Bunuel wrote:

Tough and Tricky questions: Word Problems.

The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)?

A. 36

B. 36√3

C. 24√3 + 24

D. 24√3 + 36

E. 72

Kudos for a correct solution.Attachment:

square_inscribed_in_triangle.gif

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.

Hi. I really don't get how you guys get the small triangle is 30-60-90?