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Angle Ratio Ascertainment 5 (Posted on 2016-08-06) Difficulty: 3 of 5
Consider triangle EFG with ∠EGF=90o
ET is drawn parallel to FG, such that
T and F are located on opposite sides of EG, and
FT intersects EG at S, and
ST = 2EF

Find ∠EFG/∠SFG

No Solution Yet Submitted by K Sengupta    
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solution | Comment 1 of 2
The puzzle is quite simple if seen the right way.

Consider the triangle EFT. For the law of the sines: 

ET/sin(EFT) = EF/sin(ETF)

ET=TS cos(ETS) =2EF cos(ETS)

From here: 2EF cos(ETS) sin(ETF)= EF sin(EFT)

but ETF=ETS

Then: 2 cos(ETS) sin(ETS) = sin(EFT);  sin(2ETS) = sin(EFT)

ETS= EFT/2   (angles)

The puzzle asks for the ratio EFG/SFG

EFG=EFS + SFG = EFT + ETS = 3ETS =3SFG

Then EFG/SFG=3

                  F
                /| 
            /   °|
        /      ° |
    /         °  |
/_E_____°_ |G
|        S °
|          °
|         °
|        °
|       °
|      °
|     °
|    °
|   °
|  °
| ° 
|° T


Edited on August 9, 2016, 10:40 am
  Posted by armando on 2016-08-09 04:31:42

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