In the figure, A and B are the midpoints of two sides of a regular hexagon. What fraction of the hexagon's area is gray? Express your answer as a common fraction.
+128475 Let the midpoint of the base of the triangle = (0,0)
Let the bottom left vertex of the triangle = (0, -1)
Let the top left vertex o f the hexagon be ( -1/2 , sqrt (3) / 2)
A = [ (-1 + -1/2) /2 , (sqrt (3)/2 + 0)/ 2 ] = [ -3/4 , sqrt (3) / 4) ]
So the base of the triangle = 2 and the height = y coord of A = sqrt (3) / 4
Its area = (1/2) (2) ( sqrt (3) / 4) = sqrt (3) /4
Area of the hexagon = (6) ( 1)^2 * sin (60°) = 3 sqrt (3) / 2 = 6sqrt (3) / 4
Fraction that is gray = sqrt (3) / 4 1
____________ = ____
6 sqrt (3) / 4 6
