document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204418>Question 1204418</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the area of the parallelogram with the following vertices:\r<BR>\n" );
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document.write( "1. (-2, 3), (5, 8), (3, 3), and (0, 8)\r<BR>\n" );
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document.write( "A. 16 sq units<BR>\n" );
document.write( "B. 20 sq units<BR>\n" );
document.write( "C. 34 sq units<BR>\n" );
document.write( "D. 18 sq units<BR>\n" );
document.write( "E. 25 sq units<BR>\n" );
document.write( "F. 36 sq units\r<BR>\n" );
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document.write( "2. (-2, 7), (-4, 4), (-11, 4), and (-9, 7)\r<BR>\n" );
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document.write( "A. 15 sq units<BR>\n" );
document.write( "B. 63 sq units<BR>\n" );
document.write( "C. 28 sq units<BR>\n" );
document.write( "D. 21 sq units<BR>\n" );
document.write( "E. 36 sq units<BR>\n" );
document.write( "F. 14 sq units\r<BR>\n" );
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document.write( "3. (-6, 6), (-6, 3), (-12, 3), and (-12, 6)\r<BR>\n" );
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document.write( "A. 36 sq units<BR>\n" );
document.write( "B. 54 sq units<BR>\n" );
document.write( "C. 15 sq units<BR>\n" );
document.write( "D. -45 sq units<BR>\n" );
document.write( "E. 18 sq units<BR>\n" );
document.write( "F. 24 sq units\r<BR>\n" );
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document.write( "4. (12, -3), (5, -6), (5, -3), and (12, -6)\r<BR>\n" );
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document.write( "A. 21 sq units<BR>\n" );
document.write( "B. -17 sq units<BR>\n" );
document.write( "C. 50 sq units<BR>\n" );
document.write( "D. 32.5 sq units<BR>\n" );
document.write( "E. 42 sq units<BR>\n" );
document.write( "F. 30 sq units´\r<BR>\n" );
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document.write( "5. (-2, 2), (-6, 9), (-13, 9), and (-9, 2)\r<BR>\n" );
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document.write( "A. 10 sq units<BR>\n" );
document.write( "B. 25 sq units<BR>\n" );
document.write( "C. -55 sq units<BR>\n" );
document.write( "D. 64 sq units<BR>\n" );
document.write( "E. 35 sq units<BR>\n" );
document.write( "F. 49 sq units </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840647 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=840647\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Please post <font size=4><u>one</u></font> problem at a time.\r<BR>\n" );
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document.write( "I'll focus on problem 1 only.\r<BR>\n" );
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document.write( "I'll reorder the points from<BR>\n" );
document.write( "(-2, 3), (5, 8), (3, 3), and (0, 8)<BR>\n" );
document.write( "to<BR>\n" );
document.write( "(-2, 3), (3, 3), (5, 8), and (0, 8)<BR>\n" );
document.write( "and I'll give those points the labels A,B,C,D.\r<BR>\n" );
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document.write( "A = (-2,3)<BR>\n" );
document.write( "B = (3,3)<BR>\n" );
document.write( "C = (5,8)<BR>\n" );
document.write( "D = (0,8)\r<BR>\n" );
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document.write( "Plot those points on the xy plane. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-4%2C7%2C-2%2C10%2C%0D%0Agraph%28400%2C400%2C-4%2C7%2C-2%2C10%29%2C%0D%0Acircle%28-2%2C3%2C0.05%29%2Ccircle%28-2%2C3%2C0.07%29%2Ccircle%28-2%2C3%2C0.09%29%2Ccircle%28-2%2C3%2C0.11%29%2Ccircle%28-2%2C3%2C0.13%29%2Ccircle%28-2%2C3%2C0.15%29%2Ccircle%283%2C3%2C0.05%29%2Ccircle%283%2C3%2C0.07%29%2Ccircle%283%2C3%2C0.09%29%2Ccircle%283%2C3%2C0.11%29%2Ccircle%283%2C3%2C0.13%29%2Ccircle%283%2C3%2C0.15%29%2Ccircle%285%2C8%2C0.05%29%2Ccircle%285%2C8%2C0.07%29%2Ccircle%285%2C8%2C0.09%29%2Ccircle%285%2C8%2C0.11%29%2Ccircle%285%2C8%2C0.13%29%2Ccircle%285%2C8%2C0.15%29%2Ccircle%280%2C8%2C0.05%29%2Ccircle%280%2C8%2C0.07%29%2Ccircle%280%2C8%2C0.09%29%2Ccircle%280%2C8%2C0.11%29%2Ccircle%280%2C8%2C0.13%29%2Ccircle%280%2C8%2C0.15%29%2C%0D%0Aline%28-2%2C3%2C3%2C3%29%2C%0D%0Aline%283%2C3%2C5%2C8%29%2C%0D%0Aline%285%2C8%2C0%2C8%29%2C%0D%0Aline%280%2C8%2C-2%2C3%29%2C%0D%0Alocate%28-2-0.5%2C3-0.2%2C%22A%22%29%2Clocate%283%2B0.2%2C3-0.2%2C%22B%22%29%2Clocate%285%2B0.2%2C8%2C%22C%22%29%2Clocate%280%2B0.1%2C8%2B0.5%2C%22D%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Because points A and B share the same y coordinate, it indicates segment AB is horizontal.<BR>\n" );
document.write( "The horizontal segment allows us to quickly compute the segment length.\r<BR>\n" );
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document.write( "Segment AB = 5 units long because it's the distance from -2 to 3 on the number line.<BR>\n" );
document.write( "Count out the spaces. Alternatively subtract x coordinates and use absolute value.\r<BR>\n" );
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document.write( "AB = 5 is the base.\r<BR>\n" );
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document.write( "The vertical distance from A(-2,3) to C(5,8) is 5 units because we subtract y coordinates: 8-3 = 5<BR>\n" );
document.write( "This represents the height of the parallelogram.<BR>\n" );
document.write( "Keep in mind the base and height are <u>always</u> perpendicular to each other.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "area = base*height<BR>\n" );
document.write( "area = 5*5<BR>\n" );
document.write( "area = 25 square units\r<BR>\n" );
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document.write( "<font color=black size=4>The final answer to problem 1 is <u>25 square units</u>.</font>\r<BR>\n" );
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document.write( "Another method\r<BR>\n" );
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document.write( "I'll use something called the shoelace formula.<BR>\n" );
document.write( "The naming comes from the idea we form a shoelace pattern as shown in the diagram here<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=200 src = \"https://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Shoelace3.png/270px-Shoelace3.png\"><BR>\n" );
document.write( "Image Source<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Shoelace_formula\">https://en.wikipedia.org/wiki/Shoelace_formula</a>\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "Here are alternative names for it:<BR>\n" );
document.write( "Gauss's area formula<BR>\n" );
document.write( "Surveyor's formula\r<BR>\n" );
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document.write( "It appears that tutor @erinonimus is using some variation of this formula.\r<BR>\n" );
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document.write( "To set things up, place the coordinates of A through D in four rows like this<BR>\n" );
document.write( "A = (-2,3)<BR>\n" );
document.write( "B = (3,3)<BR>\n" );
document.write( "C = (5,8)<BR>\n" );
document.write( "D = (0,8)\r<BR>\n" );
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document.write( "Let's delete the labels<BR>\n" );
document.write( "(-2,3)<BR>\n" );
document.write( "(3,3)<BR>\n" );
document.write( "(5,8)<BR>\n" );
document.write( "(0,8)\r<BR>\n" );
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document.write( "Afterward, we make a copy of the first point (-2,3) and place it at the bottom of the list<BR>\n" );
document.write( "(-2,3)<BR>\n" );
document.write( "(3,3)<BR>\n" );
document.write( "(5,8)<BR>\n" );
document.write( "(0,8)<BR>\n" );
document.write( "(-2,3)<BR>\n" );
document.write( "This will help the list form a loop.\r<BR>\n" );
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document.write( "Let's space out the x and y coordinates<BR>\n" );
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document.write( "After that is set up, draw the diagonals shown here<BR>\n" );
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document.write( "We multiply along the red diagonal pairs and add up those products<BR>\n" );
document.write( "<font color=red>-2*3</font>+<font color=red>3*8</font>+<font color=red>5*8</font>+<font color=red>0*3</font> = -6+24+40+0 = <font color=red>58</font>\r<BR>\n" );
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document.write( "Do the same for the blue diagonal pairs as well<BR>\n" );
document.write( "<font color=blue>-2*8</font>+<font color=blue>0*8</font>+<font color=blue>5*3</font>+<font color=blue>3*3</font> = -16+0+15+9 = <font color=blue>8</font>\r<BR>\n" );
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document.write( "Subtract the results<BR>\n" );
document.write( "<font color=red>58</font> - <font color=blue>8</font> = 50<BR>\n" );
document.write( "If this result was negative, then we'd take the absolute value of it to ensure it's positive.\r<BR>\n" );
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document.write( "Lastly, take half of this to get the area = (1/2)*50 = 25\r<BR>\n" );
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document.write( "<font color=black size=4>The final answer to problem 1 is <u>25 square units</u>.</font>\r<BR>\n" );
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document.write( "A word of caution: <BR>\n" );
document.write( "If you swap the positions of B and C, then the shoelace formula will yield an area of 0. <BR>\n" );
document.write( "It's a strangely fascinating situation where one half of the area cancels out with the other half (it forms a bow-tie shaped figure).\r<BR>\n" );
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document.write( "vs<BR>\n" );
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document.write( "Some more practice<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Points-lines-and-rays/Points-lines-and-rays.faq.question.1197252.html\">https://www.algebra.com/algebra/homework/Points-lines-and-rays/Points-lines-and-rays.faq.question.1197252.html</a><BR>\n" );
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