Ahhh... thank you
"A square and an equilateral triangle have the same perimeter. Let A be the area of the circle circumscribed about the square and B be the area of the circle circumscribed about the triangle. Find A/B."
It's a tough one.
You'd be far better off to post with your REAL school subject in the School Subject box. Then a tutor with that expertise would be more likely find your question.
27/32
To solve this problem, we need to find the relationship between the side length of the square and the side length of the equilateral triangle, as well as the relationship between the area of the circle circumscribed about the square and the area of the circle circumscribed about the triangle.
Let's start by assuming that the side length of the square is 's'. Since the perimeter of the square is the same as the perimeter of the equilateral triangle:
Perimeter of the square = Perimeter of the equilateral triangle
4s = 3s
This implies that the side length of the equilateral triangle is three-fourths (3/4) of the side length of the square.
Now, let's move on to finding the relationship between the area of the circle circumscribed about the square (A) and the area of the circle circumscribed about the equilateral triangle (B).
The circle circumscribed about the square passes through all the vertices of the square and its diameter is equal to the diagonal of the square. Using the Pythagorean theorem, we can find the diagonal of the square:
Diagonal of the square = s√2
Radius of the circle circumscribed about the square = (Diagonal of the square)/2 = s√2/2
Area of the circle circumscribed about the square = π * (Radius of the circle)^2 = π * (s√2/2)^2 = π * s^2/2
Similarly, the circle circumscribed about the equilateral triangle passes through all the vertices of the triangle and its diameter is equal to the side length of the triangle. The radius of this circle is equal to half the side length of the equilateral triangle:
Radius of the circle circumscribed about the equilateral triangle = (Side length of the triangle)/2 = (3s/4)/2 = 3s/8
Area of the circle circumscribed about the equilateral triangle = π * (Radius of the circle)^2 = π * (3s/8)^2 = 9πs^2/64
Now, we can find the ratio of A/B by dividing the area of the circle circumscribed about the square (A) by the area of the circle circumscribed about the triangle (B):
A/B = (π * s^2/2) / (9πs^2/64)
= (π * s^2/2) * (64/9πs^2)
= 32/9
Hence, A/B = 32/9.
I hope this explanation helps you understand how to solve the problem! Let me know if you have any further questions.