document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210474>Question 1210474</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In rectangle ABCD, corner A is folded over crease DE to point F on BC.  Find BC. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852834 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=852834\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem as stated is **unsolvable** because a necessary dimension is missing. The length of the side $BC$ (which is equal to $AD$) cannot be uniquely determined without knowing the length of the adjacent side **$CD$** (or $AB$).\r<BR>\n" );
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document.write( "Here is the general geometric relationship that must hold in this type of folding problem, which shows the dependency:\r<BR>\n" );
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document.write( "Let:<BR>\n" );
document.write( "* $BC = AD = l$ (The length you are trying to find)<BR>\n" );
document.write( "* $CD = w$ (The missing width)\r<BR>\n" );
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document.write( "### 1. Relationships from the Fold\r<BR>\n" );
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document.write( "When corner $A$ is folded over the crease $DE$ to point $F$ on $BC$, two triangles, $\triangle ADE$ and $\triangle FDE$, are congruent. This gives us the following equivalences:\r<BR>\n" );
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document.write( "* The hypotenuse of the fold: $DF = AD = l$<BR>\n" );
document.write( "* The fold sides: $AE = FE$<BR>\n" );
document.write( "* $\angle DFE = \angle DAE = 90^\circ$\r<BR>\n" );
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document.write( "### 2. Using the Pythagorean Theorem\r<BR>\n" );
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document.write( "We can set up two equations using the Pythagorean theorem in the right-angled triangles $\triangle FDC$ and $\triangle FCE$:\r<BR>\n" );
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document.write( "* **In $\triangle FDC$ (Right-angled at C):**<BR>\n" );
document.write( "    $$DF^2 = DC^2 + CF^2$$<BR>\n" );
document.write( "    $$l^2 = w^2 + CF^2 \quad \rightarrow \quad CF = \sqrt{l^2 - w^2}$$\r<BR>\n" );
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document.write( "* **In $\triangle FCE$ (Right-angled at C):**<BR>\n" );
document.write( "    $$FE^2 = CE^2 + CF^2$$<BR>\n" );
document.write( "    Since $FE = AE$, we substitute $AE$ and the expression for $CF^2$:<BR>\n" );
document.write( "    $$AE^2 = CE^2 + (l^2 - w^2)$$\r<BR>\n" );
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document.write( "### 3. Using the Side $CD$\r<BR>\n" );
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document.write( "The line segments $DE$ and $CE$ make up the side $CD$: $CD = DE + CE$.\r<BR>\n" );
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document.write( "* **In $\triangle ADE$ (Right-angled at D):**<BR>\n" );
document.write( "    $$AE^2 = AD^2 + DE^2$$<BR>\n" );
document.write( "    $$AE^2 = l^2 + DE^2$$\r<BR>\n" );
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document.write( "### Conclusion of Dependency\r<BR>\n" );
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document.write( "By setting the two expressions for $AE^2$ equal to each other:\r<BR>\n" );
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document.write( "$$CE^2 + l^2 - w^2 = l^2 + DE^2$$<BR>\n" );
document.write( "$$\rightarrow CE^2 = DE^2 + w^2$$\r<BR>\n" );
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document.write( "Since $w$ and $l$ are not given, $BC$ (which is $l$) cannot be uniquely determined.\r<BR>\n" );
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document.write( "**To solve the problem, you must know the length of $CD$ (or a relationship between $l$ and $w$).** </SPAN>\n" );
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