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<div dir="ltr" data-setdir="false">SSS has a unique solution. AAA has an unlimited number of solutions.<br></div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Math can be considered in several ways. <br></div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Most schools present a problem and one way to solve that problem. Then another problem. At the end of the course, students are expected to have memorized ways to solve a bunch of problems. It is not common to teach so that the student can solve a different problem on her own.<br></div><div dir="ltr" data-setdir="false"><br></div>Addition is useful by itself. Skill in addition is needed to do multiplication. And multiplication can be used for many things, the cost of several items, the area of a square.<div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false"><div><div dir="ltr" data-setdir="false">Triangle problems can be solved with trig. They can also be solved by carefully drawing lines and angles.</div><div dir="ltr" data-setdir="false"><br></div></div><div>There are many situations where trig can be useful in solving problems that didn't have anything to do with triangles. I'm still trying to find anything related to electricity that can be measured with a protractor. </div></div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false"><br></div></div><div id="ydp9f2ddf21yahoo_quoted_8238249916" class="ydp9f2ddf21yahoo_quoted">
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On Saturday, April 25, 2020, 4:10:40 AM PDT, Michael Paoli <michael.paoli@cal.berkeley.edu> wrote:
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<div>> From: Texx <<a shape="rect" href="mailto:texxgadget@gmail.com" rel="nofollow" target="_blank">texxgadget@gmail.com</a>><br clear="none">> Subject: Re: [conspire] Password permutations (was: Correction)<br clear="none">> Date: Thu, 16 Apr 2020 14:18:23 -0700<br clear="none"><br clear="none">> If you have 2 sides and an angle or 2 angles and a side, and you know WHICH<br clear="none">> sides & angles you have, you can reconstruct the triangle<br clear="none"><br clear="none">No, no, no. Didn't you get taught in trig, there's SAS and ASA,<br clear="none">but you don't get ASS!<br clear="none"><br clear="none">So, just think about it ...<br clear="none"><br clear="none">SAS ... you have a fixed length side, you put in an angle,<br clear="none">put put your second side of fixed length. Only one way left<br clear="none">to do a triangle - connect those two free ends, and you've got a<br clear="none">triangle - as long as your angle isn't 0 or 180 degrees, or any segments<br clear="none">of length 0, and you're good (unless you want to consider degenerate<br clear="none">triangles).<br clear="none"><br clear="none">ASA - you lay down your fixed length segment.<br clear="none">You stick your angles on each end - each less than 180 degrees,<br clear="none">you orient your angles in the plane, so, extending them out, you'd have<br clear="none">lines that intersect. There you go - one uniquely defined triangle<br clear="none">(at least to congruencies). Again, ignoring degenerate cases.<br clear="none"><br clear="none">ASS ... uh huh, put down an angle, attach a fixed length side,<br clear="none">now attach another fixed length side from that, with indeterminate<br clear="none">angle ... where can it in line drawn out other side of that first<br clear="none">angle? Unless that last angle turns out to be exactly 90 degrees,<br clear="none">there are exactly two possible triangles (again, ignoring degenerates).<br clear="none"><br clear="none">So, you get SAS and ASA, but no ASS.<br clear="none"><br clear="none">Yes, order matters.<div class="ydp9f2ddf21yqt5190125964" id="ydp9f2ddf21yqtfd29346"><br clear="none"><br clear="none"><br clear="none">_______________________________________________<br clear="none">conspire mailing list<br clear="none"><a shape="rect" href="mailto:conspire@linuxmafia.com" rel="nofollow" target="_blank">conspire@linuxmafia.com</a><br clear="none"><a shape="rect" href="http://linuxmafia.com/mailman/listinfo/conspire" rel="nofollow" target="_blank">http://linuxmafia.com/mailman/listinfo/conspire</a><br clear="none"></div></div>
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