document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1064005>Question 1064005</A>: <I><SPAN STYLE=\"word-break: break-all;\"> &#916;ABC has vertices at A(5,2), B(&#8722;3,1), and C(&#8722;2,5). Point D is located on the intersection of the altitude and AB�, in such a way that D has coordinates at approximately (&#8722;1.52,1.18).\r<BR>\n" );
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document.write( "https://cds.flipswitch.com/tools/asset/media/607812\r<BR>\n" );
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document.write( "Match each question with the correct measurement, rounded to one decimal place.\r<BR>\n" );
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document.write( "What is the approximate area of the triangle?<BR>\n" );
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document.write( "What is the approximate length of the base of the triangle for the given altitude?<BR>\n" );
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document.write( "What is the approximate length of the altitude of the triangle?<BR>\n" );
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document.write( "16.1 units<BR>\n" );
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document.write( "3.9 units<BR>\n" );
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document.write( "8.1 units<BR>\n" );
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document.write( "7.6 units<BR>\n" );
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document.write( "15.8 units<BR>\n" );
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document.write( "8.0 units<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844063 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=844063\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "Area = 15.8 square units<BR>\n" );
document.write( "Base = 8.1 units<BR>\n" );
document.write( "Altitude = 3.9 units\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Use the distance formula to determine the length of side AB<BR>\n" );
document.write( "A = (x1,y1) = (5,2) and B = (x2,y2) = (-3,1)<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%285-%28-3%29%29%5E2+%2B+%282-1%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%285-%28-3%29%29%5E2+%2B+%282-1%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+64+%2B+1+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+64+%2B+1+%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+65+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+65+%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+=+8.0623 ALIGN=MIDDLE  ALT=\"d+=+8.0623\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately \r<BR>\n" );
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document.write( "Follow similar steps to find the distance from C = (-2,5) to D = (-1.52,1.18) is roughly 3.85004 which rounds to 3.9\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "area of triangle = 0.5*base*height<BR>\n" );
document.write( "area = 0.5*AB*CD<BR>\n" );
document.write( "area = 0.5*8.1*3.9<BR>\n" );
document.write( "area = 15.795<BR>\n" );
document.write( "area = 15.8 square units.\r<BR>\n" );
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document.write( "There's rounding error going on here. <BR>\n" );
document.write( "The area of triangle ABC should be 15.5 square units exactly. <BR>\n" );
document.write( "This can be determined in the next section below.\r<BR>\n" );
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document.write( "When given the coordinates of the vertex points, we could use the <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Shoelace_formula\">shoelace formula</a> to fairly quickly find the area of any polygon. \r<BR>\n" );
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document.write( "Given list of vertex points<BR>\n" );
document.write( "(5,2)<BR>\n" );
document.write( "(-3,1)<BR>\n" );
document.write( "(-2,5)\r<BR>\n" );
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document.write( "Make a copy of the first point (5,2) and place it at the bottom of the list<BR>\n" );
document.write( "(5,2)<BR>\n" );
document.write( "(-3,1)<BR>\n" );
document.write( "(-2,5)<BR>\n" );
document.write( "(5,2)<BR>\n" );
document.write( "This will help form a loop.\r<BR>\n" );
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document.write( "Space the x and y coordinates out. Each (x,y) point gets its own row.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28300%2C200%2C-5%2C5%2C-5%2C5%2C%0D%0Alocate%28-3.3333%2C1.6666%2C5%29%2Clocate%28-1.3333%2C1.6666%2C2%29%2C%0D%0Alocate%28-3.3333%2C-0.0001%2C-3%29%2Clocate%28-1.3333%2C-0.0001%2C1%29%2C%0D%0Alocate%28-3.3333%2C-1.6668%2C-2%29%2Clocate%28-1.3333%2C-1.6668%2C5%29%2C%0D%0Alocate%28-3.3333%2C-3.3335%2C5%29%2Clocate%28-1.3333%2C-3.3335%2C2%29%2C%0D%0Alocate%28-3.3333%2C+3.3333%2Cmatrix%281%2C2%2C%22Shoelace%22%2C%22Formula%22%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( "Then draw in the diagonals as indicated below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28300%2C200%2C-5%2C5%2C-5%2C5%2C%0D%0Alocate%28-3.3333%2C1.6666%2C5%29%2Clocate%28-1.3333%2C1.6666%2C2%29%2C%0D%0Alocate%28-3.3333%2C-0.0001%2C-3%29%2Clocate%28-1.3333%2C-0.0001%2C1%29%2C%0D%0Alocate%28-3.3333%2C-1.6668%2C-2%29%2Clocate%28-1.3333%2C-1.6668%2C5%29%2C%0D%0Alocate%28-3.3333%2C-3.3335%2C5%29%2Clocate%28-1.3333%2C-3.3335%2C2%29%2C%0D%0Ared%28line%28-2.8333%2C0.9666%2C-1.4833%2C-0.3501%29%29%2C%0D%0Ared%28line%28-2.8333%2C-0.7001%2C-1.4833%2C-2.0168%29%29%2C%0D%0Ared%28line%28-2.8333%2C-2.3668%2C-1.4833%2C-3.6835%29%29%2C%0D%0Ablue%28line%28-2.8333%2C-3.6835%2C-1.4833%2C-2.3668%29%29%2C%0D%0Ablue%28line%28-2.8333%2B0.25%2C-2.0168%2C-1.4833%2C-0.7001%29%29%2C%0D%0Ablue%28line%28-2.8333%2B0.25%2C-0.3501%2C-1.4833%2C0.9666%29%29%2C%0D%0Alocate%28-3.3333%2C+3.3333%2Cmatrix%281%2C2%2C%22Shoelace%22%2C%22Formula%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply along the red diagonal pairs and add up those products.<BR>\n" );
document.write( "<font color=red>5*1</font>+(<font color=red>-3*5</font>)+(<font color=red>-2*2</font>) = 5-15-4 = <font color=red>-14</font><BR>\n" );
document.write( "Do the same for the blue diagonal pairs.<BR>\n" );
document.write( "<font color=blue>5*5</font>+(<font color=blue>-2*1</font>)+(<font color=blue>-3*2</font>) = 25-2-6 = <font color=blue>17</font>\r<BR>\n" );
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document.write( "Subtract the results.<BR>\n" );
document.write( "<font color=red>red</font> - <font color=blue>blue</font> = <font color=red>-14</font> - <font color=blue>17</font> = -31\r<BR>\n" );
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document.write( "We get a negative result. Let's take the absolute value to get |-31| = 31\r<BR>\n" );
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document.write( "Lastly, take half of this to get the area = (1/2)*31 = 15.5\r<BR>\n" );
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document.write( "The area of the triangle is exactly 15.5 square units.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> can be used to verify.\r<BR>\n" );
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document.write( "More practice on the shoelace formula is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Parallelograms/Parallelograms.faq.question.1204418.html\">here</a>\r<BR>\n" );
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document.write( "What's another way to see how the area is exactly 15.5?\r<BR>\n" );
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document.write( "My post is already quite lengthy so this section will be a brief overview rather than getting into the gritty details.<BR>\n" );
document.write( "The distance formula will show that<BR>\n" );
document.write( "AB = sqrt(65)\r<BR>\n" );
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document.write( "The equation of line AB is y = (1/8)x + 11/8<BR>\n" );
document.write( "The equation of line CD is y = -8x - 11<BR>\n" );
document.write( "Intersect those two equations to determine point D is at the exact location (-99/65, 77/65)<BR>\n" );
document.write( "Note how -99/65 = -1.52 and 77/65 = 1.18 approximately.<BR>\n" );
document.write( "So that's where your teacher is getting (-1.52, 1.18) for point D.\r<BR>\n" );
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document.write( "Use the distance formula to determine that CD = 31/sqrt(65)\r<BR>\n" );
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document.write( "Then lastly,<BR>\n" );
document.write( "area = 0.5*base*height<BR>\n" );
document.write( "area = 0.5*AB*CD<BR>\n" );
document.write( "area = 0.5*sqrt(65)*( 31/sqrt(65) )<BR>\n" );
document.write( "area = 0.5*31<BR>\n" );
document.write( "area = 15.5<BR>\n" );
document.write( "which is exact without any rounding done to it.<BR>\n" );
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