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An equilateral triangle is inscribed in a circle such that its vertices lie on the circumference of the circle. A point is selected at random from within the circle. The probability of finding the point inside the triangle is:
- a)√3 / (2π)
- b)3 √3 / (4π)
- c)2π/ (√3)
- d)4 / (3√3π)
- e)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
An equilateral triangle is inscribed in a circle such that its vertic...
To solve this problem, we can use the concept of probability and geometry. Let's break down the solution into smaller steps:
1. Understand the problem:
- We have an equilateral triangle inscribed in a circle.
- A point is selected randomly from within the circle.
- We need to find the probability of the point being inside the triangle.
2. Determine the sample space:
- The sample space is the set of all possible outcomes.
- In this case, the sample space consists of all the points inside the circle.
3. Analyze the triangle:
- An equilateral triangle has all three sides and angles equal.
- Since the triangle is inscribed in the circle, the radius of the circle is also the length of each side of the triangle.
4. Find the area of the triangle:
- The area of an equilateral triangle can be calculated using the formula: Area = (sqrt(3) / 4) * s^2, where s is the length of each side.
- In this case, the area of the triangle is (sqrt(3) / 4) * r^2, where r is the radius of the circle.
5. Find the area of the circle:
- The area of a circle can be calculated using the formula: Area = π * r^2, where r is the radius of the circle.
6. Calculate the probability:
- The probability can be calculated by dividing the area of the triangle by the area of the circle.
- P = Area of Triangle / Area of Circle
- P = [(sqrt(3) / 4) * r^2] / [π * r^2]
- P = (sqrt(3) / 4π)
7. Simplify the probability:
- P = (sqrt(3) / 4π)
- P = (3 * sqrt(3) / 4π)
- P = (3 * sqrt(3) / (4π))
8. Final answer:
- The probability of finding the point inside the triangle is 3 * sqrt(3) / (4π), which corresponds to option B.
1. Understand the problem:
- We have an equilateral triangle inscribed in a circle.
- A point is selected randomly from within the circle.
- We need to find the probability of the point being inside the triangle.
2. Determine the sample space:
- The sample space is the set of all possible outcomes.
- In this case, the sample space consists of all the points inside the circle.
3. Analyze the triangle:
- An equilateral triangle has all three sides and angles equal.
- Since the triangle is inscribed in the circle, the radius of the circle is also the length of each side of the triangle.
4. Find the area of the triangle:
- The area of an equilateral triangle can be calculated using the formula: Area = (sqrt(3) / 4) * s^2, where s is the length of each side.
- In this case, the area of the triangle is (sqrt(3) / 4) * r^2, where r is the radius of the circle.
5. Find the area of the circle:
- The area of a circle can be calculated using the formula: Area = π * r^2, where r is the radius of the circle.
6. Calculate the probability:
- The probability can be calculated by dividing the area of the triangle by the area of the circle.
- P = Area of Triangle / Area of Circle
- P = [(sqrt(3) / 4) * r^2] / [π * r^2]
- P = (sqrt(3) / 4π)
7. Simplify the probability:
- P = (sqrt(3) / 4π)
- P = (3 * sqrt(3) / 4π)
- P = (3 * sqrt(3) / (4π))
8. Final answer:
- The probability of finding the point inside the triangle is 3 * sqrt(3) / (4π), which corresponds to option B.
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Community Answer
An equilateral triangle is inscribed in a circle such that its vertic...
The required probability is the ratio of area of the triangle to the area of the circle
Let the radius of the circle be 'r'
Area of the triangle = ½ r √3 x (3r/2)
Area of circle = π r2
Probability = [½ r √3 x (3r/2)] / π r2 = 3 √3 / (4π)
Option B
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An equilateral triangle is inscribed in a circle such that its vertices lie on the circumference of the circle. A point is selected at random from within the circle. The probability of finding the point inside the triangle is:a)√3 / (2π)b)3 √3 / (4π)c)2π/ (√3)d)4 / (3√3π)e)None of theseCorrect answer is option 'B'. Can you explain this answer?
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