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# geometry problem

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The figure is drawn on a square grid as shown.

Find the marked angle in degrees.

Jan 3, 2021

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For convenience, let the  bottom  vertex of the  square  =(0,0)

The slope of one of the  intersecting lines is 2...and  its equation is y  =2x

The slope of anoteher of the intersecting lines  is   -1/2.....and its equation is

y = -1/2x  +  2

The x  intersection of these  lines  is

2x =  -1/2x  +  2

(5/2)x   = 2

x =4/5   =   .8

And the  y coordiante is     t = (2)(4/5)   = 8/5  =  1.6

We  have a triangle with  the points   ( .8, 1.6)  , (1, 2)   and (2,2)

The (squared) sides of this triangle  are

[ (.8- 1)^2  + ( 1.6 - 2)^2  ]  = .2

[ (.8 - 2)^2  + (1.6 - 2)^2 ]  = 1.6

And 1

We can use the Law of Cosines to find the angle

1 =  .2  + 1.6  - 2  ( sqrt ( .2) sqrt ( 1.6) cos  (angle)

1 =  .2  + 1.6  - 2  sqrt ( .32) cos (angle)

[ 1 - ,2  -1.6 ] / [- 2sqrt (.32) ]  = cos (angle)

arccos   ( [ 1 - .2 - 1.6 ]  / [- 2sqrt (.32) ] )   =   45°    = angle

Jan 3, 2021