document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=70800>Question 70800</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If triangle XYZ is isosceles and LINE XY = YZ, find the value(s) of \"A\" if X(3,4), Y(4,-1)AND Z(A,-2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #50597 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=50597\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem use the fact that the distance XY must equal the distance YZ.<BR>\n" );
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document.write( "The distance formula says:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28%28y2+-+y1%29%5E2+%2B+%28x2+-+x1%29%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28%28y2+-+y1%29%5E2+%2B+%28x2+-+x1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So to find the distance XY we can use the coordinates at point X ... (3,4) for x1 and y1<BR>\n" );
document.write( "and the coordinates at point Y ... (4, -1) for x2 and y2.  If we do, then we recognize<BR>\n" );
document.write( "that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1+=+3 ALIGN=MIDDLE  ALT=\"x1+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1+=+4 ALIGN=MIDDLE  ALT=\"y1+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We also recognize that that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2+=+4 ALIGN=MIDDLE  ALT=\"x2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2+=-1 ALIGN=MIDDLE  ALT=\"y2+=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Substitute these values into the distance formula to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d+=+sqrt%28%28-1-4%29%5E2+%2B+%284-3%29%5E2%29 ALIGN=MIDDLE  ALT=\"+d+=+sqrt%28%28-1-4%29%5E2+%2B+%284-3%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28%28-5%29%5E2+%2B+%281%29%5E2%29+=+sqrt%2825+%2B+1%29+=+sqrt%2826%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28%28-5%29%5E2+%2B+%281%29%5E2%29+=+sqrt%2825+%2B+1%29+=+sqrt%2826%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So now we know that side YZ must also have a length of sqrt(26).  We will use the distance<BR>\n" );
document.write( "formula again.  This time let's again use the coordinates at point Y ... (4,-1) for x2 and y2.<BR>\n" );
document.write( "And we will use the coordinates at Z ... (A, -2) for x1 and y1.  This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2+=+4 ALIGN=MIDDLE  ALT=\"x2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2+=+-1 ALIGN=MIDDLE  ALT=\"y2+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. It also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1+=+A ALIGN=MIDDLE  ALT=\"x1+=+A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1+=+-2 ALIGN=MIDDLE  ALT=\"y1+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute these values<BR>\n" );
document.write( "into the distance formula:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28%28-1+-+%28-2%29%29%5E2+%2B+%284+-+A%29%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28%28-1+-+%28-2%29%29%5E2+%2B+%284+-+A%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Simplifying terms in the parentheses leads to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28%281%29%5E2+%2B+%284-A%29%5E2%29+=+sqrt%281+%2B+%284-A%29%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28%281%29%5E2+%2B+%284-A%29%5E2%29+=+sqrt%281+%2B+%284-A%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But we know that d has to equal sqrt(26). Look carefully at it. You should be able to<BR>\n" );
document.write( "see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-A%29%5E2 ALIGN=MIDDLE  ALT=\"%284-A%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > has to be + 25 so that it can add to 1 under the radical and make<BR>\n" );
document.write( "the radical be sqrt(26)<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-A%29%5E2 ALIGN=MIDDLE  ALT=\"%284-A%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be 25, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-A%29 ALIGN=MIDDLE  ALT=\"%284-A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must either be 5 or -5 so that it squares out to +25.<BR>\n" );
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document.write( "For the first case we have 4-A = +5. Subtracting 4 from both sides tells us that -A = +1.<BR>\n" );
document.write( "Then to solve for +A multiply both sides of this equation by -1 to get A = -1.<BR>\n" );
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document.write( "For the second case we have 4-A = -5.  Subtract 4 from both sides to get -A = -9.<BR>\n" );
document.write( "Solve for +A by multiplying both sides of the equation by -1 to get A = 9<BR>\n" );
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document.write( "In summary, the answer is that A either equals 9 or -1.<BR>\n" );
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document.write( "Hope this helps you to see your way through the problem.  It might help to sketch the<BR>\n" );
document.write( "points on a graph so you can see why either answer for A produces an isosceles triangle <BR>\n" );
document.write( "that satisfies the problem. </SPAN>\n" );
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