The fraction of the image in blue with respect to the entire hexagon is [tex]\frac{1}{3} [/tex].
Estimation of the ratio of the shaded region area to the entire area
Initially we proceed to create auxiliar constructions , represented by red line segments, to re-define the hexagon as a sum of standard figures. According to the figure, we find two types of triangles (type-1 and type-2), and the following relationship between the two types:
[tex]A_{1} = A_{2}[/tex] (1)
Where:
- [tex]A_{1}[/tex] - Area of a type-1 triangle.
- [tex]A_{2} [/tex] - Area of a type-2 triangle.
The formulae for the area of the shaded region ([tex]A_{s}[/tex]) and the entire hexagon ([tex]A_{h}[/tex]) are, respectively:
Shaded region
[tex]A_{s} = 2\cdot A_{1} + 4\cdot A_{2}[/tex] (2)
Entire hexagon
[tex]A_{h} = 8\cdot A_{1} + 10\cdot A_{2}[/tex] (3)
And the fraction of the image in blue is:
[tex]\frac{A_{s}}{A_{h}} = \frac{2\cdot A_{1}+4\cdot A_{2}}{8\cdot A_{1}+10\cdot A_{2}} [/tex]
[tex]\frac{A_{s}}{A_{h}} = \frac{6\cdot A}{18\cdot A} [/tex]
[tex]\frac{A_{s}}{A_{h}} = \frac{1}{3} [/tex]
The fraction of the image in blue with respect to the entire hexagon is [tex]\frac{1}{3} [/tex]. [tex]\blacksquare[/tex]
To learn more on hexagons, we kindly invite to check this verified question: https://brainly.com/question/4083236
