document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=972914>Question 972914</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Adjacent sides of parallelogram are 7cm and 9cm. Its 1 diagonal is 8cm. Find its other diagonal.<BR>\n" );
document.write( "2. Diagonals of parallelogram are 12cm and 26cm. One of its side is 17cm. Find the other side. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #595334 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</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=595334\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. Adjacent sides of parallelogram are 7cm and 9cm. Its 1 diagonal is 8cm. Find its other diagonal.<BR>\n" );
document.write( "Two adjacent sides and that diagonal form a triangle,<BR>\n" );
document.write( "with the angle between those two sides opposite that diagonal.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The law of cosines says that for a triangle with vertices A, B, and C,<BR>\n" );
document.write( "opposite sides a, b, and c respectively,<BR>\n" );
document.write( "the measures of angles and sides are related by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "There is a triangle formed by<BR>\n" );
document.write( "two sides measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7 ALIGN=MIDDLE  ALT=\"b=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >cm and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=9 ALIGN=MIDDLE  ALT=\"c=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >cm flanking angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and a diagonal measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=8 ALIGN=MIDDLE  ALT=\"a=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >cm, connected to both of those sides.<BR>\n" );
document.write( "That triangle, with sides so similar in length, must be an acute triangle.<BR>\n" );
document.write( "Applying law of cosines to that triangle, we write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E2=7%5E2%2B9%5E2-2%2A7%2A9%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"8%5E2=7%5E2%2B9%5E2-2%2A7%2A9%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64=49%2B81-2%2A7%2A9%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"64=49%2B81-2%2A7%2A9%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A7%2A9%2Acos%28A%29=49%2B81-64 ALIGN=MIDDLE  ALT=\"2%2A7%2A9%2Acos%28A%29=49%2B81-64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=126cos%28A%29=66 ALIGN=MIDDLE  ALT=\"126cos%28A%29=66\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%29=66%2F126=11%2F21 ALIGN=MIDDLE  ALT=\"cos%28A%29=66%2F126=11%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The cosine is positive, meaning that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an acute angle.<BR>\n" );
document.write( "In fact, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=about58.4%5Eo ALIGN=MIDDLE  ALT=\"A=about58.4%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but we do not need to calculate that.<BR>\n" );
document.write( "The parallelogram looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28350%2C200%2C-4.5%2C9.5%2C-1%2C7%2C%0D%0Atriangle%28-3.667%2C5.963%2C5.333%2C5.963%2C0%2C0%29%2C%0D%0Atriangle%280%2C0%2C5.333%2C5.963%2C9%2C0%29%2C%0D%0Alocate%28-0.5%2C0%2CD%29%2Clocate%289%2C0%2CA%29%2C%0D%0Alocate%28-4%2C6.5%2CC%29%2Clocate%285.67%2C6.5%2CB%29%2C%0D%0Alocate%284.3%2C0%2C9cm%29%2Clocate%282.66%2C2.98%2C8cm%29%2C%0D%0Alocate%286.5%2C2.98%2C7cm%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The other diagonal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > is opposite the other angle of the parallelogram, the angle at D,<BR>\n" );
document.write( "which is supplementary to angle A, so that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28D%29=-cos%28A%29=-11%2F21 ALIGN=MIDDLE  ALT=\"cos%28D%29=-cos%28A%29=-11%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and is also flanked by sides measuring 7cm and 9cm.<BR>\n" );
document.write( "Those sides and the other diagonal form another triangle.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28350%2C200%2C-4.5%2C9.5%2C-1%2C7%2C%0D%0Atriangle%28-3.667%2C5.963%2C5.333%2C5.963%2C9%2C0%29%2C%0D%0Atriangle%28-3.667%2C5.963%2C0%2C0%2C9%2C0%29%2C%0D%0Alocate%28-0.5%2C0%2CD%29%2Clocate%289%2C0%2CA%29%2C%0D%0Alocate%28-4%2C6.5%2CC%29%2Clocate%285.67%2C6.5%2CB%29%2C%0D%0Alocate%284.3%2C0%2C9cm%29%2Clocate%282.66%2C2.9%2Cd%29%2C%0D%0Alocate%28-1.6%2C2.98%2C7cm%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Applying law of cosines to that other triangle, we write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2=7%5E2%2B9%5E2-2%2A7%2A9%2Acos%28D%29=49%2B81-126%2A%28-11%2F21%29=49%2B81%2B66=196 ALIGN=MIDDLE  ALT=\"d%5E2=7%5E2%2B9%5E2-2%2A7%2A9%2Acos%28D%29=49%2B81-126%2A%28-11%2F21%29=49%2B81%2B66=196\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28196%29=14 ALIGN=MIDDLE  ALT=\"d=sqrt%28196%29=14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The other diagonal measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2814cm%29 ALIGN=MIDDLE  ALT=\"highlight%2814cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "2. Diagonals of parallelogram are 12cm and 26cm. One of its side is 17cm.Yhe <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of the other side in cm.<BR>\n" );
document.write( "Since the diagonals of a parallelogram bisect each other,<BR>\n" );
document.write( "the longer diagonal will be split into two congruent segments,<BR>\n" );
document.write( "each one measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26cm%2F2=13cm ALIGN=MIDDLE  ALT=\"26cm%2F2=13cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the shorter diagonal will be split into two congruent segments,<BR>\n" );
document.write( "each one measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12cm%2F2=6cm ALIGN=MIDDLE  ALT=\"12cm%2F2=6cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, the parallelogram, divided into two pairs of congruent triangles, will look like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28550%2C176%2C-1%2C26.5%2C-1.4%2C8.4%2C%0D%0Atriangle%280%2C0%2C17%2C0%2C12.41%2C3.464%29%2Ctriangle%2824.84%2C6.928%2C17%2C0%2C12.41%2C3.464%29%2C%0D%0Atriangle%280%2C0%2C17%2C0%2C7.84%2C6.928%29%2Ctriangle%2824.84%2C6.928%2C7.84%2C6.928%2C12.41%2C3.464%29%2C%0D%0Alocate%2815.8%2C6.9%2C17%29%2Clocate%286.2%2C1.8%2C6%29%2Clocate%2818.5%2C5.15%2C6%29%2C%0D%0Alocate%289%2C5.6%2C13%29%2Clocate%283.9%2C3.7%2Cx%29%2C%0D%0Alocate%2811.1%2C4.15%2Cgreen%28X%29%29%2Clocate%2812.4%2C4.5%2Cred%28Y%29%29%2C%0D%0Ared%28arc%2812.41%2C3.464%2C2.1%2C2.1%2C-143%2C-16%29%29%2C%0D%0Agreen%28arc%2812.41%2C3.464%2C3%2C3%2C164%2C217%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The law of cosines says that for a triangle with vertices A, B, and C,<BR>\n" );
document.write( "opposite sides a, b, and c respectively,<BR>\n" );
document.write( "the measures of angles and sides are related by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "To find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we can apply law of cosines to the left side triangle and write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=6%5E2%2B13%5E2-2%2A6%2A13%2Acos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"x%5E2=6%5E2%2B13%5E2-2%2A6%2A13%2Acos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but we do not know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28X%29 ALIGN=MIDDLE  ALT=\"green%28X%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"cos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28Y%29 ALIGN=MIDDLE  ALT=\"red%28Y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28X%29 ALIGN=MIDDLE  ALT=\"green%28X%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are supplementary,<BR>\n" );
document.write( "and that because of that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28red%28Y%29%29=-cos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"cos%28red%28Y%29%29=-cos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "We can find about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28red%28Y%29%29 ALIGN=MIDDLE  ALT=\"cos%28red%28Y%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"cos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by applying law of cosines to the top triangle:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17%5E2=6%5E2%2B13%5E2-2%2A6%2A13%2Acos%28red%28Y%29%29 ALIGN=MIDDLE  ALT=\"17%5E2=6%5E2%2B13%5E2-2%2A6%2A13%2Acos%28red%28Y%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17%5E2=6%5E2%2B13%5E2%2B2%2A6%2A13%2Acos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"17%5E2=6%5E2%2B13%5E2%2B2%2A6%2A13%2Acos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=289=36%2B169%2B156%2Acos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"289=36%2B169%2B156%2Acos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28green%28X%29%29=%28289-36-169%29%2F156 ALIGN=MIDDLE  ALT=\"cos%28green%28X%29%29=%28289-36-169%29%2F156\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28green%28X%29%29=%28289-36-169%29%2F156=84%2F156=7%2F13 ALIGN=MIDDLE  ALT=\"cos%28green%28X%29%29=%28289-36-169%29%2F156=84%2F156=7%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Now we get back to the left triangle to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=6%5E2%2B13%5E2-2%2A6%2A13%2Acos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"x%5E2=6%5E2%2B13%5E2-2%2A6%2A13%2Acos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=36%2B169-156cos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"x%5E2=36%2B169-156cos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=36%2B169-156cos%28green%28X%29%29 ALIGN=MIDDLE  ALT=\"x%5E2=36%2B169-156cos%28green%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=36%2B169-156%287%2F13%29 ALIGN=MIDDLE  ALT=\"x%5E2=36%2B169-156%287%2F13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=121 ALIGN=MIDDLE  ALT=\"x%5E2=121\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28121%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28121%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=11 ALIGN=MIDDLE  ALT=\"x=11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The other side measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2811cm%29 ALIGN=MIDDLE  ALT=\"highlight%2811cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );