If the arcs of the same length in two circles subtend angles `75^(@)` and `120^(@)` at the centre find the ratio of their radii

Answer

To solve the problem of finding the ratio of the radii of two circles that subtend angles of \(75^\circ\) and \(120^\circ\) at the center for the same arc length, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
Let \( r_1 \) be the radius of the first circle and \( r_2 \) be the radius of the second circle. The arc length \( l \) is the same for both circles.

2. **Use the Arc Length Formula:**
The formula for arc length is given by:
\[ l = r \theta \]
where \( \theta \) is in radians.

3. **Set Up Equations for Both Circles:**
For the first circle with angle \( 75^\circ \):
\[ l = r_1 \theta_1 \]
For the second circle with angle \( 120^\circ \):
\[ l = r_2 \theta_2 \]

4. **Convert Angles from Degrees to Radians:**
We need to convert the angles from degrees to radians using the conversion factor \( \frac{\pi}{180} \):
- For \( 75^\circ \):
\[ \theta_1 = 75^\circ \times \frac{\pi}{180} = \frac{75\pi}{180} = \frac{5\pi}{12} \text{ radians} \]
- For \( 120^\circ \):
\[ \theta_2 = 120^\circ \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3} \text{ radians} \]

5. **Equate the Arc Lengths:**
Since both arc lengths are equal, we can set the two equations equal to each other:
\[ r_1 \theta_1 = r_2 \theta_2 \]

6. **Rearrange to Find the Ratio of Radii:**
We can rearrange this equation to find the ratio of the radii:
\[ \frac{r_1}{r_2} = \frac{\theta_2}{\theta_1} \]

7. **Substitute the Values of Angles:**
Substitute the values of \( \theta_1 \) and \( \theta_2 \):
\[ \frac{r_1}{r_2} = \frac{\frac{2\pi}{3}}{\frac{5\pi}{12}} \]

8. **Simplify the Ratio:**
Cancel out \( \pi \) from the numerator and denominator:
\[ \frac{r_1}{r_2} = \frac{2/3}{5/12} = \frac{2}{3} \times \frac{12}{5} = \frac{24}{15} = \frac{8}{5} \]

9. **Final Result:**
Thus, the ratio of the radii \( r_1 : r_2 \) is:
\[ r_1 : r_2 = 8 : 5 \]