If the arcs of the same lengths m two circles subtend angles `65o`and `110o`at the centre, find the ratio of their radii.

Explanation

To solve the problem, we need to find the ratio of the radii of two circles given that the arcs of the same lengths subtend angles of 65° and 110° at the center of the respective circles.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The length of an arc \( s \) in a circle is related to the radius \( r \) and the angle \( \theta \) (in radians) subtended at the center by the formula:
\[ s = r \cdot \theta \]
However, since we are given angles in degrees, we can still use this relationship directly for our calculations since the arcs are equal.

2. **Setting Up the Equations**:
Let:
- \( s_1 \) be the length of the arc in the first circle with radius \( r_1 \) and angle \( \theta_1 = 65° \).
- \( s_2 \) be the length of the arc in the second circle with radius \( r_2 \) and angle \( \theta_2 = 110° \).

Since the arcs are of the same length, we have:
\[ s_1 = s_2 \]
Therefore, we can write:
\[ r_1 \cdot \theta_1 = r_2 \cdot \theta_2 \]

3. **Substituting the Angles**:
Substitute \( \theta_1 \) and \( \theta_2 \) into the equation:
\[ r_1 \cdot 65° = r_2 \cdot 110° \]

4. **Rearranging for the Ratio**:
We want to find the ratio \( \frac{r_1}{r_2} \):
\[ \frac{r_1}{r_2} = \frac{110°}{65°} \]

5. **Simplifying the Ratio**:
To simplify \( \frac{110°}{65°} \):
\[ \frac{110}{65} = \frac{110 \div 5}{65 \div 5} = \frac{22}{13} \]

6. **Final Result**:
Thus, the ratio of the radii \( r_1 \) to \( r_2 \) is:
\[ \frac{r_1}{r_2} = \frac{22}{13} \]

### Final Answer:
The ratio of the radii of the two circles is \( \frac{22}{13} \).