document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189507>Question 1189507</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram below, Point X is the intersection of the two diagonals TW and UV of the cubical box illustrated. Given that UX is 2 √2 cm, what is the square of the area of triangle XYZ in cm^4\r<BR>\n" );
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document.write( "Diagram: https://imgur.com/a/miA6pQf </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820962 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=820962\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Given info:<BR>\n" );
document.write( "X is the intersection of lines TW and UV<BR>\n" );
document.write( "UX = 2*sqrt(2)\r<BR>\n" );
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document.write( "The 3D figure is a cube, which has 6 square faces.<BR>\n" );
document.write( "That means the 2D quadrilateral VWUT is a square.\r<BR>\n" );
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document.write( "Draw out VWUT on a separate sheet of paper or off to the side. <BR>\n" );
document.write( "Include the diagonals TW and UV, along with the point X in the middle. \r<BR>\n" );
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document.write( "Recall that a square has its diagonals bisect each other. <BR>\n" );
document.write( "This property applies to rectangles as well (any square is a rectangle but not vice versa). <BR>\n" );
document.write( "So this means UX = XV = 2*sqrt(2)\r<BR>\n" );
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document.write( "We can then say:<BR>\n" );
document.write( "UV = UX+XV<BR>\n" );
document.write( "UV = 2*sqrt(2) + 2*sqrt(2)<BR>\n" );
document.write( "UV = 4*sqrt(2)<BR>\n" );
document.write( "The other diagonal TW is the same length as UV.\r<BR>\n" );
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document.write( "Now focus on right triangle VWU. <BR>\n" );
document.write( "We found the hypotenuse to be UV = 4*sqrt(2)<BR>\n" );
document.write( "Let's say the side lengths of the cube are x units<BR>\n" );
document.write( "So VW = x and UW = x also.\r<BR>\n" );
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document.write( "Apply the pythagorean theorem to find x.<BR>\n" );
document.write( "a^2 + b^2 = c^2<BR>\n" );
document.write( "(VW)^2 + (UW)^2 = (UV)^2<BR>\n" );
document.write( "x^2+x^2 = (4*sqrt(2))^2<BR>\n" );
document.write( "2x^2 = 32<BR>\n" );
document.write( "x^2 = 32/2<BR>\n" );
document.write( "x^2 = 16<BR>\n" );
document.write( "x = 4<BR>\n" );
document.write( "The sides of the cube are 4 cm.<BR>\n" );
document.write( "This means YZ = 4 which we'll use later. This is the base of triangle XYZ.\r<BR>\n" );
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document.write( "Let's add two points to this diagram. <BR>\n" );
document.write( "Plot point A directly below point X such that point A is on segment VW. <BR>\n" );
document.write( "Plot point B on segment YZ that is directly across from point A. <BR>\n" );
document.write( "We'll make segment AB to be parallel to VY and WZ.\r<BR>\n" );
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document.write( "By construction, points A and B are midpoints of VW and YZ respectively.\r<BR>\n" );
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document.write( "Turn your attention to triangle BAX. <BR>\n" );
document.write( "This is a right triangle with legs of XA = 2 (half the cube's side length) and AB = 4. <BR>\n" );
document.write( "The hypotenuse BX is the height of triangle XYZ.\r<BR>\n" );
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document.write( "We'll use the pythagorean theorem again.<BR>\n" );
document.write( "a^2 + b^2 = c^2<BR>\n" );
document.write( "(XA)^2 + (AB)^2 = (BX)^2<BR>\n" );
document.write( "BX = sqrt( (XA)^2 + (AB)^2 )<BR>\n" );
document.write( "BX = sqrt( (2)^2 + (4)^2 )<BR>\n" );
document.write( "BX = sqrt( 20 )<BR>\n" );
document.write( "BX = sqrt(4*5)<BR>\n" );
document.write( "BX = sqrt(4)*sqrt(5)<BR>\n" );
document.write( "BX = 2*sqrt(5)\r<BR>\n" );
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document.write( "We found that: <BR>\n" );
document.write( "YZ = 4<BR>\n" );
document.write( "BX = 2*sqrt(5)<BR>\n" );
document.write( "which represent the base and height of the triangle XYZ. \r<BR>\n" );
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document.write( "We conclude things by using the formula below<BR>\n" );
document.write( "area = (1/2)*base*height<BR>\n" );
document.write( "area = (1/2)*YZ*BX<BR>\n" );
document.write( "area = (1/2)*4*2*sqrt(5)<BR>\n" );
document.write( "area = <font color=red>4*sqrt(5)</font>\r<BR>\n" );
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document.write( "Final Answer:<BR>\n" );
document.write( "<font color=red>4*sqrt(5) square cm</font><BR>\n" );
document.write( "This value is exact. \r<BR>\n" );
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document.write( "If you were to use your calculator to compute the approximate area, then you'll get roughly 4*sqrt(5) = 8.94427<BR>\n" );
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