Question 996872: The diagonals of rectangle BANK intersect at S such that angle ANS is 120 degrees, segment AB is 8 units. Find the length of segment AN. Express your answer in simplest radical form. Will someone please show me how to do this? You do not have to simplify it in simplest radical form because I know how to do that. I included it just so you know the length is supposed to be in a radical.
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! I assume your rectangle is labeled in this manner
top left is B, top right is A, bottom right is N and bottom left is K
all 4 interior angles of a rectangle are 90 degrees each, I think your angle is really angle ASN since ANS is 90 degrees.
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triangle BAS and triangle KSN are equilateral triangles
triangle BSK and triangle ANS are isosceles triangles
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side AS is 8 units and the altitude of triangle ANS drawn to side AN bisects angle ASN and side AN, therefore
sin(60) = (AN/2) / 8
16*sin(60) = AN
note that sin(60) = (1/2) * sqrt(3)
AN = 8 * sqrt(3)
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