document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1110560>Question 1110560</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find the area of a triangle using a base and altitude with vertices (-6,3), (4,13) and (10,-5) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #725569 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=725569\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can pick any of the sides to be the base.<BR>\n" );
document.write( "you then need to find a perpendicular to that base that intersect with the intersection of the other 2 sides.\r<BR>\n" );
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document.write( "it helps to sketch your triangle first.\r<BR>\n" );
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document.write( "pick your base.\r<BR>\n" );
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document.write( "i picked the line formed by (-6,3) and (10,-5)\r<BR>\n" );
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document.write( "the equation for that line is y = -1/2 * x\r<BR>\n" );
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document.write( "now you want to find the line that's perpendicular to that line and intersects at the point of intersection of the other 2 lines.\r<BR>\n" );
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document.write( "from your sketch you should see that the point of intersection of the other 2 lines is at (4,13).\r<BR>\n" );
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document.write( "the line perpendicular to your base line will have a slope that is a negative reciprocal of the slope of your base line.\r<BR>\n" );
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document.write( "that makes the slope equal to 2.\r<BR>\n" );
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document.write( "you then use the point (4,13) to find the equation.\r<BR>\n" );
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document.write( "the equation for that line will be y = 2x + 5.\r<BR>\n" );
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document.write( "you now have 2 lines and you need to find the length of each line.\r<BR>\n" );
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document.write( "the length of your base will be the length of the line between the points (-6,3) and (10,-5).\r<BR>\n" );
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document.write( "that length will be sqrt((-16)^2 + 8^2) = sqrt(320).\r<BR>\n" );
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document.write( "next you need to find the intersection of the line perpendicular to the base with the base.\r<BR>\n" );
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document.write( "that requires you to solve those 2 equations simultaneously.\r<BR>\n" );
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document.write( "the 2 equations are\r<BR>\n" );
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document.write( "y = -1/2 * x\r<BR>\n" );
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document.write( "y = 2x + 5\r<BR>\n" );
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document.write( "subtract the first equation from the second to get:\r<BR>\n" );
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document.write( "0 = 2.5 * x + 5\r<BR>\n" );
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document.write( "solve for x to get x = -2\r<BR>\n" );
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document.write( "replace x in either of the original equations to get y = 1.\r<BR>\n" );
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document.write( "your point of intersection is (-2,1)\r<BR>\n" );
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document.write( "you can now solve for the length of the altitude because that line goes between the points (4,13) and (-2,1)\r<BR>\n" );
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document.write( "the length is sqrt(6^2 + 12^2) = sqrt(180).\r<BR>\n" );
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document.write( "you find the area by using the formula 1/2 * base * height.\r<BR>\n" );
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document.write( "this formula becomes area = 1/2 * sqrt(320) * sqrt(180).\r<BR>\n" );
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document.write( "solve for area to get area = 120 square units.\r<BR>\n" );
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document.write( "here's my sketch drawn by hand and final diagram using the equations of the lines.\r<BR>\n" );
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document.write( "in the final diagram, the perpendicular of the triangle is red and the triangle itself is blue.\r<BR>\n" );
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document.write( "the lines themselves go on indefinitely, but it's the line segments between the points shown that you want.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/022601.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/022602.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "to find the equation of a line from 2 points, you first find the slope and then solve for the y-intersection using the slope intercept form of the equation for a straight line.\r<BR>\n" );
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document.write( "the general form of the slope intercept equation is y = mx + b.\r<BR>\n" );
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document.write( "m is the slope<BR>\n" );
document.write( "b is the y-intercept\r<BR>\n" );
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document.write( "for the base line, the points were (-6,3) and (10,-5)\r<BR>\n" );
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document.write( "the slope was (-5-3) / (10-(-6) which became -8/16 which became -1/2.\r<BR>\n" );
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document.write( "you then use one of the points to solve for the y-intercept.\r<BR>\n" );
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document.write( "i used the point (-6,3).\r<BR>\n" );
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document.write( "replace y with 3 and x with -6 and y = mx + b became 3 = -1/2 * (-6) + b\r<BR>\n" );
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document.write( "simplify to get 3 = 3 + b\r<BR>\n" );
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document.write( "solve for b to get b = 0\r<BR>\n" );
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document.write( "the equation for the base line became y = -1/2 * x\r<BR>\n" );
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document.write( "once you know the point where the perpendicular line intersect with the base line, you use the same procedure to solve for the line perpendicular to the base.\r<BR>\n" );
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document.write( "the equation for that line was calculated to be y = 2x + 5\r<BR>\n" );
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document.write( "the formula for the length of each line is length = sqrt((x2-x1)^2 + (y2-y1)^2)\r<BR>\n" );
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document.write( "the line perpendicular to the base line went between the points (4,13) and (-2,1)\r<BR>\n" );
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document.write( "the length of the line became sqrt((1-13)^2 + (-2-4)^2).\r<BR>\n" );
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document.write( "this became sqrtZ((-12)^2+ (-6)^2) which became sqrt(144+36) which became sqrt(120).\r<BR>\n" );
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document.write( "similar procedure was use to find the length of the base line, which was sqrt(320).\r<BR>\n" );
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document.write( "finally, the area became 1/2 * base * height which became 1/2 * sqrt(180) * sqrt(320) which became 1/2 * sqrt(180 * 320) which became 1/2 * 240 which became 120.\r<BR>\n" );
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document.write( "all you needed was the length of the base line and the length of the line perpendicular to it.\r<BR>\n" );
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document.write( "in order to graph the triangle using equations, i needed to find the equation for each of the lines.\r<BR>\n" );
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document.write( "it was more work, but the procedure to find the equation for each line was just a repetition of the procedure i showed you for one of the  lines.\r<BR>\n" );
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document.write( "without the sketch, you might have needed to find the intersections algebraically, so the sketch helped to eliminate the need for additional line equations and was useful to give you a picture of what your triangle would look like generally.\r<BR>\n" );
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