document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036432>Question 1036432</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If each side of a scalene traingle is tripled then prove that ratio of their area will be 1:9. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #651089 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=651089\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the ratio of the area is equal to the square of the ratio of the sides.<BR>\n" );
document.write( "if the ratio of the sides is 3 to 1, then the ratio of the area is 3^2 to 1 which makes it 9 to 1.\r<BR>\n" );
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document.write( "i am assuming that the two scalene triangles are similar.<BR>\n" );
document.write( "this means that all of their corresponding sides are proportional.<BR>\n" );
document.write( "this includes the height of each of the triangles.\r<BR>\n" );
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document.write( "if the larger triangle has sides that are 3 times the size of the smaller triangle, then the height is 3 times the size as well.\r<BR>\n" );
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document.write( "the area of any triangle is equal to 1/2 * base * height.\r<BR>\n" );
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document.write( "we'll call the base of the smaller triangle b and we'll call the height of the smaller triangle h.\r<BR>\n" );
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document.write( "we'll call the base of the larger triangle 3b and we'll call the height of the smaller triangle 3h.\r<BR>\n" );
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document.write( "the area of the smaller triangle is 1/2 * b * h<BR>\n" );
document.write( "the area of the larger triangle is 1/2 * 3 * b * 3 * h.\r<BR>\n" );
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document.write( "simplify the formula and you get the area of the larger triangle is equal to 1/2 * 9 * b * h.\r<BR>\n" );
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document.write( "the ratio of the area of the larger triangle is equal to the square of the ratio of the sides of the smaller triangle.\r<BR>\n" );
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document.write( "this applies to all triangles, not just scalene triangles.\r<BR>\n" );
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document.write( "there is a formula that will calculate the area of a triangle based on the size of the sides alone.\r<BR>\n" );
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document.write( "that is heron's formula.\r<BR>\n" );
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document.write( "heron's formula says that the area of any triangle can be derived from the length of the sides alone.\r<BR>\n" );
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document.write( "the formula states that A = sqrt(s * (s-a) * (s-b) * (s-c))\r<BR>\n" );
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document.write( "A is equal to the area of the triangle.<BR>\n" );
document.write( "s is equal to (a + b + c) / 2.<BR>\n" );
document.write( "a,b,c are the length of the sides of the triangle.\r<BR>\n" );
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document.write( "here's a reference to the formula.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/geometry/herons-formula.html\" target = \"_blank\">https://www.mathsisfun.com/geometry/herons-formula.html</a><BR>\n" );
document.write( "there are many other, more detailed references, that can be found on the web.\r<BR>\n" );
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document.write( "if two scalene triangle, or any triangle for that matter, are similar, with their sides in the ratio of 3 to 1, you can use this formula to prove that the areas would be in the ratio of 9 to 1.\r<BR>\n" );
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document.write( "here's how i did it.\r<BR>\n" );
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document.write( "you are given that heron's formula states that:\r<BR>\n" );
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document.write( "A = sqrt(s * (s-a) * (s-b) * (s-c))\r<BR>\n" );
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document.write( "A is equal to the area of the triangle.<BR>\n" );
document.write( "s is equal to (a + b + c) / 2.\r<BR>\n" );
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document.write( "if the two triangle are similar, and their sides are in a ratio of 3 to 1, then every side of the larger triangle is 3 * the size of the smaller triangle.\r<BR>\n" );
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document.write( "heron's formula for the larger triangle becomes:\r<BR>\n" );
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document.write( "A = sqrt(3*s * (3*s - 3*a) * (3*s - 3*b) * (3*s - 3*c))\r<BR>\n" );
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document.write( "this formula can be simplified to:\r<BR>\n" );
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document.write( "A = sqrt(3 * s * 3 * (s - a) * 3 * (s - b) * 3 * s-c))\r<BR>\n" );
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document.write( "this formula can be further simplified to:\r<BR>\n" );
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document.write( "A = sqrt(3 * 3 * 3 * 3 * s * (s - a) * (s - b) * (s - c))\r<BR>\n" );
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document.write( "since sqrt(a * b) = sqrt(a) * sqrt(b), this formula can be further simplified to:\r<BR>\n" );
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document.write( "A = sqrt(3*3) * sqrt(3*3) * sqrt(s * (s-a) * (s-b) * s-c))\r<BR>\n" );
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document.write( "since sqrt(3*3) = sqrt(9) which is equal to 3, then this formula can be further simplified to:\r<BR>\n" );
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document.write( "A = 3 * 3 * sqrt(s * (s-a) * (s-b) * s-c)) which can be further simplified to:\r<BR>\n" );
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document.write( "A = 9 * sqrt(s * (s-a) * (s-b) * s-c)).\r<BR>\n" );
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document.write( "you might also ask where i got 3 * s from.\r<BR>\n" );
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document.write( "you are given that s = (a + b + c) / 2\r<BR>\n" );
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document.write( "since the larger triangle has sides 3 times the length of the sides of the original triangle, then you get:\r<BR>\n" );
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document.write( "s1 = (3 * a + 3 * b + 3 * c) / 2 which can be simplified to:\r<BR>\n" );
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document.write( "s1 = 3 * (a + b + c) / 2.\r<BR>\n" );
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document.write( "since s = (a + b + c) / 2, it follow that:\r<BR>\n" );
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document.write( "s1 = 3 * s because s1 is equal to 3 * (a + b + c) / 2.\r<BR>\n" );
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document.write( "since s1 = 3 * s, then that's the reason why 3 * s was used in the formula for the area of the larger triangle.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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