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/sci/ - Science & Math

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10559430 No.10559430 [Reply] [Original]

[math]
\text{Right triangle }ABC\text{ has right angle at }C\text{ and }\angle BAC =\theta\text{; the point }D\text{ is}
\\
\text{chosen on }AB\text{ so that }|AC|=|AD|=1\text{; the point }E\text{ is chosen on }BC\text{ so}
\\
\text{that }\angle CDE = \theta\text{. The perpendicular to }BC\text{ at }E\text{ meets }AB\text{ at }F\text{. Evaluate}
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\lim_{\theta\rightarrow 0} |EF|\text{.}
[/math]

>> No.10559433

Previous Thread >>10556517

>> No.10559462

1/3? This seems too simple

>> No.10559560

The two sides AD and AC collapse onto one another as theta tends to 0.
That means that the angle ACD tends to 90 degrees, hence ECD tends to 0, CE and ED collapse into CD.
Because CD tends to 0, E tends to C, and the perpendicular collapses onto AC and intersects AD in the point F, which tends to A.
Therefore the answer is 1.

I think that's right, although I haven't drawn it yet. Also don't know at what level of formality to give a mix of analysis and geometry like this.

>> No.10559591

>>10559430
1/3. Weird how easy is this compared to previous. Just straight school geometry.

>> No.10559617

>>10559560
>E tends to C
It doesn't though, E can also tend to B. And when that happens, you'd expect to EF to go to 0. The right answer would be something between 0 and 1

>> No.10559626

>>10559617
It can't tend to B because the angle CDE is theta by construction, so E always lies below the perpendicular of BC that intersects D. That perpendicular certainly collapses onto AC.

>> No.10559628

>>10559626
(Okay not always, but for small theta)

>> No.10559634

>>10559430
Using sine rule: sina/A=sinb/B=sinc/C through a couple of manipulations we get EF(theta) = 1 - cos(theta)sin(theta)/cos(theta/2)sin(3/2theta). Finding limit yields EF -> 1/3.

>> No.10559638

>>10559626
Okay but I'm still not convinced E tends to C.

>> No.10559648

>>10559430
Are EF and ED the same line?

>> No.10560191

>>10559430
1

>> No.10560945

based putnamposter

>> No.10562109

blergh, geometry

(polite sage because I'm adding nothing of value.)