A point O lies inside a square ABCD such that OB=7 and OD=6, OC=2 find the value of OA?

Options

A. 9
B. 6
C. 5
D. 12

Answer

To find the value of OA given the distances OB, OD, and OC, we can use the property of a square concerning the distances from a point inside the square to its vertices. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
We have a square ABCD with a point O inside it. The distances from point O to the vertices are given:
- OB = 7
- OD = 6
- OC = 2

We need to find the distance OA.

### Step 2: Use the Property of the Square
There is a property that states for any point O inside a square ABCD:
\[ OA^2 + OC^2 = OB^2 + OD^2 \]

### Step 3: Substitute the Known Values
We can substitute the known distances into the equation:
- OA^2 + OC^2 = OB^2 + OD^2
- OA^2 + 2^2 = 7^2 + 6^2

### Step 4: Calculate the Squares
Now, we calculate the squares of the distances:
- OC^2 = 2^2 = 4
- OB^2 = 7^2 = 49
- OD^2 = 6^2 = 36

### Step 5: Set Up the Equation
Substituting these values into the equation gives:
\[ OA^2 + 4 = 49 + 36 \]

### Step 6: Simplify the Right Side
Now, simplify the right side:
\[ OA^2 + 4 = 49 + 36 = 85 \]

### Step 7: Solve for OA^2
Now, isolate OA^2:
\[ OA^2 = 85 - 4 \]
\[ OA^2 = 81 \]

### Step 8: Take the Square Root
Finally, take the square root to find OA:
\[ OA = \sqrt{81} = 9 \]

### Conclusion
Thus, the value of OA is 9.

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