document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027673>Question 1027673</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The figure shows Traingle ABC inscribed in a circle with centre O. If Angle CBO is twice of Angle CAO and angle BAO is one and a half times of angle CBO, find angle CAO.\r<BR>\n" );
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document.write( "This is the image of the corresponding question.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642911 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=642911\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> there are a couple of ways to solve this.\r<BR>\n" );
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document.write( "one finds the central angle of each triangle formed.\r<BR>\n" );
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document.write( "the other finds the number of degrees of the intercepted arcs on the circle.\r<BR>\n" );
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document.write( "the sum of the central angles of the circle is equal to 360 degrees.\r<BR>\n" );
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document.write( "an inscribed angle of the circle is equal to 1/2 the degrees of the intercepted arc on the circle.\r<BR>\n" );
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document.write( "the sum of the degrees of the intercepted arcs on the circle is equal to 360 degrees.\r<BR>\n" );
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document.write( "both these methods yield the same answer.\r<BR>\n" );
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document.write( "that answer is that angle CAO = 15 degrees and angle CBO = 30 degrees and angle BAO = 45 degrees.\r<BR>\n" );
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document.write( "you are given that angle CBO is twice angle CAO.\r<BR>\n" );
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document.write( "if we let x = angle CAO, then angle CBO must be equal to 2x.\r<BR>\n" );
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document.write( "you are also given that BAO is equal to 1.5 times angle CBO.\r<BR>\n" );
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document.write( "since angle CBO is equal to 2x, this mean that angle BAO must be equal to 3x.\r<BR>\n" );
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document.write( "you have:\r<BR>\n" );
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document.write( "angle CAO = x<BR>\n" );
document.write( "angle CBO = 2x<BR>\n" );
document.write( "angle BAO = 3x.\r<BR>\n" );
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document.write( "these angles are the base angles of isosceles triangles.\r<BR>\n" );
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document.write( "the isosceles triangles are triangle CAO, CBO, and BAO.\r<BR>\n" );
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document.write( "since the base angles of isosceles triangles are equal, this means that:\r<BR>\n" );
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document.write( "angle ACO = x<BR>\n" );
document.write( "angle BCO = 2x<BR>\n" );
document.write( "angle ABO = 3x\r<BR>\n" );
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document.write( "since the sum of the angles of any triangle is equal to 180 degrees, this means that:\r<BR>\n" );
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document.write( "angle COA = 180 - 2x<BR>\n" );
document.write( "angle COB = 180 - 4x<BR>\n" );
document.write( "angle BOA = 180 - 6x\r<BR>\n" );
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document.write( "the sum of these 3 central angles of the circle must be equal to 360 degrees.\r<BR>\n" );
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document.write( "this means that 540 - 12x = 360 degrees.\r<BR>\n" );
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document.write( "subtract 360 from both sides of this equation and add 12x to both sides of this equation to get 540 - 360 = 12x\r<BR>\n" );
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document.write( "solve for x to get x = (540 - 360) / 12 = 15 degrees.\r<BR>\n" );
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document.write( "this means that:\r<BR>\n" );
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document.write( "angle CAO = x = 15 degrees.<BR>\n" );
document.write( "angle CBO = 2x = 30 degrees.<BR>\n" );
document.write( "angle BAO = 3x = 45 degrees.\r<BR>\n" );
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document.write( "the central angles become:\r<BR>\n" );
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document.write( "angle COA = 180 - 2x = 180 - 30 = 150<BR>\n" );
document.write( "angle COB = 180 - 4x = 180 - 60 = 120<BR>\n" );
document.write( "angle BOA = 180 - 6x = 180 - 90 = 90\r<BR>\n" );
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document.write( "the sum of these central angles is equal to 360 degrees, as it should be.\r<BR>\n" );
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document.write( "you could also have solved this by extending the radii of the circle to form diameters and then looking at the intercepted arcs.\r<BR>\n" );
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document.write( "these intercepted arcs are equal to twice the inscribed angle.\r<BR>\n" );
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document.write( "when you do that you get 24 * x = 360 degrees.\r<BR>\n" );
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document.write( "when you solve for x, you get x = 360 / 24 = 15 degrees.\r<BR>\n" );
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document.write( "it's the same answer.\r<BR>\n" );
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document.write( "i drew the both diagrams for you to see.\r<BR>\n" );
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document.write( "the first is using the central angle to find.\r<BR>\n" );
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document.write( "the second is using the intercepted arc angle to find x.\r<BR>\n" );
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