The locus of the mid-points of the chords of the circle of lines radiùs r which subtend an angle `pi/4` at any point on the circumference of the circle is a concentric circle with radius equal to (a) `(r)/(2)` (b) `(2r)/(3)` (c) `(r )/(sqrt(2))` (d) `(r )/(sqrt(3))`

Options

A. `(r)/(2)`
B. `(2r)/(3)`
C. `(r )/(sqrt(2))`
D. `(r )/(sqrt(3))`

Explanation

To solve the problem, we need to find the locus of the midpoints of the chords of a circle of radius \( r \) that subtend an angle of \( \frac{\pi}{4} \) at any point on the circumference of the circle.

### Step-by-Step Solution:

1. **Understanding the Circle and Chords**:
- We have a circle centered at the origin \( O \) with radius \( r \).
- Let \( A \) and \( B \) be two points on the circumference of the circle such that the chord \( AB \) subtends an angle of \( \frac{\pi}{4} \) at any point \( P \) on the circumference.

2. **Angle at the Center**:
- The angle subtended by the chord \( AB \) at the center \( O \) of the circle is twice the angle subtended at any point on the circumference. Therefore, the angle \( AOB = 2 \times \frac{\pi}{4} = \frac{\pi}{2} \).

3. **Finding the Midpoint**:
- Let \( M \) be the midpoint of the chord \( AB \).
- The distance from the center \( O \) to the midpoint \( M \) can be found using the cosine of half the angle \( AOB \).
- The distance \( OM \) can be calculated as:
\[ OM = r \cos\left(\frac{AOB}{2}\right) = r \cos\left(\frac{\pi}{4}\right) = r \cdot \frac{1}{\sqrt{2}} = \frac{r}{\sqrt{2}}. \]

4. **Locus of Midpoints**:
- Since \( M \) is at a constant distance \( \frac{r}{\sqrt{2}} \) from the center \( O \), the locus of all such midpoints \( M \) forms a circle.
- This circle is concentric with the original circle and has a radius of \( \frac{r}{\sqrt{2}} \).

5. **Conclusion**:
- Therefore, the radius of the locus of the midpoints of the chords is \( \frac{r}{\sqrt{2}} \).

### Final Answer:
The radius of the concentric circle is \( \frac{r}{\sqrt{2}} \), which corresponds to option (c).