document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190941>Question 1190941</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me find the answe tho these questions: A triangle ABC with A(1, 1) and B(-1,4). The gradients of AB, AV and BC are -3m, 3m and m respectively.<BR>\n" );
document.write( "a) Find the value of m <BR>\n" );
document.write( "b) Find the coordinates of C<BR>\n" );
document.write( "c) Show that AC = 2AB [10 Marks] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #822687 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=822687\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part a)\r<BR>\n" );
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document.write( "I'm assuming you meant to say AC instead of AV.\r<BR>\n" );
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document.write( "A = (1,1)<BR>\n" );
document.write( "B = (-1,4)<BR>\n" );
document.write( "Gradient = slope<BR>\n" );
document.write( "Slope = (change in y)/(change in x)<BR>\n" );
document.write( "Slope = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "Slope of AB = (4-1)/(-1-1)<BR>\n" );
document.write( "Slope of AB = (3)/(-2)<BR>\n" );
document.write( "Slope of AB = -3/2\r<BR>\n" );
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document.write( "Set this equal to the stated slope of AB (which was -3m) and solve for m<BR>\n" );
document.write( "-3/2 = -3m<BR>\n" );
document.write( "m = (-3/2)*(-1/3)<BR>\n" );
document.write( "<font color=red>m = 1/2</font>\r<BR>\n" );
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document.write( "Part b)\r<BR>\n" );
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document.write( "Let C be located at (x,y). <BR>\n" );
document.write( "The goal is to find the numbers that replace x and y.\r<BR>\n" );
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document.write( "A = (1,1)<BR>\n" );
document.write( "C = (x,y)<BR>\n" );
document.write( "Slope of AC = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "Slope of AC = (y-1)/(x-1)<BR>\n" );
document.write( "Set this equal to 3m. Then plug in m = 1/2\r<BR>\n" );
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document.write( "Slope of AC = 3m<BR>\n" );
document.write( "(y-1)/(x-1) = 3m<BR>\n" );
document.write( "(y-1)/(x-1) = 3(1/2)<BR>\n" );
document.write( "(y-1)/(x-1) = 3/2\r<BR>\n" );
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document.write( "Let's now cross multiply and isolate y<BR>\n" );
document.write( "(y-1)/(x-1) = 3/2<BR>\n" );
document.write( "2(y-1) = 3(x-1)<BR>\n" );
document.write( "2y-2 = 3x-3<BR>\n" );
document.write( "2y = 3x-3+2 <BR>\n" );
document.write( "2y = 3x-1 <BR>\n" );
document.write( "y = (3x-1)/2 <BR>\n" );
document.write( "y = (3/2)x - (1/2)<BR>\n" );
document.write( "y = 1.5x - 0.5<BR>\n" );
document.write( "This is the equation of line AC.\r<BR>\n" );
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document.write( "Next, we'll find the slope of line BC<BR>\n" );
document.write( "B = (-1,4)<BR>\n" );
document.write( "C = (x,y)<BR>\n" );
document.write( "Slope of BC = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "Slope of BC = (y-4)/(x-(-1))<BR>\n" );
document.write( "Slope of BC = (y-4)/(x+1)\r<BR>\n" );
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document.write( "Set this equal to the stated slope of m<BR>\n" );
document.write( "m = (y-4)/(x+1)\r<BR>\n" );
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document.write( "Then we'll plug in these items<BR>\n" );
document.write( "m = 1/2 = 0.5<BR>\n" );
document.write( "y = 1.5x - 0.5\r<BR>\n" );
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document.write( "We end up with this<BR>\n" );
document.write( "m = (y-4)/(x+1)<BR>\n" );
document.write( "0.5 = (1.5x-0.5-4)/(x+1)<BR>\n" );
document.write( "0.5 = (1.5x-4.5)/(x+1)\r<BR>\n" );
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document.write( "It's a lot of steps to take in (there might be a faster more efficient way, so I'll let another tutor step in for that). <BR>\n" );
document.write( "However, it shouldn't be too bad once you get enough practice with algebra.\r<BR>\n" );
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document.write( "From here, we isolate x to get...<BR>\n" );
document.write( "0.5 = (1.5x-4.5)/(x+1)<BR>\n" );
document.write( "0.5(x+1) = 1.5x-4.5<BR>\n" );
document.write( "0.5x+0.5 = 1.5x-4.5<BR>\n" );
document.write( "0.5+4.5 = 1.5x-0.5x<BR>\n" );
document.write( "5 = x<BR>\n" );
document.write( "x = 5<BR>\n" );
document.write( "Then use this x value to find its paired y value<BR>\n" );
document.write( "y = 1.5x-0.5<BR>\n" );
document.write( "y = 1.5*5-0.5<BR>\n" );
document.write( "y = 7.5-0.5<BR>\n" );
document.write( "y = 7<BR>\n" );
document.write( "Point C is located at (x,y) = <font color=red>(5, 7)</font><BR>\n" );
document.write( "Don't forget to use parenthesis when writing the coordinates.\r<BR>\n" );
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document.write( "We have these three point locations<BR>\n" );
document.write( "A = (1,1)<BR>\n" );
document.write( "B = (-1,4)<BR>\n" );
document.write( "C = (5,7)<BR>\n" );
document.write( "I'll let you use the slope formula to confirm that -3m, 3m and m represent the slopes of AB, AC, and BC respectively (where m = 1/2 = 0.5 calculated earlier).<BR>\n" );
document.write( "A tool like GeoGebra is very handy to quickly confirm we have the correct values so far.\r<BR>\n" );
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document.write( "Part c)\r<BR>\n" );
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document.write( "Focus on points A and C. Use the distance formula to find the length of the segment AC.<BR>\n" );
document.write( "A = (x1,y1) = (1,1)<BR>\n" );
document.write( "C = (x2,y2) = (5,7)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{(1-5)^2 + (1-7)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{(-4)^2 + (-6)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{16 + 36}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{52}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{4*13}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{4}*\sqrt{13}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = 2\sqrt{13}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d \approx 7.21\"><BR>\n" );
document.write( "Segment AC is exactly <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?2\sqrt{13}\"> units long.<BR>\n" );
document.write( "If we can show that segment AB is exactly <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sqrt{13}\"> units long, then we'd be able to wrap things up.\r<BR>\n" );
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document.write( "Use the distance formula for points A and B<BR>\n" );
document.write( "A = (x1,y1) = (1,1)<BR>\n" );
document.write( "B = (x2,y2) = (-1,4)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{(1-(-1))^2 + (1-4)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{(1+1)^2 + (1-4)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{(2)^2 + (-3)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{4 + 9}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d = \sqrt{13}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large d \approx 3.61\"><BR>\n" );
document.write( "We see that AB is exactly <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sqrt{13}\"> units long, so the proof is concluded.<BR>\n" );
document.write( "We have shown that AC is twice as long as AB.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "a) <font color=red>m = 1/2</font><BR>\n" );
document.write( "b) Point C is at <font color=red>(5, 7)</font><BR>\n" );
document.write( "c) Proof is shown above<BR>\n" );
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