document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210476>Question 1210476</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram, ABCD is a square with side length 2, and BEF is an equilateral triangle with side length 3.  Find DF. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852832 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=852832\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The given information allows us to determine the exact angles between the sides of the square and the triangle:\r<BR>\n" );
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document.write( "* **Square ABCD** side length: $AB = BC = 2$.<BR>\n" );
document.write( "* **Equilateral Triangle BEF** side length: $BE = BF = 3$.<BR>\n" );
document.write( "* **Connecting Angle:** $\angle ABE = 30^\circ$.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 1. Distance between A and F ($\mathbf{AF}$)\r<BR>\n" );
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document.write( "The distance $AF$ can be found by analyzing $\triangle ABF$.\r<BR>\n" );
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document.write( "### Finding Angle $\angle ABF$\r<BR>\n" );
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document.write( "The angle $\angle ABF$ is the sum of the angle $\angle ABE$ and the angle $\angle EBF$, assuming the triangle is positioned sequentially adjacent to $AB$:<BR>\n" );
document.write( "$$\angle ABF = \angle ABE + \angle EBF$$<BR>\n" );
document.write( "* $\angle ABE = 30^\circ$ (Given)<BR>\n" );
document.write( "* $\angle EBF = 60^\circ$ (Angle of an equilateral triangle)\r<BR>\n" );
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document.write( "$$\angle ABF = 30^\circ + 60^\circ = 90^\circ$$\r<BR>\n" );
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document.write( "Since $\angle ABF = 90^\circ$, $\triangle ABF$ is a **right-angled triangle**.\r<BR>\n" );
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document.write( "### Using the Pythagorean Theorem\r<BR>\n" );
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document.write( "We use the Pythagorean theorem: $AF^2 = AB^2 + BF^2$.\r<BR>\n" );
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document.write( "$$AF^2 = 2^2 + 3^2$$<BR>\n" );
document.write( "$$AF^2 = 4 + 9$$<BR>\n" );
document.write( "$$AF^2 = 13$$<BR>\n" );
document.write( "$$AF = \mathbf{\sqrt{13}}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Distance between C and F ($\mathbf{CF}$)\r<BR>\n" );
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document.write( "The distance $CF$ can be found by analyzing $\triangle CBF$.\r<BR>\n" );
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document.write( "### Finding Angle $\angle CBF$\r<BR>\n" );
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document.write( "First, we find $\angle EBC$, assuming $E$ is *inside* the $\angle ABC$ corner:<BR>\n" );
document.write( "$$\angle EBC = \angle ABC - \angle ABE$$<BR>\n" );
document.write( "* $\angle ABC = 90^\circ$ (Angle of a square)<BR>\n" );
document.write( "* $\angle ABE = 30^\circ$ (Given)\r<BR>\n" );
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document.write( "$$\angle EBC = 90^\circ - 30^\circ = 60^\circ$$\r<BR>\n" );
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document.write( "For $\angle CBF$, we assume the points $C, E, F$ are positioned such that $\angle CBF = \angle EBC + \angle EBF$ (a typical non-collinear configuration).<BR>\n" );
document.write( "$$\angle CBF = \angle EBC + \angle EBF$$<BR>\n" );
document.write( "$$\angle CBF = 60^\circ + 60^\circ = 120^\circ$$\r<BR>\n" );
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document.write( "### Using the Law of Cosines\r<BR>\n" );
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document.write( "We use the Law of Cosines: $CF^2 = CB^2 + BF^2 - 2(CB)(BF) \cos(\angle CBF)$.\r<BR>\n" );
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document.write( "$$CF^2 = 2^2 + 3^2 - 2(2)(3) \cos(120^\circ)$$<BR>\n" );
document.write( "$$CF^2 = 4 + 9 - 12 \left(-\frac{1}{2}\right)$$<BR>\n" );
document.write( "$$CF^2 = 13 - (-6)$$<BR>\n" );
document.write( "$$CF^2 = 19$$<BR>\n" );
document.write( "$$CF = \mathbf{\sqrt{19}}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "Given that $\angle ABF=90^\circ$ simplifies the calculation significantly, it is highly likely that the intended question was to find $AF$.\r<BR>\n" );
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document.write( "The answer is:<BR>\n" );
document.write( "* The distance **$AF$** is **$\sqrt{13}$**.<BR>\n" );
document.write( "* The distance **$CF$** is **$\sqrt{19}$**. </SPAN>\n" );
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