If the two regoins above have the same area, what is the

[Mariela MGMAT Student]

12/37
If the two regoins above have the same area, what is the ratio t:s?

in the diagram the triangle has each side equal to t and the square has each side equal to s. The square has each vertex labeled as a 90 degree angle.
A 2:3
B 16:3
C 4: sqrt of 3
D 2:3^1/4
E 4:3^1/4
Ans is d

I know the formula for the area of triangle which is 1/2 base * heigt is = s^2. the area of the triangle is 1/2 t*t?3 but that as far as i go.

[esledge]

Hi Mariela,

This is kind of hard to do without the pictures, but judging by the answer I believe I am picturing it correctly. I think you got hung up on applying the triangle formula (maybe the height specifically):

You have an equilateral triangle with side t and a square with side s. Their areas are equal.

The short answer:
You could remember that in an equilateral triangle, height = base * sqrt(3)/2.
Area of triangle = (1/2) * base * height = (1/2) * t * (t* sqrt(3)/2) = t^2*sqrt(3)/4
Area of square = s^2
Therefore,
t^2 * sqrt(3)/4 = s^2
t^2/s^2 = 4/sqrt(3)
(t/s)^2 = 4/sqrt(3) = 4/(3^1/2)
t/s = 2/(3^1/4)

The long answer:
What if you forget the height of an equilateral triangle? You can cut the triangle into two equal halves: each will be a 30-60-90 triangle. Use the ratio of the sides to figure out the height
1: sqrt(3) : 2
(t/2)1: (t/2)sqrt(3) : (t/2)2 <----multiplying by t/2, just because that is what will make our long side in this ratio = t, as in our triangle

The 30-60-90 triangles thus have sides (t/2: t * sqrt(3)/2 : t) and the solution from there is the same as above.