document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175969>Question 1175969</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Equilateral triangle XYZ is insicribed in equilateral triangle ABC as shown. What is the ratio of the area of triangle XYZ to the area of triangle ABC.\r<BR>\n" );
document.write( "\n" );
document.write( "Image - https://media.discordapp.net/attachments/802751414874537995/815059748683513877/OHRfG3q16hmGvCHR81U6QM1nwb84dF8berXqGYa8IdHzVTpAzWfBvzhdaAHswAAAA5JREFU0Xxt6teoZhr4D065Y756MQvRAAAAA.png </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #801783 by </B><A HREF=/tutors/aboutme.mpl?userid=Solver92311><B>Solver92311(821)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Solver92311><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=Solver92311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=801783\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Solver92311&illustration_name=Diagram_1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Angle B is a 60-degree angle by virtue of the fact that it is a vertex of an equilateral triangle and therefore is the vertex of an equiangular triangle whose measure must be one-third of 180 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since angle BYZ is right, angle BZY must be 30 degrees (180 - 90 - 60 = 30)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore triangle BYZ is a 30-60-90 right triangle, and by similar logic and the fact that triangle XYZ is equilateral as a given, triangles BYZ, AXY, and CZX are all congruent.  Congruency demands that the three triangles mentioned must be equal in area.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let the measure of segment BZ be 1 unit.  Then by the properties of a 30-60-90 right triangle, segments BY, CZ, and AX must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2}\"> unit, and YZ, ZX, and XY must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\sqrt{3}}{2}\"> units.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since segment BC is the sum of segments BZ and ZC, the measure of BC must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{3}{2}\"> units, so if you drop a perpendicular to AC from vertex B at M, triangle BCM is both a 30-60-90 right triangle and is one-half of the area of triangle ABC.  By the properties of a 30-60-90 right triangle, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large MC\ =\ \frac{3}{4}\"> units and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large BM\ =\ \frac{3\sqrt{3}}{4}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Area of triangle ABC: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\(\frac{BM * MC}{2}\)\ =\ 2\(\frac{\(\frac{3 \sqrt{3}}{4}\)\(\frac{3}{4}\)}{2}\)\ =\ \frac{9 \sqrt{3}}{16}  \"> units².\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Area of triangle BYZ: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\(\frac{1}{2}\)\(\frac{\sqrt{3}}{2}\)}{2}\ =\ \frac{\sqrt{3}}{8}\">.  And the sum of the areas of triangles BYZ, AXY, and CZX is 3 times the area of BYZ, to wit: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{3 \sqrt{3}}{8}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The area of triangle XYZ is the difference between the area of triangle ABC and the sum of the areas of the three triangles just calculated, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{9 \sqrt{3}}{16}\ -\ \frac{3 \sqrt{3}}{8}\ =\ \frac{3 \sqrt{3}}{16}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then the ratio of the area of XYZ to ABC is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\frac{3 \sqrt{3}}{16}}{\frac{9 \sqrt{3}}{16}}\ =\ \frac{1}{3}\">\r<BR>\n" );
document.write( "\n" );
document.write( "																<BR>\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Solver92311&illustration_name=darwinfish.jpg\">\r<BR>\n" );
document.write( "\n" );
document.write( "From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> <BR>\n" );
document.write( "I > Ø<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );