A point O lies inside a square ABCD such that OA=3, OB=4 and OD=5, find the value of OC.

Options

A. A)`root(3)(3)`
B. B)6.67
C. C)`root(3)(2)`
D. D)4`root()(2)`

Answer

To solve the problem, we will use the property of a point inside a square. The property states that for any point \( O \) inside a square \( ABCD \):

\[ OA^2 + OC^2 = OB^2 + OD^2 \]

Given:
- \( OA = 3 \)
- \( OB = 4 \)
- \( OD = 5 \)

We need to find \( OC \).

### Step-by-step Solution:

1. **Calculate \( OA^2 \)**:
\[ OA^2 = 3^2 = 9 \]

2. **Calculate \( OB^2 \)**:
\[ OB^2 = 4^2 = 16 \]

3. **Calculate \( OD^2 \)**:
\[ OD^2 = 5^2 = 25 \]

4. **Substitute the values into the property**:
Using the property \( OA^2 + OC^2 = OB^2 + OD^2 \):
\[ 9 + OC^2 = 16 + 25 \]

5. **Simplify the right side**:
\[ 16 + 25 = 41 \]
So, we have:
\[ 9 + OC^2 = 41 \]

6. **Isolate \( OC^2 \)**:
\[ OC^2 = 41 - 9 \]
\[ OC^2 = 32 \]

7. **Calculate \( OC \)**:
\[ OC = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \]

Thus, the value of \( OC \) is \( 4\sqrt{2} \).

### Final Answer:
\[ OC = 4\sqrt{2} \]