Question

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

Find: area of the square

Answer 1

Answer:

9 units²

Step-by-step 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:

$\implies 3 + 2\sqrt{3} =(2 + \sqrt{3} )x$

$\implies x=\dfrac{3 + 2\sqrt{3}}{2 + \sqrt{3} }$

$\implies x=\sqrt{3}$

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