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help im bad at geo

 

In the diagram below, $ABCDEFGH$ is a cube. Find $\sin \angle GAC$.  Find $\sin \angle HAC$.

 

 Aug 16, 2023
 #3
avatar+128732 
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GAC  forms a right triangle with the right  angle  at G

The other two sides are equal sides of the cube such that  GC = GA

So this forms an isosceles right riangle....so angle GAC = 45°....and the  sin 45°  = sqrt (2) /  2

 

HAC is a little more difficult to  see.....let's call the side of the  cube = 1  = HA

AC =  the diagonal  across the face of the  cube  = sqrt (2)

And  HC is the major diagonal of the  cube =  sqrt (3)

 

Then HAC must form a right triangle  because

 

HA^2 + AC^2  = HC^2

 1^2    +  (sqrt 2)^2  = (sqrt 3)^2

  1    +   2  =  3

And angle HAC is opposite the  hypotenuse in this  triangle  so its  measure = 90°....and sin HAC = sin 90° = 1

 

cool cool cool

 Aug 16, 2023

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