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The area of a circle whose radius is 8 cm is trisected by two concentric circles. The ratio of radii of the concentric circles in ascending order is
- a)1 : 2 : 3
- b)2 : 3 : 5
- c)1 : √2 : √3
- d)√2 :√3:√5
Correct answer is option 'C'. Can you explain this answer?
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Explanation:
The area of the circle whose radius is 8 cm is given by the formula A = πr^2, where r is the radius of the circle.
Given that the area is trisected by two concentric circles, the areas of the three circles are in the ratio 1:1:1.
Let the radii of the concentric circles be r1, r2, and r3 respectively.
We know that the area of a circle is directly proportional to the square of its radius.
Calculations:
- The area of the circle with radius r1 is πr1^2.
- The area of the circle with radius r2 is πr2^2.
- The area of the circle with radius r3 is πr3^2.
Since the areas are in the ratio 1:1:1, we have:
πr1^2 : πr2^2 : πr3^2 = 1 : 1 : 1
r1^2 : r2^2 : r3^2 = 1 : 1 : 1
Taking the square root of both sides:
r1 : r2 : r3 = 1 : 1 : 1
Since we need to find the ratio of the radii in ascending order, we simplify the ratio to:
r1 : r2 : r3 = 1 : 1 : 1
To express the ratio in terms of square roots, we have:
r1 : r2 : r3 = √1 : √1 : √1
r1 : r2 : r3 = √1 : √1 : √1
r1 : r2 : r3 = 1 : √1 : √1
r1 : r2 : r3 = 1 : √2 : √3
Therefore, the ratio of the radii of the concentric circles in ascending order is 1 : √2 : √3, which corresponds to option 'C'.
The area of the circle whose radius is 8 cm is given by the formula A = πr^2, where r is the radius of the circle.
Given that the area is trisected by two concentric circles, the areas of the three circles are in the ratio 1:1:1.
Let the radii of the concentric circles be r1, r2, and r3 respectively.
We know that the area of a circle is directly proportional to the square of its radius.
Calculations:
- The area of the circle with radius r1 is πr1^2.
- The area of the circle with radius r2 is πr2^2.
- The area of the circle with radius r3 is πr3^2.
Since the areas are in the ratio 1:1:1, we have:
πr1^2 : πr2^2 : πr3^2 = 1 : 1 : 1
r1^2 : r2^2 : r3^2 = 1 : 1 : 1
Taking the square root of both sides:
r1 : r2 : r3 = 1 : 1 : 1
Since we need to find the ratio of the radii in ascending order, we simplify the ratio to:
r1 : r2 : r3 = 1 : 1 : 1
To express the ratio in terms of square roots, we have:
r1 : r2 : r3 = √1 : √1 : √1
r1 : r2 : r3 = √1 : √1 : √1
r1 : r2 : r3 = 1 : √1 : √1
r1 : r2 : r3 = 1 : √2 : √3
Therefore, the ratio of the radii of the concentric circles in ascending order is 1 : √2 : √3, which corresponds to option 'C'.
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The area of a circle whose radius is 8 cm is trisected by two concentric circles. The ratio of radii of the concentric circles in ascending order isa)1 : 2 : 3b)2 : 3 : 5c)1 : √2 : √3d)√2 :√3:√5Correct answer is option 'C'. Can you explain this answer?
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