2AAB%2AAD%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"BD%5E2=AB%5E2%2BAD%5E2-2%2AAB%2AAD%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With the length in cm,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD%5E2=20%2B49-2%2AAB%2A7%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"BD%5E2=20%2B49-2%2AAB%2A7%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > however <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB%2Acos%28A%29=AP=2cm ALIGN=MIDDLE  ALT=\"AB%2Acos%28A%29=AP=2cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so we did not need the measure of angle A, or the length of AB, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD%5E2=20%2B49-2%2A7%2A2=20%2B49-28=41 ALIGN=MIDDLE  ALT=\"BD%5E2=20%2B49-2%2A7%2A2=20%2B49-28=41\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD=sqrt%2841%29 ALIGN=MIDDLE  ALT=\"BD=sqrt%2841%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm=6.403cm ALIGN=MIDDLE  ALT=\"cm=6.403cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );