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The diameter of the smaller circle is equal to the side of a square and the diagonal of this square is equal to the diameter of the bigger circle. The area of the smaller circle to the bigger circle is in the ratio:
- a)1 : 2
- b)2 : 3
- c)1 : √2
- d)1 : 4
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The diameter of the smaller circle is equal to the side ofa square and...
Let diameter of smaller circle = a = side of square
►so diagonal of square = a√2 = diameter of bigger circle
►Area of smaller circle = πa2 and area of bigger circle = π(a√2)2
►Required ratio = πa2 : π(a√2)2 = 1 : 2
Most Upvoted Answer
The diameter of the smaller circle is equal to the side ofa square and...
Let's assume that the diameter of the smaller circle is "d" and the side length of the square is also "d".
Since the diagonal of the square is equal to the diameter of the bigger circle, we can use the Pythagorean theorem to find the length of the diagonal of the square.
Using the Pythagorean theorem, we have: (side length of square)^2 + (side length of square)^2 = (diagonal of square)^2
Substituting "d" for the side length of the square, we have: d^2 + d^2 = (diagonal of square)^2
Simplifying this equation, we get: 2d^2 = (diagonal of square)^2
Taking the square root of both sides, we get: sqrt(2d^2) = diagonal of square
Simplifying further, we get: sqrt(2)d = diagonal of square
Now let's find the area of the smaller circle. The formula for the area of a circle is A = πr^2, where "r" is the radius.
Since the diameter of the smaller circle is "d", the radius is half of that, which is d/2.
Therefore, the area of the smaller circle is: A = π(d/2)^2 = π(d^2/4)
Similarly, let's find the area of the bigger circle. The diameter of the bigger circle is equal to the diagonal of the square, which is sqrt(2)d.
Therefore, the radius of the bigger circle is half of that, which is (sqrt(2)d)/2 = sqrt(2)d/2.
Therefore, the area of the bigger circle is: A = π(sqrt(2)d/2)^2 = π(2d^2/4) = πd^2/2
Now, let's find the ratio of the area of the smaller circle to the area of the bigger circle:
(Area of smaller circle)/(Area of bigger circle) = (π(d^2/4))/(πd^2/2)
Simplifying this equation, we get: (d^2/4)/(d^2/2) = (d^2/4)*(2/d^2) = 2/4 = 1/2
Therefore, the area of the smaller circle to the area of the bigger circle is in the ratio 1 : 2.
Answer: a) 1 : 2
Since the diagonal of the square is equal to the diameter of the bigger circle, we can use the Pythagorean theorem to find the length of the diagonal of the square.
Using the Pythagorean theorem, we have: (side length of square)^2 + (side length of square)^2 = (diagonal of square)^2
Substituting "d" for the side length of the square, we have: d^2 + d^2 = (diagonal of square)^2
Simplifying this equation, we get: 2d^2 = (diagonal of square)^2
Taking the square root of both sides, we get: sqrt(2d^2) = diagonal of square
Simplifying further, we get: sqrt(2)d = diagonal of square
Now let's find the area of the smaller circle. The formula for the area of a circle is A = πr^2, where "r" is the radius.
Since the diameter of the smaller circle is "d", the radius is half of that, which is d/2.
Therefore, the area of the smaller circle is: A = π(d/2)^2 = π(d^2/4)
Similarly, let's find the area of the bigger circle. The diameter of the bigger circle is equal to the diagonal of the square, which is sqrt(2)d.
Therefore, the radius of the bigger circle is half of that, which is (sqrt(2)d)/2 = sqrt(2)d/2.
Therefore, the area of the bigger circle is: A = π(sqrt(2)d/2)^2 = π(2d^2/4) = πd^2/2
Now, let's find the ratio of the area of the smaller circle to the area of the bigger circle:
(Area of smaller circle)/(Area of bigger circle) = (π(d^2/4))/(πd^2/2)
Simplifying this equation, we get: (d^2/4)/(d^2/2) = (d^2/4)*(2/d^2) = 2/4 = 1/2
Therefore, the area of the smaller circle to the area of the bigger circle is in the ratio 1 : 2.
Answer: a) 1 : 2
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The diameter of the smaller circle is equal to the side ofa square and...
Let diameter of smaller circle = a = side of square
►so diagonal of square = a√2 = diameter of bigger circle
►Area of smaller circle = πa2 and area of bigger circle = π(a√2)2
►Required ratio = πa2 : π(a√2)2 = 1 : 2
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