Question

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

Explanation:

Step1: Find total area of all triangles

The figure is made up of 12 congruent equilateral triangles with each triangle having area \(a\) \(cm^{2}\). So the total area of all 12 triangles is \(12\times a=12a\) \(cm^{2}\).

Step2: Analyze the hexagon's relation

The hexagon area is \(b\) \(cm^{2}\). The entire shaded - region can be thought of in terms of the triangle and hexagon areas. The hexagon is part of the overall figure. The area of the six triangles around the hexagon is \(6a\) (since there are 6 triangles around the hexagon), and adding the area of the hexagon \(b\), the area of the shaded region is \(6a + b\) \(cm^{2}\). Also, we can note that the two - part structure of the figure (in terms of symmetry) implies that the area of the entire shaded region can be considered as composed of two such 'hexagon + surrounding triangles' parts in a sense, and since the area of one such part is \(6a + b\), we can also think of it as \(12a- 6a + b=6a + b\).

Answer:

12a \(cm^{2}\)
(6a + b) \(cm^{2}\)
2b \(cm^{2}\) (because the entire shaded region can be thought of as composed of two non - overlapping parts of area \(b\) in a symmetric way)