GeminiHeat wrote:
A rectangle is inscribed in a hexagon that has all sides of equal length and all angles of equal measure, as shown in the figure above. If the total area of the hexagon is x, then the sum of the areas of the two shaded regions is
A. x/6
B. x/4
C. x/3
D. x/2
E. 2x/3
Key property: A regular hexagon consists of 6 equilateral trianglesSo we can divide the given hexagon into 6 equilateral triangles as follows:
This means, the area of the entire hexagon =
the area of 6 equilateral trianglesWe'll now label the four portions that comprise the shaded area as follows:
So....
The area of region A = 1/2 of an equilateral triangle
The area of region B = 1/2 of an equilateral triangle
The area of region C = 1/2 of an equilateral triangle
The area of region D = 1/2 of an equilateral triangle
So, the total shaded area = (1/2) + (1/2) + (1/2) + (1/2)
= 2
So, the shaded area is equal to the area of 2 equilateral triangles.
Since the hexagon consists of 6 equilateral triangles, the shaded portion = 2/6 of the entire hexagon.
So, if the hexagon has an area of x, the area of the shaded portion = 2/6 of x = 1/3 of x = x/3
Answer: C