In a circle with centre `O ,\ A B\ a n d\ C D` are two diameters perpendicular to each other. The length of chord `A C` is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understand the Geometry**:
- Since AB and CD are diameters of the circle and are perpendicular, they intersect at the center O. This means that angle AOC = 90 degrees.

2. **Identify the Radii**:
- Let the length of diameter AB be denoted as d. The radius (r) of the circle is half of the diameter. Therefore, \( AO = \frac{d}{2} \) and \( OC = \frac{d}{2} \).

3. **Apply the Pythagorean Theorem**:
- In triangle AOC, we can apply the Pythagorean theorem because it is a right triangle (angle AOC = 90 degrees).
- According to the Pythagorean theorem:
\[ AC^2 = AO^2 + OC^2 \]

4. **Substitute the Values**:
- Since \( AO = OC = \frac{d}{2} \), we substitute these values into the equation:
\[ AC^2 = \left(\frac{d}{2}\right)^2 + \left(\frac{d}{2}\right)^2 \]
- This simplifies to:
\[ AC^2 = \frac{d^2}{4} + \frac{d^2}{4} = \frac{2d^2}{4} = \frac{d^2}{2} \]

5. **Take the Square Root**:
- To find AC, we take the square root of both sides:
\[ AC = \sqrt{\frac{d^2}{2}} = \frac{d}{\sqrt{2}} \]

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