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Let point A be at (1,1). Let point B be at (2,3) and point C at (4,0).

 

Let theta = angle BAC. Then we can write cos theta = frac{x}{sqrt{2}} for some value of x. What is x?

 May 9, 2019
 #1
avatar+100519 
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Distance from A to C  =  sqrt [ (1- 4)^2 + (1 - 0)^2 ]  =  sqrt [ (-3)^2 + (1)^2 ]  = sqrt (10)

Distance from A to B  =  sqrt [ (1 - 2)^2 + (1 - 3)^2 ]  =  sqrt [ (-1)^2 + (-2)^2 ]  =  sqrt (5)

Distance fromB to C =  sqrt [ (4 - 2)^2 + (3 - 0)^2 ]  =  sqrt [ 2^2 + 3^2 ]  =  sqrt [13]

 

So........using the Law of Cosines we have that

 

[sqrt (13)]^2  - [sqrt(10)]^2 - [sqrt (5)]^2   

________________________________  = cos theta

     -  2 sqrt (10)*sqrt(5)

 

 

[ 13 - 10 - 5 ]

__________   =   cos theta

-2 sqrt (50)

 

[-2]

________    = cos theta

-2 sqrt (50)

 

 

1

_______    =  cos theta

sqrt (50)

 

1

_______   =  cos theta

5 sqrt (2)

 

So  x  =  1/5

 

 

 

cool cool cool

 May 10, 2019

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