document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205618>Question 1205618</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" src=\"https://i.ibb.co/P1MwGxB/Untitled-drawing.png\" height=\"200px\">\r<BR>\n" );
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document.write( "Point X is the intersection of the two diagonals TW and UV of the cubical box illustrated. The shortest distance from point T to point Z is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4sqrt%283%29 ALIGN=MIDDLE  ALT=\"4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm. Find the area in square centimetres of triangle XYZ. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842530 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=842530\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The space diagonal connects two opposite corners of a cube. <BR>\n" );
document.write( "One example of a space diagonal is us connecting points T and Z. Think of a rope going through a room. \r<BR>\n" );
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document.write( "In contrast, segments TW and VU are face diagonals since they are contained entirely on one face or plane.<BR>\n" );
document.write( "These two ropes are glued entirely to one wall.<BR>\n" );
document.write( "The face diagonals intersect at point X. <BR>\n" );
document.write( "It turns out that this is the midpoint of segments TW and VU. <BR>\n" );
document.write( "I'll leave the proof for the reader to determine.\r<BR>\n" );
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document.write( "All sides of a cube are the same length.<BR>\n" );
document.write( "s = side length of the cube\r<BR>\n" );
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document.write( "Draw a segment to connect points V and Z.<BR>\n" );
document.write( "Focus on right triangle VWZ.<BR>\n" );
document.write( "leg1 = VW = s<BR>\n" );
document.write( "leg2 = WZ = s<BR>\n" );
document.write( "hypotenuse = VZ = unknown<BR>\n" );
document.write( "Use the Pythagorean theorem to determine that hypotenuse VZ = s*sqrt(2)<BR>\n" );
document.write( "I'll leave the steps and scratch work for the student to do.\r<BR>\n" );
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document.write( "Move your focus to right triangle TVZ.<BR>\n" );
document.write( "leg1 = TV = s<BR>\n" );
document.write( "leg2 = VZ = s*sqrt(2)<BR>\n" );
document.write( "Use the Pythagorean theorem to determine that hypotenuse TZ = s*sqrt( 3 ).<BR>\n" );
document.write( "I'll leave the steps and scratch work for the student to do.<BR>\n" );
document.write( "Because we're told that TZ = 4*sqrt(3), this must mean that s = 4.<BR>\n" );
document.write( "The cube has a side length of 4 units.\r<BR>\n" );
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document.write( "Define these two new points<BR>\n" );
document.write( "P = midpoint of YZ<BR>\n" );
document.write( "Q = midpoint of VW\r<BR>\n" );
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document.write( "Right triangle PQX has these side lengths<BR>\n" );
document.write( "PQ = horizontal leg = 4<BR>\n" );
document.write( "QX = vertical leg = 2 (since X is at the midpoint)<BR>\n" );
document.write( "PX = unknown hypotenuse\r<BR>\n" );
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document.write( "Use the Pythagorean theorem one last time<BR>\n" );
document.write( "(PQ)^2 + (QX)^2 = (PX)^2<BR>\n" );
document.write( "(4)^2 + (2)^2 = (PX)^2<BR>\n" );
document.write( "(PX)^2 = 20<BR>\n" );
document.write( "I skipped a few steps and will let the student fill in the details.<BR>\n" );
document.write( "PX = 2*sqrt(5)\r<BR>\n" );
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document.write( "Triangle XYZ has base YZ = 4 and height PX = 2*sqrt(5)\r<BR>\n" );
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document.write( "area = 0.5*base*height<BR>\n" );
document.write( "area = 0.5*YZ*PX<BR>\n" );
document.write( "area = 0.5*4*2*sqrt(5)<BR>\n" );
document.write( "<font color=red>area = 4*sqrt(5) square cm</font><BR>\n" );
document.write( "4*sqrt(5) = 8.944271909999 approximately <BR>\n" );
document.write( "Round this however needed.<BR>\n" );
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