SOLUTION: Hi, Thank you very much for what you do, it really means a lot. The small square is inside the circle and the four corners of the square are touching the circle. The circle is

Algebra ->  Triangles -> SOLUTION: Hi, Thank you very much for what you do, it really means a lot. The small square is inside the circle and the four corners of the square are touching the circle. The circle is      Log On


   

Question 486247: Hi,
Thank you very much for what you do, it really means a lot.
The small square is inside the circle and the four corners of the square are touching the circle. The circle is inside the big square and is touching the big square, I think, on the mid-point of the sides. They say the area of the big square is 4x^2 and ask for the area of the small square.
I think the area of a square is (side * side) and if I know what one side is that gives me the diameter of the circle which gives me the hypotenuse of the complementary isosceles triangles inside the small square. Then I think if I know the hypotenuse of the isosceles that should give the length of one of the sides of the small square by the Pythagorean theorem. But what is the side of the big square?
Thanks,
Dave
Answer by cleomenius(959) About Me  (Show Source):
You can put this solution on YOUR website!
The area of your big square will be the side of the square squared, so if the area of your big square is 4a^2, a side will be 2a.
You are correct in that this is also the diameter of the circle.
You are correct that this will be the hypotenuse of triangle formed from the inner square (2a).
What is formed by the diagonal of the circle, is two 45 45 90 degree triangles, we need to find the length of the sides; after which we can determine the area of the inner triangle.
In a 45 90 45 triangle, the sides are s and the hypotenuse is ssqrt%282%29
Therefore 2a = ssqrt%282%29.
Worked out, s = (2asqrt%282%29)/2.
To find the area of the inner square we will square (2asqrt%282%29)/2.
We obtain 2A^2.
Cleomenius.