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The largest equilateral triangle that can be inscribed inside a circle of radius 1 cm has a side (in cm) of __________.
Correct answer is '1.732'. Can you explain this answer?
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The largest equilateral triangle that can be inscribed inside a circle...
Let O be the centre of the circle . So OA=OB=OC=1cm.
To find : The side of the triangle .
Construction : Draw OD being the perpendicular bisector of side BC.
Proof : We know the angle ODB would be equal to 90degree by construction.
Now , angle B = 60degree {given}
angle OBD=30degree {half of 60degree as BO
bisects angle B}
In triangle OBD,
cos30degree= BD/OB
BD=√3/2{As OB=1}
BC=2BD=√3=1.732cm
Hence,AB=BC=CA=1.732cm. (Proved)
To find : The side of the triangle .
Construction : Draw OD being the perpendicular bisector of side BC.
Proof : We know the angle ODB would be equal to 90degree by construction.
Now , angle B = 60degree {given}
angle OBD=30degree {half of 60degree as BO
bisects angle B}
In triangle OBD,
cos30degree= BD/OB
BD=√3/2{As OB=1}
BC=2BD=√3=1.732cm
Hence,AB=BC=CA=1.732cm. (Proved)
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The largest equilateral triangle that can be inscribed inside a circle...
The Concept of Inscribed Equilateral Triangle
When dealing with circles and inscribed shapes, the largest equilateral triangle that can fit inside a circle (circumcircle) has its vertices on the circle.
Understanding the Circle and Triangle Relationship
- A circle of radius 1 cm has a diameter of 2 cm.
- The circumradius (R) of the inscribed equilateral triangle is equal to the radius of the circle.
Calculating the Side Length
For an equilateral triangle inscribed in a circle, the relationship between the side length (s) and the circumradius (R) is given by the formula:
s = R × √3
- Here, R = 1 cm (the radius of the circle).
- Substitute R into the formula:
s = 1 × √3
- This simplifies to:
s = √3
Numerical Value of the Side Length
- The numerical value of √3 is approximately 1.732.
- Therefore, the side length of the largest equilateral triangle that can be inscribed inside a circle of radius 1 cm is 1.732 cm.
Conclusion
In conclusion, the largest equilateral triangle inscribed in a circle of radius 1 cm has a side length of approximately 1.732 cm. This relationship highlights the geometric properties of triangles and circles, demonstrating the elegance of mathematical relationships in geometry.
When dealing with circles and inscribed shapes, the largest equilateral triangle that can fit inside a circle (circumcircle) has its vertices on the circle.
Understanding the Circle and Triangle Relationship
- A circle of radius 1 cm has a diameter of 2 cm.
- The circumradius (R) of the inscribed equilateral triangle is equal to the radius of the circle.
Calculating the Side Length
For an equilateral triangle inscribed in a circle, the relationship between the side length (s) and the circumradius (R) is given by the formula:
s = R × √3
- Here, R = 1 cm (the radius of the circle).
- Substitute R into the formula:
s = 1 × √3
- This simplifies to:
s = √3
Numerical Value of the Side Length
- The numerical value of √3 is approximately 1.732.
- Therefore, the side length of the largest equilateral triangle that can be inscribed inside a circle of radius 1 cm is 1.732 cm.
Conclusion
In conclusion, the largest equilateral triangle inscribed in a circle of radius 1 cm has a side length of approximately 1.732 cm. This relationship highlights the geometric properties of triangles and circles, demonstrating the elegance of mathematical relationships in geometry.
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The largest equilateral triangle that can be inscribed inside a circle of radius 1 cm has a side (in cm) of __________.Correct answer is '1.732'. Can you explain this answer? for IIT JAM 2025 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The largest equilateral triangle that can be inscribed inside a circle of radius 1 cm has a side (in cm) of __________.Correct answer is '1.732'. Can you explain this answer? covers all topics & solutions for IIT JAM 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The largest equilateral triangle that can be inscribed inside a circle of radius 1 cm has a side (in cm) of __________.Correct answer is '1.732'. Can you explain this answer?.
The largest equilateral triangle that can be inscribed inside a circle of radius 1 cm has a side (in cm) of __________.Correct answer is '1.732'. Can you explain this answer? for IIT JAM 2025 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The largest equilateral triangle that can be inscribed inside a circle of radius 1 cm has a side (in cm) of __________.Correct answer is '1.732'. Can you explain this answer? covers all topics & solutions for IIT JAM 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The largest equilateral triangle that can be inscribed inside a circle of radius 1 cm has a side (in cm) of __________.Correct answer is '1.732'. Can you explain this answer?.
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