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Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle?
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Two circles of an equal radii are drawn without any overlap in a semi ...
Understanding the Problem
To determine the radius of the largest two circles that can fit within a semicircle of radius 2 cm, we must analyze the geometry involved. The two circles are inscribed within the semicircle without overlapping, maximizing the use of available space.
Geometry Involved
- The semicircle has a radius of 2 cm.
- The centers of both circles will lie on the diameter of the semicircle.
- The distance from the center of the semicircle to the top of the circles must equal the radius of the semicircle.
Calculating Radius of the Circles
1. Position of Circles:
- The two circles will be tangent to each other and to the semicircle.
- Let the radius of each circle be r.
2. Distance from Center:
- The distance from the center of the semicircle to the center of each circle is 2 - r (since the semicircle's radius is 2 cm).
3. Setting Up the Equation:
- The total distance between the centers of the two circles (which is 2r) must be equal to the diameter of the semicircle minus the radius of one circle:
- 2 - r = r + r, leading to 2 = 2r + r, or 2 = 3r.
4. Solving for r:
- Rearranging gives r = 2/3 cm.
Conclusion
The radius of each of the largest two circles that can fit within the semicircle of radius 2 cm is:
- Radius of each circle: 2/3 cm.
This configuration ensures that both circles are maximally utilized within the given space of the semicircle.
To determine the radius of the largest two circles that can fit within a semicircle of radius 2 cm, we must analyze the geometry involved. The two circles are inscribed within the semicircle without overlapping, maximizing the use of available space.
Geometry Involved
- The semicircle has a radius of 2 cm.
- The centers of both circles will lie on the diameter of the semicircle.
- The distance from the center of the semicircle to the top of the circles must equal the radius of the semicircle.
Calculating Radius of the Circles
1. Position of Circles:
- The two circles will be tangent to each other and to the semicircle.
- Let the radius of each circle be r.
2. Distance from Center:
- The distance from the center of the semicircle to the center of each circle is 2 - r (since the semicircle's radius is 2 cm).
3. Setting Up the Equation:
- The total distance between the centers of the two circles (which is 2r) must be equal to the diameter of the semicircle minus the radius of one circle:
- 2 - r = r + r, leading to 2 = 2r + r, or 2 = 3r.
4. Solving for r:
- Rearranging gives r = 2/3 cm.
Conclusion
The radius of each of the largest two circles that can fit within the semicircle of radius 2 cm is:
- Radius of each circle: 2/3 cm.
This configuration ensures that both circles are maximally utilized within the given space of the semicircle.
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Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle?
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Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle?.
Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two circles of an equal radii are drawn without any overlap in a semi circle of radius 2cm if these are the largest Portugal circles that the semi circle can accommodate what is the radius and centimetre of the ecircle?.
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