document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210147>Question 1210147</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In \triangle ABC, we have AB = AC = 4 and \angle BAC = 45^\circ.  If M is the midpoint of BC, then find AM^2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851369 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=851369\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $\triangle ABC$ be a triangle with $AB = AC = 4$ and $\angle BAC = 45^\circ$. Let $M$ be the midpoint of $BC$. We want to find $AM^2$.\r<BR>\n" );
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document.write( "We can use the Law of Cosines to find $BC$.<BR>\n" );
document.write( "$$BC^2 = AB^2 + AC^2 - 2(AB)(AC)\cos(\angle BAC)$$<BR>\n" );
document.write( "$$BC^2 = 4^2 + 4^2 - 2(4)(4)\cos(45^\circ)$$<BR>\n" );
document.write( "$$BC^2 = 16 + 16 - 32\left(\frac{\sqrt{2}}{2}\right)$$<BR>\n" );
document.write( "$$BC^2 = 32 - 16\sqrt{2}$$<BR>\n" );
document.write( "$$BC = \sqrt{32 - 16\sqrt{2}} = 4\sqrt{2 - \sqrt{2}}$$\r<BR>\n" );
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document.write( "Since $M$ is the midpoint of $BC$, we can use Apollonius' Theorem:<BR>\n" );
document.write( "$$AB^2 + AC^2 = 2(AM^2 + BM^2)$$<BR>\n" );
document.write( "$$4^2 + 4^2 = 2(AM^2 + BM^2)$$<BR>\n" );
document.write( "$$32 = 2(AM^2 + BM^2)$$<BR>\n" );
document.write( "$$16 = AM^2 + BM^2$$\r<BR>\n" );
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document.write( "We know $BM = \frac{1}{2}BC$, so $BM^2 = \frac{1}{4}BC^2 = \frac{1}{4}(32 - 16\sqrt{2}) = 8 - 4\sqrt{2}$.\r<BR>\n" );
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document.write( "Substituting $BM^2$ into the equation from Apollonius' Theorem:<BR>\n" );
document.write( "$$16 = AM^2 + (8 - 4\sqrt{2})$$<BR>\n" );
document.write( "$$AM^2 = 16 - (8 - 4\sqrt{2})$$<BR>\n" );
document.write( "$$AM^2 = 8 + 4\sqrt{2}$$\r<BR>\n" );
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document.write( "Thus, $AM^2 = 8 + 4\sqrt{2}$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{8 + 4 \sqrt{2}}$<BR>\n" );
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