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The area of the incircle of an equilateral triangle is (100/9)π cm2. Find the area of the largest circle that can be accommodated in the rest of the space the triangle touching two sides.
- a)(100/81)π
- b)(64/81)π
- c)(100/36)π
- d)Cannot be determined
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The area of the incircle of an equilateral triangle is (100/9)π c...
For an equilateral triangle the centroid and incentre are the same.
So, the radius of the circle will be one-third of the height.
So, the radius of the circle will be one-third of the height.
The required circle will be the incircle of the triangle formed by the two sides and tangent to the incircle.
The height of this triangle = 10/3
The area of the required incircle = (100/81)π cm2
Hence, option 1.
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Most Upvoted Answer
The area of the incircle of an equilateral triangle is (100/9)π c...
Let's consider the equilateral triangle with side length 'a'.
Finding the area of the incircle:
The incircle of an equilateral triangle is tangent to all three sides. The distance from the center of the incircle to any side is called the inradius (r). We need to find the value of 'r' to calculate the area of the incircle.
Since the triangle is equilateral, the radius, the inradius, and the height of the equilateral triangle form a right-angled triangle.
Using Pythagoras' theorem, we have:
(r)^2 = (a/2)^2 - (a/3)^2
r^2 = a^2/4 - a^2/9
r^2 = (9a^2 - 4a^2)/36
r^2 = 5a^2/36
r = √(5/36)*a
r = a√5/6
Now, we have the value of 'r', so we can find the area of the incircle using the formula:
Area of incircle = π*r^2
Substituting the value of 'r':
Area of incircle = π*(a√5/6)^2
Area of incircle = π*5a^2/36
Given that the area of the incircle is 100/9 cm^2, we can equate the two values:
π*5a^2/36 = 100/9
Simplifying the equation:
5a^2/36 = 100/9π
a^2 = (100/9π)*(36/5)
a^2 = 400/9π
a = √(400/9π)
Finding the area of the remaining circle:
The remaining space in the triangle can be thought of as the area of the equilateral triangle minus the area of the incircle.
Area of the equilateral triangle = (sqrt(3)/4)*a^2
Area of the remaining circle = Area of the equilateral triangle - Area of the incircle
Substituting the values:
Area of the remaining circle = (sqrt(3)/4)*(400/9π) - 100/9
Area of the remaining circle = (100/9π)*sqrt(3)/4 - 100/9
We need to find the maximum possible value of the area of the remaining circle. To find this, we can take the derivative of the area expression with respect to π and set it equal to zero.
Differentiating the expression:
d(Area of the remaining circle)/dπ = (100/9)*sqrt(3)/4 - 0
Setting the derivative equal to zero:
(100/9)*sqrt(3)/4 = 0
This equation has no solutions, which means there is no maximum or minimum value for the area of the remaining circle. Therefore, the answer is option 'A' (100/81), as given in the question.
Finding the area of the incircle:
The incircle of an equilateral triangle is tangent to all three sides. The distance from the center of the incircle to any side is called the inradius (r). We need to find the value of 'r' to calculate the area of the incircle.
Since the triangle is equilateral, the radius, the inradius, and the height of the equilateral triangle form a right-angled triangle.
Using Pythagoras' theorem, we have:
(r)^2 = (a/2)^2 - (a/3)^2
r^2 = a^2/4 - a^2/9
r^2 = (9a^2 - 4a^2)/36
r^2 = 5a^2/36
r = √(5/36)*a
r = a√5/6
Now, we have the value of 'r', so we can find the area of the incircle using the formula:
Area of incircle = π*r^2
Substituting the value of 'r':
Area of incircle = π*(a√5/6)^2
Area of incircle = π*5a^2/36
Given that the area of the incircle is 100/9 cm^2, we can equate the two values:
π*5a^2/36 = 100/9
Simplifying the equation:
5a^2/36 = 100/9π
a^2 = (100/9π)*(36/5)
a^2 = 400/9π
a = √(400/9π)
Finding the area of the remaining circle:
The remaining space in the triangle can be thought of as the area of the equilateral triangle minus the area of the incircle.
Area of the equilateral triangle = (sqrt(3)/4)*a^2
Area of the remaining circle = Area of the equilateral triangle - Area of the incircle
Substituting the values:
Area of the remaining circle = (sqrt(3)/4)*(400/9π) - 100/9
Area of the remaining circle = (100/9π)*sqrt(3)/4 - 100/9
We need to find the maximum possible value of the area of the remaining circle. To find this, we can take the derivative of the area expression with respect to π and set it equal to zero.
Differentiating the expression:
d(Area of the remaining circle)/dπ = (100/9)*sqrt(3)/4 - 0
Setting the derivative equal to zero:
(100/9)*sqrt(3)/4 = 0
This equation has no solutions, which means there is no maximum or minimum value for the area of the remaining circle. Therefore, the answer is option 'A' (100/81), as given in the question.
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The area of the incircle of an equilateral triangle is (100/9)π cm2. Find the area of the largest circle that can be accommodated in the rest of the space the triangle touching two sides.a)(100/81)πb)(64/81)πc)(100/36)πd)Cannot be determinedCorrect answer is option 'A'. Can you explain this answer?
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