Question

Square DEFG is inscribed in equilateral triangle ABC with a side length of 3+2√3.

Find: area of the square

Answer 1

9 units²

Explanation:

**Refer to the attached diagram**

The angles in the right triangle FAD (and HBE) are 30-60-90 which means that its sides will be in the ratio 1 : √3 : 2

Let x = the smallest side

Therefore, the sides of the right triangle FAD are in the ratio x : √3x : 2x

This means that:

• the side length of the square = √3x
• the side length of the triangle = x + √3x + x = (2 + √3)x

We are told that the side length of the triangle is 3 + 2√3, so:

`\implies 3 + 2√(3) =(2 + √(3) )x`

`\implies x=(3 + 2√(3))/(2 + √(3) )`

`\implies x=√(3)`

As the side length of the square is √3x, and x = √3,

⇒ side length of the square = √3 x √3 = 3

Now we know the side length of the square, we can calculate the area:

⇒ area of a square = side length x side length = 3 x 3 = 9 units²