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Let T be a point inside square EFGH such that TE = 1, TF = 2, and TG = 3. Find angle ETF.

 

 Apr 9, 2020
 #1
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Angle ETF = 120 degrees.

 Apr 10, 2020
 #2
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Let  T  =  (x , y)

 

Let  the  side of the  square  =  s

 

We  have  this system  [ we are only interested in  s ]

 

1  =  x^2  + ( s - y)^2

4  = x^2  + y^2

9  = ( s - x)^2  + y^2

 

This  is a little sticky to solve,  so using WolframAlpha,  we get that s =  √ [ 5 + √8]

 

Using the Law  of Cosines we  have that

 

s^2   = ET^2 + FT^2  - 2(ET * FT) cos  ETF  

 

5 +√8  = 1^2 + 2^2  -  2(1*2)  cos ETF

 

5 + √8 =  5 -  4 cos ETF

 

√8 / [ -4]  = cos ETF

 

cos ETF  = - 2√ 2 /  [ 2 * 2]  =  -√2/2  = -1/√2  

 

arccos (-1/√2)  = ETF  = 135°

 

 

cool cool cool

 Apr 10, 2020
edited by CPhill  Apr 10, 2020

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