Question

six equilateral triangles are connected to create a regular hexagon. the area of the hexagon is (24a^2 - 18) square units. which is an equivalent expression for the area of the hexagon based on the area of a triangle?

• (6(4a^2 - 3))
• (6(8a^2 - 9))
• (6a(12a - 9))
• (6a(18a - 12))

Explanation:

Identify the given area and geometric relationship

The total area of the regular hexagon is \(24a^2 - 18\). Since the hexagon is composed of six identical equilateral triangles, the area of one triangle is:

$$ \text{Area of one triangle} = \frac{24a^2 - 18}{6} = 4a^2 - 3 $$

Express the total area in terms of the triangle's area

The total area of the hexagon is six times the area of one triangle:

$$ \text{Total Area} = 6(4a^2 - 3) $$

Match with the given options

Comparing the factored expression \(6(4a^2 - 3)\) with the choices:

• \(6(4a^2 - 3)\)
• \(6(8a^2 - 9)\)
• \(6a(12a - 9)\)
• \(6a(18a - 12)\)

The correct expression is \(6(4a^2 - 3)\).

Answer:

• (A) \(6(4a^2 - 3)\) (Correct answer)
• (B) \(6(8a^2 - 9)\)
• (C) \(6a(12a - 9)\)
• (D) \(6a(18a - 12)\)