To find the ratio of the radii of the circles given the areas in the ratio of 4:9:25, we need to use the relationship between the area of a circle and its radius.
The area (A) of a circle is calculated using the formula: A = πr2, where r is the radius of the circle.
Let the areas of the three circles be represented as follows:
- Area of circle 1: A1 = 4k
- Area of circle 2: A2 = 9k
- Area of circle 3: A3 = 25k
Here, k is a constant that we can use to express the areas in proportion to each other.
Now, using the area formula:
- For the first circle: A1 = πr12 = 4k
- For the second circle: A2 = πr22 = 9k
- For the third circle: A3 = πr32 = 25k
From the area equations, we can derive the radii:
- For the first circle: r1 = √(A1/π) = √(4k/π)
- For the second circle: r2 = √(A2/π) = √(9k/π)
- For the third circle: r3 = √(A3/π) = √(25k/π)
Now, to find the ratio of their radii:
Taking the square roots of the ratios of the areas:
- Ratio of radii: r1:r2:r3 = √4 : √9 : √25
Calculating the square roots:
- This simplifies to: 2 : 3 : 5
Thus, the ratio of the radii of the three circles is 2:3:5.