document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=936586>Question 936586</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Explain the different triangle and similarity relationships of the Geometric Mean. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #570085 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=570085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you start with right triangle ABC where angle ABC is the right angle.\r<BR>\n" );
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document.write( "you then drop a perpendicular from B to D.\r<BR>\n" );
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document.write( "this creates two other triangles\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "right triangle ADB where angle ADB is the right angle.\r<BR>\n" );
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document.write( "right triangle BDC where angle BDC is the right angle.\r<BR>\n" );
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document.write( "in triangle ABC, if you assume that angle BCA is equal to x, then angle BAC must be equal to 90 - x because the acute angles in a right triangle are complementary.\r<BR>\n" );
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document.write( "in right triangle ADB, since angle DAB is the same angle as angle BAC in right triangle ABC, then angle DAB is equal to 90 - x and angle DBA is equal to x.\r<BR>\n" );
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document.write( "in right triangle BDC, since angle DCB is the same angle as angle ACB in triangle ABC, then angle DCB is equal to x and angle DBC is equal to 90 - x.\r<BR>\n" );
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document.write( "what this all means is that:\r<BR>\n" );
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document.write( "triangle ABC is similar to triangle ADB which is similar to triangle BDC\r<BR>\n" );
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document.write( "while difficult to see in diagram 1, this is eaier to see in diagrams 2 and 3 and 4 where the similar triangle have been oriented so that their corresponding sides and corresponding angles are line up the same way.\r<BR>\n" );
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document.write( "since these triangle are similar, then their corresponding sides are proportional.\r<BR>\n" );
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document.write( "that similarity relationship is what creates the geometric mean theorem.\r<BR>\n" );
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document.write( "what the geometric mean theorem states is based on diagram 1 in the picture.\r<BR>\n" );
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document.write( "first rule is that AD / AB = AB / AC\r<BR>\n" );
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document.write( "this rule is based on the similarity relationships in diagrams 2 and 3.\r<BR>\n" );
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document.write( "AB in diagram 2 and AD in diagram 3 are corresponding sides of similar triangles.\r<BR>\n" );
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document.write( "AC in diagram 2 and AB in diagram 3 are corresponding sides of similar triangles.\r<BR>\n" );
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document.write( "since the triangle are similar, the corresponding sides are in proportion and you get:\r<BR>\n" );
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document.write( "AD in diagram 3 is to AB in diagram 2 as AB in diagram 3 is to AC in diagram 2.\r<BR>\n" );
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document.write( "algebraically, this is shown as:\r<BR>\n" );
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document.write( "AD / AB = AB / AC\r<BR>\n" );
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document.write( "second rule is that DC / BC = BC / AC\r<BR>\n" );
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document.write( "this rule is based on the similarity relationships in diagrams 2 and 4.\r<BR>\n" );
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document.write( "BC in diagram 2 and DC in diagram 4 are corresponding sides of similar triangles.\r<BR>\n" );
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document.write( "AC in diagram 2 and BC in diagram 4 are corresponding sides of similar triangles.\r<BR>\n" );
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document.write( "since the triangle are similar, the corresponding sides are in proportion and you get:\r<BR>\n" );
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document.write( "DC in diagram 4 is to BC in diagram 2 as BC in diagram 4 is to AC in diagram 2.\r<BR>\n" );
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document.write( "algebraically, this is shown as:\r<BR>\n" );
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document.write( "DC / BC = BC / AC\r<BR>\n" );
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document.write( "the third rule is that AD / BD = BD / CD\r<BR>\n" );
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document.write( "this rule is based on the similarity relationships in diagrams 3 and 4.\r<BR>\n" );
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document.write( "DB in diagram 3 and DC in diagram 4 are corresponding sides of similar triangles.\r<BR>\n" );
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document.write( "DA in diagram 3 and DB in diagram 4 are corresponding sides of similar triangles.\r<BR>\n" );
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document.write( "since the triangle are similar, the corresponding sides are in proportion and you get:\r<BR>\n" );
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document.write( "DA in diagram 3 is to DB in diagram 4 as DB in diagram 3 is to <BR>\n" );
document.write( "DC in diagram 4\r<BR>\n" );
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document.write( "algebraically, this is shown as:\r<BR>\n" );
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document.write( "DA / DB = DB / DC\r<BR>\n" );
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document.write( "since DA is the same as AD and DB is the same as BD and DC is the same as CD, the ration can also be shown as:\r<BR>\n" );
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document.write( "AD / BD = BD / DC\r<BR>\n" );
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document.write( "the diagrams are shown below:\r<BR>\n" );
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document.write( "the rules and how they are visualized in diagram 1 are shown below:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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