Special for Experts question (Similarity)

[abstract aurora]

Please Dont use trigonometry.

[foggy berry]

Why can't i use trigonometry

[abstract aurora]

Because İ need a solution to understand and İ don't know trigonometry

Use similarity

[runic trellis]

Maybe show what u did we can see where you are stuck

[plucky surge]

If T lies in the plane (MNP) then the answer is arctan(1/sqrt(2)) which cannot be expressed as an angle of some rational number of degrees. So, by the nature of the answer itself you need to use trigonometry.

Or you can give an answer in the form: angle TKN equals the angle which is opposite to the side of length 1 in the right triangle with sides 1, sqrt(2), sqrt(3). That's without trigonometry

[abstract aurora]

This question can must be solved without trigonometry because it has solution without trigonometry but I am not able yo have it now

[runic trellis]

Yes u can do without u need to draw based on info given so u can find adjacent angels and from there you will find which triangle is similar to which…. Then you can use that relationship to find the angel K

[plucky surge]

oh, I thought initially that it is a 3D problem. You diagram sort of hints at that. But now I see that it is from plane geometry, and yes it does have a solution without trigo.

[plucky surge]

One possible solution is to ||choose a point A as the midpoint of MK. Then triangle TNA is equilateral and ∠TKN is an inscribed angle in a circle centered at A. Hence, ∠TKN=60°/2=30°.||

Another even easier solution is to note that ||MTNK is cyclic (since ∠TMK+∠TNK=180°), then ∠TKN=∠TMN=30°||

[abstract aurora]

Thanks. BUT why TMK + TNK = 180°

Ok I figured it THANK YOU!

.close