In a circle with centre O, AB and CD are two diameters perpendicular to each other. The length of chord AC is

Options

A. `2 AB`
B. `sqrt(2) AB `
C. `1/2 AB`
D. `1/(sqrt2) AB`

Answer

To find the length of the chord AC in a circle with center O, where AB and CD are two diameters that are perpendicular to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circle and Diameters**:
- We have a circle with center O.
- AB and CD are two diameters that intersect at O and are perpendicular to each other.

2. **Label the Points**:
- Let A, B, C, and D be points on the circle such that:
- A and B are endpoints of diameter AB.
- C and D are endpoints of diameter CD.
- The points A, C, B, and D can be visualized on the coordinate plane. For simplicity, we can place:
- O at the origin (0, 0).
- A at (-R, 0) and B at (R, 0) (where R is the radius).
- C at (0, R) and D at (0, -R).

3. **Finding Coordinates of Points A and C**:
- The coordinates of point A are (-R, 0).
- The coordinates of point C are (0, R).

4. **Using the Distance Formula**:
- To find the length of chord AC, we can use the distance formula:
\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Substituting the coordinates of A and C:
\[ AC = \sqrt{(0 - (-R))^2 + (R - 0)^2} \]
\[ AC = \sqrt{(R)^2 + (R)^2} \]
\[ AC = \sqrt{2R^2} \]
\[ AC = R\sqrt{2} \]

5. **Relating to Diameter AB**:
- The diameter AB is given as 2R.
- We can express AC in terms of the diameter:
\[ AC = \frac{1}{\sqrt{2}} \times 2R \]

6. **Final Answer**:
- Therefore, the length of chord AC is:
\[ AC = \frac{2R}{\sqrt{2}} = R\sqrt{2} \]

### Conclusion:
The length of chord AC is \( R\sqrt{2} \), which can also be expressed as \( \frac{2R}{\sqrt{2}} \).