document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209058>Question 1209058</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram, AB = BC and BD=DC=CE. AB=4 cm. Find the length of AE, in cm.<BR>\n" );
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document.write( "Answer: <BR>\n" );
document.write( "Exact length = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm<BR>\n" );
document.write( "Approximate length = 4.47214 cm<BR>\n" );
document.write( "This approximate value will slightly vary depending how you round it.\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Let's place point B at the origin.<BR>\n" );
document.write( "4 units above B is point A(0,4)\r<BR>\n" );
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document.write( "AB = BC = 4<BR>\n" );
document.write( "Since BC = 4, we move 4 units to the right of B to arrive at C(4,0)\r<BR>\n" );
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document.write( "BD = DC tells us that D is the midpoint of BC, so BD = DC = CE = 2\r<BR>\n" );
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document.write( "From point C move 2 units up to arrive at E(4,2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-1%2C5%2C-1%2C5%2C%0D%0Agraph%28400%2C400%2C-1%2C5%2C-1%2C5%29%2C%0D%0Aline%280%2C4%2C0%2C0%29%2Cline%280%2C0%2C4%2C0%29%2Cline%284%2C0%2C4%2C2%29%2Cline%282%2C0%2C4%2C2%29%2Cline%280%2C4%2C4%2C0%29%2C%0D%0Alocate%280.2%2C-0.2%2B0.5%2C%22B%22%29%2Clocate%284.2%2C-0.2%2C%22C%22%29%2Clocate%282.2-0.5%2C-0.2%2B0.5%2C%22D%22%29%2Clocate%283.2-0.2%2C0.8%2B0.6%2C%22F%22%29%2C%0D%0A%0D%0Ablue%28+locate%280.2%2B0.1%2C3.8%2B0.5%2C%22A%280%2C4%29%22%29%2Clocate%284.2%2C2.1%2C%22E%284%2C2%29%22%29%2Cline%280%2C4%2C4%2C2%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can use the distance formula to find out how far it is from A(0,4) to E(4,2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%280-4%29%5E2+%2B+%284-2%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%280-4%29%5E2+%2B+%284-2%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in (x1,y1) = (0,4) and (x2,y2) = (4,2)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28-4%29%5E2+%2B+%282%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28-4%29%5E2+%2B+%282%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+16+%2B+4+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+16+%2B+4+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+20+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+20+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%284%2A5%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%284%2A5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%284%29%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%284%29%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+2%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"d+=+2%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+4.47214 ALIGN=MIDDLE  ALT=\"d+=+4.47214\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Segment AE is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm long. This approximates to 4.47214 cm.\r<BR>\n" );
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document.write( "A slight alternate route:\r<BR>\n" );
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document.write( "From point E, draw a horizontal line until reaching the y axis. This forms right triangle EGA where G is on the same level as E and directly below point A.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-1%2C5%2C-1%2C5%2C%0D%0Agraph%28400%2C400%2C-1%2C5%2C-1%2C5%29%2C%0D%0Aline%280%2C4%2C0%2C0%29%2Cline%280%2C0%2C4%2C0%29%2Cline%284%2C0%2C4%2C2%29%2Cline%282%2C0%2C4%2C2%29%2Cline%280%2C4%2C4%2C0%29%2Cline%280%2C2%2C4%2C2%29%2C%0D%0Alocate%280.2%2C-0.2%2B0.5%2C%22B%22%29%2Clocate%284.2%2C-0.2%2C%22C%22%29%2Clocate%282.2-0.5%2C-0.2%2B0.5%2C%22D%22%29%2Clocate%283.2-0.2%2C0.8%2B0.6%2C%22F%22%29%2Clocate%280%2B0.1%2C2%2B0.3%2C%22G%22%29%2C%0D%0A%0D%0Ablue%28+locate%280.2%2B0.1%2C3.8%2B0.5%2C%22A%280%2C4%29%22%29%2Clocate%284.2%2C2.1%2C%22E%284%2C2%29%22%29%2Cline%280%2C4%2C4%2C2%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's get rid of any points or lines we don't need.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-1%2C5%2C-1%2C5%2C%0D%0Agraph%28400%2C400%2C-1%2C5%2C-1%2C5%29%2C%0D%0Aline%280%2C4%2C0%2C2%29%2Cline%280%2C2%2C4%2C2%29%2C%0D%0Alocate%280%2B0.1%2C2%2B0.3%2C%22G%22%29%2C%0D%0A%0D%0Ablue%28+locate%280.2%2B0.1%2C3.8%2B0.5%2C%22A%280%2C4%29%22%29%2Clocate%284.2%2C2.1%2C%22E%284%2C2%29%22%29%2Cline%280%2C4%2C4%2C2%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We have a right triangle with horizontal leg of GE = 4 and vertical leg GA = 2<BR>\n" );
document.write( "Use the Pythagorean theorem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to determine <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2%2B2%5E2+=+c%5E2 ALIGN=MIDDLE  ALT=\"4%5E2%2B2%5E2+=+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > solves to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+sqrt%2820%29+=+2%2Asqrt%285%29+=+4.47214 ALIGN=MIDDLE  ALT=\"c+=+sqrt%2820%29+=+2%2Asqrt%285%29+=+4.47214\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the hypotenuse of this right triangle. And it's also the distance from A to E.\r<BR>\n" );
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document.write( "As you can probably tell (or know by now), the distance formula is a modified version of the Pythagorean theorem.<BR>\n" );
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