document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1101251>Question 1101251</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many triangles XYZ can be formed with angle x =30,x=3 and y=8<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #716112 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=716112\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The other tutor didn't read the problem correctly; the given numbers are not the lengths of the three sides of the triangle.<br><BR>\n" );
document.write( "(By chance, his answer is right; but not for the right reason.)<br><BR>\n" );
document.write( "This problem is one where we need to determine whether there are 0, 1, or 2 triangles that can be formed, given one angle and two sides, one of which is opposite the given angle.<br><BR>\n" );
document.write( "I always draw the picture for this type of problem the same way: the side with unknown length is horizontal, with the given angle at the left, opening upward.<br><BR>\n" );
document.write( "So with the given information being angle X = 30 degrees, side y=8 and side x=3, I have XY horizontal, with X at the left, and segment XZ pointing upward making angle YXZ 30 degrees.<br><BR>\n" );
document.write( "Segment XZ is y=8; and I'm wanting segment ZY to be x=3.<br><BR>\n" );
document.write( "But if triangle YXZ were a right triangle, with angle X being 30 degrees and side XZ having length 8, segment YZ would have length 4.<br><BR>\n" );
document.write( "But the length of segment YZ is only 3; it is not long enough to reach side XY.<br><BR>\n" );
document.write( "So no triangles can be formed with the given measurements.<br><BR>\n" );
document.write( "Formally, the shortest distance from point Z to line XY is the length of the segment ZY that is perpendicular to XY, making XYZ a right triangle.<BR>\n" );
document.write( "If XYZ is a right triangle, then ZY/XZ is the sine of 30 degrees, which is 1/2.<br><BR>\n" );
document.write( "That means there would be exactly one triangle that could be formed if YZ were exactly 4 (half the length of XZ); if YZ is shorter than 4, then no triangles can be formed (YZ is not long enough to reach XY); and if YZ is longer than 4, then two different triangles can be formed, because YZ could reach XZ in two different places.<BR>\n" );
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document.write( "Note that it could be possible for a similar problem to say that YZ was longer than XZ; but if you draw the picture of that case, it really doesn't make any sense. </SPAN>\n" );
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