document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209520>Question 1209520</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (39) Square SQUR has sides of length x. If triangle SQE is equilateral, find the area of triangle QAU.<BR>\n" );
document.write( "Link to diagram: https://ibb.co/C58rZ09R </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849138 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849138\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the area of triangle QAU:\r<BR>\n" );
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document.write( "**1. Visualize the problem:**\r<BR>\n" );
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document.write( "It's helpful to draw a diagram.  You have a square SQUR with side length *x*.  Triangle SQE is equilateral, meaning all its sides are also of length *x*.  Point A is likely somewhere related to the geometry of the square and triangle.  We need more information about the location of point A to solve the problem.  Let's assume A is the intersection of lines RE and QU.\r<BR>\n" );
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document.write( "**2. Find the coordinates of the points (assuming a coordinate system):**\r<BR>\n" );
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document.write( "Let's place the square in a coordinate system.  Put S at the origin (0,0), Q at (x,0), U at (x,x), and R at (0,x).\r<BR>\n" );
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document.write( "Since SQE is equilateral, point E will be at (x/2, (x√3)/2).\r<BR>\n" );
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document.write( "**3. Find the equations of lines RE and QU:**\r<BR>\n" );
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document.write( "* **Line RE:**  We have points R(0,x) and E(x/2, (x√3)/2).  The slope of RE is:<BR>\n" );
document.write( "   m_RE = [(x√3)/2 - x] / [x/2 - 0] = (√3 - 2)x / x = √3 - 2\r<BR>\n" );
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document.write( "   The equation of line RE (using point-slope form) is:<BR>\n" );
document.write( "   y - x = (√3 - 2)(x - 0)<BR>\n" );
document.write( "   y = (√3 - 2)x + x<BR>\n" );
document.write( "   y = (√3 - 1)x\r<BR>\n" );
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document.write( "* **Line QU:** We have points Q(x,0) and U(x,x).  The slope of QU is undefined since it is a vertical line. The equation of the line is x=x.\r<BR>\n" );
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document.write( "**4. Find the coordinates of point A (the intersection of RE and QU):**\r<BR>\n" );
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document.write( "Since QU is the vertical line x = x, the x-coordinate of point A is *x*.  Substitute *x* for x in the equation of line RE to find the y-coordinate:<BR>\n" );
document.write( "y = (√3 - 1)x\r<BR>\n" );
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document.write( "So, point A is at (x, (√3 - 1)x).\r<BR>\n" );
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document.write( "**5. Find the area of triangle QAU:**\r<BR>\n" );
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document.write( "The base of triangle QAU is QU, which has length *x*.  The height of the triangle is the perpendicular distance from A to the line QU. Since QU is a vertical line, the height is the horizontal distance from x to x, which is 0. The height of the triangle is the difference in the y-coordinates of A and Q. Thus the height is (√3 - 1)x - 0 = (√3 - 1)x.\r<BR>\n" );
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document.write( "Area of triangle QAU = (1/2) * base * height<BR>\n" );
document.write( "Area = (1/2) * x * (√3 - 1)x<BR>\n" );
document.write( "Area = (x²(√3 - 1))/2\r<BR>\n" );
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document.write( "Therefore, the area of triangle QAU is (x²(√3 - 1))/2.<BR>\n" );
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