the figure consists of 12 congruent equilateral triangles. the area of one equilateral triangle is a cm²…

the figure consists of 12 congruent equilateral triangles. the area of one equilateral triangle is a cm². the area of the hexagon, shaded slightly darker, is b cm². which expressions represent the area of the entire shaded region, including the light and dark shading? select three options. 12a cm² 2b cm² (6a - b) cm² (12a + 2b) cm² (6a + b) cm²

the figure consists of 12 congruent equilateral triangles. the area of one equilateral triangle is a cm². the area of the hexagon, shaded slightly darker, is b cm². which expressions represent the area of the entire shaded region, including the light and dark shading? select three options. 12a cm² 2b cm² (6a - b) cm² (12a + 2b) cm² (6a + b) cm²

Answer

Answer:

A. $12a\ cm^{2}$, B. $2b\ cm^{2}$, E. $(6a + b)\ cm^{2}$

Explanation:

Step1: Find area by triangle - count

The figure has 12 congruent equilateral - triangles with each area $a\ cm^{2}$. So the total area is $12\times a=12a\ cm^{2}$.

Step2: Analyze hexagon and outer - triangles

The hexagon area is $b\ cm^{2}$. The outer part of the shaded region (excluding the hexagon) is made up of 6 equilateral triangles with area $a\ cm^{2}$ each. The total area of the shaded region is the area of the 6 outer triangles plus the area of the hexagon, which is $6a + b\ cm^{2}$. Also, the hexagon can be thought of as half of the total shaded region in a certain sense, and the total shaded region can be expressed as $2b\ cm^{2}$ since the whole figure's shaded part is symmetrically related to the hexagon.