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Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB is times the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle and the inner circle respectively, what is the length of the side of the triangle?
- a)√2
- b)√3
- c)2
- d)3
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Two concentric circles have their centers at point O such that a line ...
Given
- Two concentric circles with center at O
- Let’s assume the radius of the outer circle to be R and that of inner circle to be r
- AB is a chord to the outer circle and a tangent to the inner circle
-
- An equilateral triangle with Area = R/r
- Assuming the side length of the equilateral triangle to be a, we have
To Find:
- Value of a.
Approach
- Since
to find the value of a, we need to find the value of R/r - We know that AB is a tangent to the inner circle, hence AB will make an angle of 90o with the radius of the inner circle, i.e. the line joining the center O to the point of tangency. Let’s call the point of tangency as C.
- So, OC will be perpendicular to AB.
- However, as AB is also a chord to the external circle, the line perpendicular from the center O will bisect the chord, i.e. OC will bisect the chord AB.
- Therefore,
- Therefore,
- As we know that BC =
and triangle OCB is a right angled triangle, we can use Pythagoras theorem in triangle OCB to calculate the ratio of R/r
-
Working Out
- Using Pythagoras theorem in triangle OCB, we have
(rejecting the negative root since the ratio of radii cannot be negative)
2.
Therefore, a=2 , as a being the length of a triangle, cannot be negative.
Hence the correct answer is Option C .
Most Upvoted Answer
Two concentric circles have their centers at point O such that a line ...
Let the radius of the inner circle be r. Then the length of AB is 3r.
The ratio of the radii of the outer circle to the inner circle is 3r/r = 3.
Let the side length of the equilateral triangle be s. The area of an equilateral triangle is (sqrt(3)/4)*s^2.
We are given that this area is equal to 3, so (sqrt(3)/4)*s^2 = 3.
Multiplying both sides by 4/sqrt(3), we get s^2 = (4*3)/sqrt(3) = 12/sqrt(3).
Taking the square root of both sides, we get s = sqrt(12/sqrt(3)) = sqrt(12)*sqrt(sqrt(3))/sqrt(sqrt(3)).
Simplifying, we get s = sqrt(12)*sqrt(sqrt(3))/sqrt(3).
The square root of 12 is 2*sqrt(3), so s = 2*sqrt(3)*sqrt(sqrt(3))/sqrt(3) = 2*sqrt(sqrt(3)).
Therefore, the length of the side of the equilateral triangle is 2*sqrt(sqrt(3)). Answer: \boxed{2\sqrt{\sqrt{3}}}.
The ratio of the radii of the outer circle to the inner circle is 3r/r = 3.
Let the side length of the equilateral triangle be s. The area of an equilateral triangle is (sqrt(3)/4)*s^2.
We are given that this area is equal to 3, so (sqrt(3)/4)*s^2 = 3.
Multiplying both sides by 4/sqrt(3), we get s^2 = (4*3)/sqrt(3) = 12/sqrt(3).
Taking the square root of both sides, we get s = sqrt(12/sqrt(3)) = sqrt(12)*sqrt(sqrt(3))/sqrt(sqrt(3)).
Simplifying, we get s = sqrt(12)*sqrt(sqrt(3))/sqrt(3).
The square root of 12 is 2*sqrt(3), so s = 2*sqrt(3)*sqrt(sqrt(3))/sqrt(3) = 2*sqrt(sqrt(3)).
Therefore, the length of the side of the equilateral triangle is 2*sqrt(sqrt(3)). Answer: \boxed{2\sqrt{\sqrt{3}}}.
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Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB istimes the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle andthe inner circle respectively, what is the length of the side of the triangle?a)√2b)√3c)2d)3Correct answer is option 'C'. Can you explain this answer?
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Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB istimes the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle andthe inner circle respectively, what is the length of the side of the triangle?a)√2b)√3c)2d)3Correct answer is option 'C'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB istimes the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle andthe inner circle respectively, what is the length of the side of the triangle?a)√2b)√3c)2d)3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB istimes the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle andthe inner circle respectively, what is the length of the side of the triangle?a)√2b)√3c)2d)3Correct answer is option 'C'. Can you explain this answer?.
Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB istimes the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle andthe inner circle respectively, what is the length of the side of the triangle?a)√2b)√3c)2d)3Correct answer is option 'C'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB istimes the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle andthe inner circle respectively, what is the length of the side of the triangle?a)√2b)√3c)2d)3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB istimes the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle andthe inner circle respectively, what is the length of the side of the triangle?a)√2b)√3c)2d)3Correct answer is option 'C'. Can you explain this answer?.
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