If the ratio of the areas of two circles is `9:16,` find the ratio of their radii.

Answer

To find the ratio of the radii of two circles given the ratio of their areas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the area of a circle**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

2. **Set up the ratio of the areas**:
We are given that the ratio of the areas of two circles is \( 9:16 \). We can express this as:
\[ \frac{A_1}{A_2} = \frac{9}{16} \]
where \( A_1 \) is the area of the first circle and \( A_2 \) is the area of the second circle.

3. **Substitute the area formula into the ratio**:
Using the area formula, we can write:
\[ \frac{\pi r_1^2}{\pi r_2^2} = \frac{9}{16} \]
Here, \( r_1 \) and \( r_2 \) are the radii of the first and second circles, respectively.

4. **Cancel out \( \pi \)**:
Since \( \pi \) is common in both the numerator and the denominator, we can cancel it out:
\[ \frac{r_1^2}{r_2^2} = \frac{9}{16} \]

5. **Take the square root of both sides**:
To find the ratio of the radii, we take the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{9}{16}} \]

6. **Calculate the square root**:
The square root of \( \frac{9}{16} \) is:
\[ \frac{r_1}{r_2} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4} \]

7. **State the final answer**:
Therefore, the ratio of the radii of the two circles is:
\[ \frac{r_1}{r_2} = \frac{3}{4} \]