The figure above shows a square inscribed within an equilateral triang

OFFICIAL SOLUTION:

D) If you look at small triangles to the left and to the right of the square, you’ll notice that they are 90°-60°-30° triangles. Furthermore, the triangles have equal sides, so these triangles are equal.

The sides of a 90°-60°-30° triangle are in 2:√3:1 ratio. If we denote the smallest side in this triangle by y, then x : y = √3 : 1. Therefore y = 12 / √3.Let’s simplify to exclude a root sign in the denominator. 12 / √3 = (12 × √3) / (√3 × √3) = 4√3.

The base side of the large equilateral triangle equals to the sum of the three segments. y + x + y = 4√3 + 12 + 4√3 = 12 + 8√3.

All sides of an equilateral triangle are equal. Therefore the perimeter of the triangle is 3 × (12 + 8√3) = 36 + 24√3. The correct answer is D.

Alternative solution:

Is there a way to solve the problem if I don’t remember the properties of a 90°-60°-30° triangle?

Yes, there is. The small triangle on top of the square is equilateral as well, because all its angles are 60°. Therefore all its sides are equall x (12 inches).

The hypotenyses of the right triangles to the left and to the right of the square are both equal √(x² + y²) according to Pythagorean theorem.

Therefore the two sides of the large equilateral triangle equal x + √(x² + y²). The base side of the large equilateral triangle equals x + 2y. All sides in an equilateral triangle must be equal. Therefore x + √(x² + y²) = x + 2y.

x = √3y (because we deal with positive values only)
y = 4√3 so we can easily calculate the perimeter.

Excellent problem.

First derived that the square had made an equilateral triangle above it and then used the 30-60-90 ratio to find the length of the remaining portion.
Hence the actual perimeter is 3*(12+8\sqrt{3}) = 36+24\sqrt{3}

Option D is the answer.......

Dear Bunuel,
Hi. I really don't get how you guys get the small triangle is 30-60-90?
Would you explain? Thank you

Every angle of an equilateral triangle is 60 degrees so the rightmost angle of the small triangle is 60 (since it is also the angle of the large equilateral triangle). There is also a 90 degree angle made by the side of the square. So the leftover angle must be 30 degrees. Hence the 30-60-90 triangle.

Hello!

Each angle of a square is 90 degrees; so the adjacent angle [one of the angle in small triangle] is 180-90 = 90 degrees.
Every angle of an equilateral triangle is 60 degrees and we have the same angle common in the small triangle - hence it is 60 .
So the third angle is 180-(60=90) = 30 degrees.

Hence it is 30 60 90 triangle and the sides are in the ratio 1: sqrt3 :2 and we can proceed further.

How did you get 8√3?