To find the ratio of the radii of the three circles whose areas are in the ratio of 4:9:25, we start by using the relationship between the area of a circle and its radius.
The area A of a circle is given by the formula:
A = πr²
where r is the radius of the circle. Therefore, if we have three circles with areas in the ratio 4:9:25, we can denote their areas as:
- A₁ = 4 (for the first circle)
- A₂ = 9 (for the second circle)
- A₃ = 25 (for the third circle)
Next, we can express the radii in terms of their respective areas:
- For the first circle: A₁ = πr₁² → r₁² = A₁ / π → r₁ = √(A₁ / π)
- For the second circle: A₂ = πr₂² → r₂² = A₂ / π → r₂ = √(A₂ / π)
- For the third circle: A₃ = πr₃² → r₃² = A₃ / π → r₃ = √(A₃ / π)
Now, to find the ratio of the radii, we can find the ratios of r₁, r₂, and r₃ directly from the areas:
r₁ : r₂ : r₃ = √(A₁) : √(A₂) : √(A₃) = √(4) : √(9) : √(25) = 2 : 3 : 5
Thus, the ratio of the radii of the three circles is 2:3:5.