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2025-11-17T05:35:42Z

by freem.ai...

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{{DISPLAYTITLE:Solve angle problem}}

''This conversation was exported from ChatGPT''<ref>{{cite web|title=Solve angle problem|url=https://chatgpt.com/share/6919ca73-5614-8009-bb7f-6ff22c9d2193|publisher=ChatGPT|access-date=2025-11-16}}</ref>

== gpt-5-t-mini ==
=== User: solve this ===
solve this

=== Assistant: Answer: x80∘\displaystyle x80^\circx80∘. ===
Answer: x=80∘\displaystyle x=80^\circx=80∘.

Sketch of a short solution (clean numeric / trig setup):

Place the square with A=(0,1),B=(1,1),C=(1,0),D=(0,0)A=(0,1),B=(1,1),C=(1,0),D=(0,0)A=(0,1),B=(1,1),C=(1,0),D=(0,0). Let E=(e,0)E=(e,0)E=(e,0) and F=(1,1−t)F=(1,1-t)F=(1,1−t) (so t∈(0,1)t\in(0,1)t∈(0,1)).
Angle at AAA between AE→=(e,−1)\overrightarrow{AE}=(e,-1)AE=(e,−1) and AF→=(1,−t)\overrightarrow{AF}=(1,-t)AF=(1,−t) is 45∘45^\circ45∘, so

cos⁡45∘=(e)(1)+(−1)(−t)e2+11+t2=e+te2+11+t2.\cos45^\circ=\frac{(e)(1)+(-1)(-t)}{\sqrt{e^2+1}\sqrt{1+t^2}}=\frac{e+t}{\sqrt{e^2+1}\sqrt{1+t^2}}.cos45∘=e2+11+t2(e)(1)+(−1)(−t)=e2+11+t2e+t.
Angle at FFF between the vertical BFBFBF and FE→=(e−1,−(1−t))\overrightarrow{FE}=(e-1,-(1-t))FE=(e−1,−(1−t)) gives

cos⁡70∘=1−t(1−e)2+(1−t)2.\cos70^\circ=\frac{1-t}{\sqrt{(1-e)^2+(1-t)^2}}.cos70∘=(1−e)2+(1−t)21−t.
Solving these two equations (one can do it numerically or with algebraic manipulation) yields

e≈0.17632698,t≈0.70020754.e\approx0.17632698,\qquad t\approx0.70020754.e≈0.17632698,t≈0.70020754.
Finally the desired angle at EEE (between EA→=(−e,1)\overrightarrow{EA}=(-e,1)EA=(−e,1) and EF→=(1−e, 1−t)\overrightarrow{EF}=(1-e,\,1-t)EF=(1−e,1−t)) evaluates to

∠AEF≈80∘.\angle AEF \approx 80^\circ.∠AEF≈80∘.
So x=80∘x=80^\circx=80∘.

== References ==
<references />

[[Category:ChatGPT Shares]]
[[Category:openai]]

Openai/6919ca73-5614-8009-bb7f-6ff22c9d2193 - Revision history

Lukegao: by freem.ai...