<p>Some are claiming that this should be an easy problem for anyone applying to a top school. What do you think?</p>
<p>
[quote]
The perpendicular dropped from the vertex of the right triangle upon the hypothenuse divides it into two segments of 9 and 16 feet respectively. Find the lengths of the perpendicular and the two legs of the triangle."
[/quote]
</p>
<p>Please no calculator or trig! And (please) show how you solve the problem.</p>
<p>well since im not planning on Ivys…its safe to say that this problem has officially murdered me before i even read it…</p>
<p>ALL HAIL XIGGI</p>
<p>All you have to do is sketch the diagram and use trigonometric functions to solve for the sides and the perpendicular line, since you know that the right angle divides into 2 45 degree angles.</p>
<p>I attempted a diagram, but the wording kinda screws me over.</p>
<p>The perpendicular is 12 feet, and the sides are 15 and 20?</p>
<p>15, 11 = legs of triangle</p>
<p>12 = perpendicular</p>
<p>method: just set up 3 equations.</p>
<p>[photo</a> (2) - Minus.com](<a href=“http://minus.com/mbgDxSQsxC#1f]photo”>http://minus.com/mbgDxSQsxC#1f)</p>
<p>Then, set up three equations.
a²=9²+c²
b²=16²+c²
25²=a²+b²</p>
<p>substitute a² and b².
25²=9²+c²+16²+c²
25²-16²-9²=2c²
288=2c²
144=c²
c=12</p>
<p>Plug in c=12 to find a and b.
a= 9 b= 20</p>
<p>As any old geometry teacher might remember, the altitude to a hypotenuse is equal in length to the geometric mean of the two segments formed. </p>
<p>So in this case: </p>
<p>9/h = h/16</p>
<p>h^2 = 144</p>
<p>h = 12</p>
<p>Then, the pythag thm gives you the rest.</p>
<p>BTW, there are a bunch of similar rules that can be discovered from the three similar triangles you get when you drop that altitude to the hypot. (The two little triangles formed are similar to the big triangle you started with and to each other as well.) From the various geometric means that you discover, there is a way to derived the pythag thm. That’s why geometry books often teach geometric means right before they teach pythag thm.</p>
<p>
</p>
<p>wow… I never heard of this but this is really cool. How come my geometry teacher never taught me this?..</p>
<p>h²=ab
when h is the altitude of a triangle
a and b are the two segments formed.</p>
<p>Thanks for your contribution…</p>
<p>You are welcome! It’s kind of obscure but it IS cool. And if I may ramble on in a continued nerdy way:</p>
<p>It’s called the geometric mean because if you insert it between a and b, it forms a geometric sequence:</p>
<p>a, h, b is a geometric sequence if the ratio of consecutive terms is constant. In this case,</p>
<p>a/h = h/b.</p>
<p>Other types of sequences have other types of means! For example, x is the arithmetic mean of a and b if a, x, b forms an arithmetic sequence: the difference of consecutive terms is constant. So: </p>
<p>x - a = b- x
2x = a + b
x = (a + b)/2 </p>
<p>Ohhh. That’s why they call the average two numbers the “arithmetic mean” </p>
<p>And hang in for one more…it has a surprise ending:</p>
<p>z is the “harmonic mean” of a and b if a, z, b form a harmonic sequence. A harmonic sequence is one where the difference of the reciprocals of consecutive terms is constant.</p>
<p>So 1/a - 1/z = 1/z - 1/b</p>
<p>or 1/a + 1/b = 2/z</p>
<p>or (a+b)/ab = 2/z</p>
<p>or z = 2ab/(a+b) Look familiar? It’s Xiggi’s formula for averaging average speed!</p>
<p>OK, end of nerd -fest.</p>
<p>HAHAHAAHAHA. I really appreciate that…
I did not know about geometric mean or the harmonic mean…
Wow… Maybe I’m not really understanding this too well. Can you explain why the harmonic mean equals Xiggi’s formula?</p>
<p>Well, Xiggi explains it best(and explains it about once every two or three weeks) so I’ll try to give him a night off:</p>
<p>Xiggi’s formula applies when you have traveled the same distance at two different speeds and you want the average speed for your trip.</p>
<p>Since RT=D, T = D/R. So going one way, T1=D/R1 and the other way T2=D/R2. </p>
<p>(Those are subscripts, not exponents!) </p>
<p>Total distance: 2D</p>
<p>Total time: T1+T2 = D/R1 +D/R2.</p>
<p>Average speed: 2D/(D/R1 + D/R2) which cleans up to be 2(R1)(R2)/(R1+R2)</p>
<p>Oh I see… I did see Xiggi explain how his formula was derived, but I wanted to know why harmonic sequence is identical to Xiggi’s formula…</p>
<p>Maybe I’m just not getting this. I’m so stupid :(</p>
<p>Let me think about it and try to come up with another way to answer.</p>
<p>Sent from my Droid using CC App</p>
<p>pckeller, you would have made Mr. Golfman proud. My old Algebra 2 teacher would always say “then you reach into your bag of tricks…” and, let me tell you, his bag had a Home Depot’s worth of nifty tricks.</p>
<p>I had hoped that the discussion might last a bit longer, but thanks to pckeller for pointing in the right direction. Here’s a helpful link to a full discussion about the proportional or geometric mean. I believe this could come in handy for a few SAT problems.</p>
<p>[Mean</a> Proportional](<a href=“http://regentsprep.org/Regents/math/geometry/GP12/LMeanP.htm]Mean”>http://regentsprep.org/Regents/math/geometry/GP12/LMeanP.htm)</p>
<p>By the way, I mentioned “prehistoric SAT” because this question was part of a 1869 entrance exam at MIT. You can find a few links to that year’s entrance exams at both Harvard and MIT in this CC discussion:</p>
<p><a href=“http://talk.collegeconfidential.com/parents-forum/1214877-1869-harvard-entrance-exam.html[/url]”>http://talk.collegeconfidential.com/parents-forum/1214877-1869-harvard-entrance-exam.html</a></p>
<p>If I can remember back to freshman year, my geometry teacher called the same method pckeller mentioned the “heartbeat monitor”.</p>
<p>If you can imagine the (blip?) the heartbeat monitor makes, it helps you remember the order the numbers of the equation go.</p>
<p>9/a = a/16
heartbeat monitor</p>
<p>It’s hard to describe without visual help, but I find it an easy way to remember the geometric mean and I hope it helps you guys.</p>