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In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle?
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In the adjoining diagram ABCD is a square with side 'a' cm. In the dia...
Problem:
In the adjoining diagram, ABCD is a square with side 'a' cm. In the diagram, the area of the larger circle with center 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centers are P, Q, R, and S. What is the ratio between the side of the square and the radius of a smaller circle? Explain in detail.
Solution:
Step 1: Analyzing the given information
- In the diagram, ABCD is a square with side 'a' cm.
- There are four smaller circles with centers P, Q, R, and S, all having equal radii.
- The larger circle has center O and its area is equal to the sum of the areas of the four smaller circles.
Step 2: Finding the area of the larger circle
- The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius.
- Since the radius of the larger circle is not given, we need to find it in terms of 'a' (side of the square).
- The diagonal of a square is equal to a√2, and it also acts as the diameter of the larger circle.
- Therefore, the radius of the larger circle is (a√2)/2 = a√2/2.
Step 3: Finding the area of each smaller circle
- The area of a circle with radius 'r' can be calculated using the formula A = πr^2.
- Since the four smaller circles have equal radii, let's assume the radius of each smaller circle to be 'r'.
- The area of each smaller circle is then πr^2.
Step 4: Setting up the equation
- The area of the larger circle is equal to the sum of the areas of the four smaller circles.
- Therefore, π(a√2/2)^2 = 4πr^2.
Step 5: Solving the equation
- Simplifying the equation, (a^2/2) = 4r^2.
- Dividing both sides of the equation by 4, we get (a^2/8) = r^2.
- Taking the square root of both sides, we have a/√8 = r.
Step 6: Finding the ratio
- The ratio between the side of the square and the radius of the smaller circle is a/√8 : a.
- Simplifying further, we get 1/√8 : 1.
- Rationalizing the denominator, the ratio becomes √8/8 : 1/8.
- Simplifying the ratio, we have √8 : 1.
Answer:
The ratio between the side of the square and the radius of the smaller circle is √8 : 1.
In the adjoining diagram, ABCD is a square with side 'a' cm. In the diagram, the area of the larger circle with center 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centers are P, Q, R, and S. What is the ratio between the side of the square and the radius of a smaller circle? Explain in detail.
Solution:
Step 1: Analyzing the given information
- In the diagram, ABCD is a square with side 'a' cm.
- There are four smaller circles with centers P, Q, R, and S, all having equal radii.
- The larger circle has center O and its area is equal to the sum of the areas of the four smaller circles.
Step 2: Finding the area of the larger circle
- The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius.
- Since the radius of the larger circle is not given, we need to find it in terms of 'a' (side of the square).
- The diagonal of a square is equal to a√2, and it also acts as the diameter of the larger circle.
- Therefore, the radius of the larger circle is (a√2)/2 = a√2/2.
Step 3: Finding the area of each smaller circle
- The area of a circle with radius 'r' can be calculated using the formula A = πr^2.
- Since the four smaller circles have equal radii, let's assume the radius of each smaller circle to be 'r'.
- The area of each smaller circle is then πr^2.
Step 4: Setting up the equation
- The area of the larger circle is equal to the sum of the areas of the four smaller circles.
- Therefore, π(a√2/2)^2 = 4πr^2.
Step 5: Solving the equation
- Simplifying the equation, (a^2/2) = 4r^2.
- Dividing both sides of the equation by 4, we get (a^2/8) = r^2.
- Taking the square root of both sides, we have a/√8 = r.
Step 6: Finding the ratio
- The ratio between the side of the square and the radius of the smaller circle is a/√8 : a.
- Simplifying further, we get 1/√8 : 1.
- Rationalizing the denominator, the ratio becomes √8/8 : 1/8.
- Simplifying the ratio, we have √8 : 1.
Answer:
The ratio between the side of the square and the radius of the smaller circle is √8 : 1.
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In the adjoining diagram ABCD is a square with side 'a' cm. In the dia...
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In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle??
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In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle?? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle?? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle??.
In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle?? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle?? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle??.
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In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle??, a detailed solution for In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle?? has been provided alongside types of In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centres are P, Q, R and S. What is the ratio between the side of square and radius of a smaller circle?? theory, EduRev gives you an
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