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Find the area of the circle passing through centers of three circles with radius 2 m, 3 m and 10 m placed in such a way that each circle touches the other two circles externally.
  • a)
    225 π/4
  • b)
    36 π
  • c)
    169 π / 4
  • d)
    Cannot be determined
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Find the area of the circle passing through centers of three circles ...
To find the area of the circle passing through the centers of three circles, we can use the concept of the circumcircle. The circumcircle is the circle that passes through the vertices of a triangle. In this case, the centers of the three circles form a triangle, and the circumcircle of this triangle will pass through the centers of the three circles.

1. Drawing the Diagram:
Draw three circles with radii 2 m, 3 m, and 10 m, such that each circle touches the other two circles externally. Label the centers of the circles as A, B, and C.

2. Finding the Length of the Triangle Sides:
The sides of the triangle formed by the centers of the circles can be calculated as follows:

- The distance between the centers of the circles with radii 2 m and 3 m is the sum of their radii, which is 5 m. This is the length of side AB.
- The distance between the centers of the circles with radii 2 m and 10 m is the sum of their radii, which is 12 m. This is the length of side AC.
- The distance between the centers of the circles with radii 3 m and 10 m is the sum of their radii, which is 13 m. This is the length of side BC.

3. Applying Heron's Formula:
Heron's formula can be used to find the area of a triangle when the lengths of its sides are known. The formula is given by:

Area = √(s(s-a)(s-b)(s-c))

where s is the semi-perimeter of the triangle and is calculated as:

s = (a+b+c)/2

In this case, a = 5 m, b = 12 m, and c = 13 m.

Plugging in the values, we get:

s = (5+12+13)/2 = 15

Area = √(15(15-5)(15-12)(15-13))
= √(15*10*3*2)
= √900
= 30 m^2

4. Finding the Radius of the Circumcircle:
The radius of the circumcircle of a triangle can be calculated using the formula:

Radius = (abc)/(4*Area)

where a, b, and c are the lengths of the sides of the triangle.

Plugging in the values, we get:

Radius = (5*12*13)/(4*30)
= 13/2
= 6.5 m

5. Calculating the Area of the Circumcircle:
The area of a circle can be calculated using the formula:

Area = π * radius^2

Plugging in the value of the radius, we get:

Area = π * (6.5)^2
= π * 42.25
= 132.25 π m^2

Therefore, the correct answer is option 'C', which is 169 π / 4.
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Community Answer
Find the area of the circle passing through centers of three circles ...
As per the given condition above diagram will be formed. So lines joining the centers of three circles will form right angle triangle as 5, 12, and 13 which is a Pythagorean triplet. So the circle passing through the centers of these circles will be passing through the vertex of this right angle triangle. So the diameter of such circle is hypotenuse of triangle which is 13 cm. Thus radius 6.5 cm and hence area will be π (13/2)2 = 169 π /4
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Find the area of the circle passing through centers of three circles with radius 2 m, 3 m and 10 m placed in such a way that each circle touches the other two circles externally.a)225 π/4b)36 πc)169 π / 4d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer?
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