document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=548532>Question 548532</A>: <I><SPAN STYLE=\"word-break: break-all;\"> an isosceles triangle is inscribed in the cirle x2+y2-6x-8x=0 with vertex at the origin and one of the equal side along the axis of x.equation of the other side through the origin is  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #357125 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=357125\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume that you meant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2-6x-8x=0 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2-6x-8x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Being contrary, I tried to solve it as an algebra problem, pretending not to know geometry, but it gets complicated. It's easy to make arithmetic mistakes, and it's painful to ignore the geometric meaning of all the intermediate results.<BR>\n" );
document.write( "I'll try the other way, using all geometry, and as little algebra as possible.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2-6x-8x=0 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2-6x-8x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the equation for a circle.<BR>\n" );
document.write( "I will not use algebra to \"complete the squares\" to find the center and radius. This circle is too easy to need that.<BR>\n" );
document.write( "We can see that the point (0,0) satisfies the equation for the circle, and the problem tells us that point is the vertex of the isosceles triangle.<BR>\n" );
document.write( "We can also see that points (6,0) and (0,8) are part of the circle.<BR>\n" );
document.write( "Those 3 points, along with the center of the circle make two isosceles triangles, with bases on the axes, and vertex at the center of their circle. Their altitudes are on lines parallel to the axes, bisecting the bases (perpendicular bisectors). Those lines are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%280%2B6%29%2F2=3 ALIGN=MIDDLE  ALT=\"x=%280%2B6%29%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%280%2B8%29%2F2=4 ALIGN=MIDDLE  ALT=\"y=%280%2B8%29%2F2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. They intersect at the center of the circle: the point (3,4).<BR>\n" );
document.write( "I know what the radius is, but I don't care, because I don't need it.<BR>\n" );
document.write( "About the isosceles triangle the problem refers to, geometry tells us that its altitude goes through the center of the circle. The side on the x-axis and that altitude make an angle whose tangent we can calculate from the coordinates of the center of the circle<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan+%28alpha%29=4%2F3 ALIGN=MIDDLE  ALT=\"tan+%28alpha%29=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The side on the x-axis and the congruent side of the isosceles triangle make an angle that measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aalpha ALIGN=MIDDLE  ALT=\"2%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > and its tangent is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282%2Aalpha%29=2%2Atan+%28alpha%29%2F%281-%28tan+%28alpha%29%29%5E2%29=2%2A%284%2F3%29%2F%281-%284%2F3%29%5E2%29=%288%2F3%29%2F%281-%2816%2F9%29%29=%288%2F3%29%2F%28-7%2F9%29=-24%2F7 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is the slope of the line we were looking for. Since that lines goes through the origin, its equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-24%2F7%29%2Ax ALIGN=MIDDLE  ALT=\"y=%28-24%2F7%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or maybe the equivalent equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24x%2B7y=0 ALIGN=MIDDLE  ALT=\"24x%2B7y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be preferred.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C10%2C-4%2C10%2C%0D%0Agrid%281%29%2C%0D%0Aline%28-7.5%2C-10%2C7.5%2C10%29%2C%0D%0Acircle%283%2C4%2C5%29%2C%0D%0Aline%286%2C0%2C3%2C4%29%2Cline%280%2C8%2C3%2C4%29%2Cline%283%2C0%2C3%2C4%29%2Cline%280%2C4%2C3%2C4%29%2C%0D%0Ared%28line%2810%2C-240%2F7%2C-10%2C240%2F7%29%29%2C%0D%0Ablue%28line%28-42%2F25%2C144%2F25%2C6%2C0%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The line asked for is the red line. I drew the base of the problem triangle in blue. </SPAN>\n" );
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