document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207840>Question 1207840</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The vertices of triangle ABC are A(1, 1), B(9, 3), and C(3, 5). \r<BR>\n" );
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document.write( "(a) Find the perimeter of triangle ABC. \r<BR>\n" );
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document.write( "(b) Find the perimeter of the triangle that is formed by joining the midpoints of the three sides of triangle ABC. \r<BR>\n" );
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document.write( "(c) Compute the ratio of the perimeter in part (a) to the perimeter in part (b). \r<BR>\n" );
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document.write( "(d) What theorem from geometry provides the answer for part (c) without using the results in (a) and (b)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845967 by </B><A HREF=/tutors/aboutme.mpl?userid=mananth><B>mananth(16946)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mananth><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=mananth><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=845967\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The vertices of triangle ABC are A(1, 1), B(9, 3), and C(3, 5).<BR>\n" );
document.write( "(a) Find the perimeter of triangle ABC.\r<BR>\n" );
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document.write( "l(AB)+l(BC)+l(AC) = perimeter<BR>\n" );
document.write( "Use distance formula <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=l%28AB%29+=+sqrt%28%281-9%29%5E2%2B%281-3%29%5E2%29 ALIGN=MIDDLE  ALT=\"l%28AB%29+=+sqrt%28%281-9%29%5E2%2B%281-3%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "l(AB) = 2sqrt(17)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=l%28BC%29+=sqrt%28%289-3%29%5E2%2B%283-5%29%5E2%29 ALIGN=MIDDLE  ALT=\"l%28BC%29+=sqrt%28%289-3%29%5E2%2B%283-5%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "l(AB) = 2sqrt(10)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=l%28C%2CA%29+=sqrt%28%283-1%29%5E2%2B%285-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"l%28C%2CA%29+=sqrt%28%283-1%29%5E2%2B%285-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "l(C,A) =2sqrt(5)\r<BR>\n" );
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document.write( "Perimeter of   ABC: = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%2817%29+%2B+2sqrt%2810%29+%2B+2sqrt%285%29 ALIGN=MIDDLE  ALT=\"2sqrt%2817%29+%2B+2sqrt%2810%29+%2B+2sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(b) Find the perimeter of the triangle that is formed by joining the midpoints of the three sides of triangle ABC.\r<BR>\n" );
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document.write( "Use mid point formula let M(AB),M(BC),M(AC) be the three points\r<BR>\n" );
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document.write( "mid point (AB)= (1+9)/2 = 5  , (3+1)/2 = 2  => (5,2)\r<BR>\n" );
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document.write( "mid point (BC) = (9+3)/2)= 6 and (3+5)/ 2=4  => (6,4)\r<BR>\n" );
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document.write( "mid point (AC)= (1+3)/2 =2 and (1+5)/2 = 3 => (2,3)\r<BR>\n" );
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document.write( "We know the co ordinates of mid points of the triangle which forms a triangle when joined.\r<BR>\n" );
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document.write( "use the same formula as above to find the the three lengths joining the mid points\r<BR>\n" );
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document.write( "I used calculator to find the three lengths  (M stands for mid point.)\r<BR>\n" );
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document.write( "M(AB) +M(BC)+M(AC) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29%2Bsqrt%2817%29%2Bsqrt%2810%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29%2Bsqrt%2817%29%2Bsqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(c) Compute the ratio of the perimeter in part (a) to the perimeter in part (b)./\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%2817%29+%2B+2sqrt%2810%29+%2B+2sqrt%285%29 ALIGN=MIDDLE  ALT=\"2sqrt%2817%29+%2B+2sqrt%2810%29+%2B+2sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >/ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29%2Bsqrt%2817%29%2Bsqrt%2810%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29%2Bsqrt%2817%29%2Bsqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =  2\r<BR>\n" );
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document.write( "(d) What theorem from geometry provides the answer for part (c) without using the results in (a) and (b)?\r<BR>\n" );
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document.write( "The midpoint theorem states that if a segment is formed by connecting the midpoints of two of the sides of a triangle, then that segment must be parallel to the third side. the segment is half the length of the third side.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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