ratio of a square inscribed in equilateral triangle & the triangle itself.?

Answer 1

You're doing some brain math questions or IQ math aren't you? You're earlier question and this are both of that variety. They're interesting enough questions, and although I hate math, these are pretty cool.

You should give them a shot. They're good questions.

I think the answer is 4:7

Answer 2

Well, the sides of the square all have to be equal. Also, you know the ratio of a 30-60-90 triangle as "1-2-√(3)", so you can set up the drawing like that:

Let the sides of the equilateral triangle be 1, and let x be the distance that the square. Thus, the square will have sides of 1-2x. We also know that the side of the square is √(3x). Set these equal, and see what happens:

1 - 2x = √(3x)
(1 - 2x)² = 3x
1 - 4x + 4x² = 3x
4x² - 7x + 1 = 0

Using the quadratic formula:

x = (7 ± √(49 -16))/8
x = (7 ± √33)/8

Since x has to be much less than 1, the only answer is (7 - √33)/8 ≈ 0.15692966918274641751867356647263

Therefore, the sides of the triangle are:

0.68614066...:1

Answer 3

Starting with a square ABCD whose sides have length 1... The base of the square AB sits on the base of the triangle XYZ (base XY). Point C is somewhere on XZ, and D is somewhere on YZ.

Now look at triangle DBY. This is a right triangle with DB=1 and angle Y = 60°. So BY = cotg(60°) = 1/sqrt(3). Similarly, you also have XA = 1/sqrt(3).

You have side XY = XA + AB + BY = 1 + 2/sqrt(3). That's the length ratio of the triangle to the square. You'll want to square this if you want the area ratio.