If O is the origin and the coordinates of A and B are (51, 65) and (75, 81), respectively. Then `OA xx OB cos angle AOB` is equal to

Answer

To solve the problem, we need to find the value of \( OA \times OB \times \cos(\angle AOB) \). Here are the steps to find the solution:

### Step 1: Calculate the distances \( OA \) and \( OB \)

The distance \( OA \) can be calculated using the distance formula:
\[ OA = \sqrt{(x_A - x_O)^2 + (y_A - y_O)^2} \]
Given \( O(0,0) \) and \( A(51, 65) \):
\[ OA = \sqrt{(51 - 0)^2 + (65 - 0)^2} = \sqrt{51^2 + 65^2} \]

Calculating \( 51^2 \) and \( 65^2 \):
\[ 51^2 = 2601 \quad \text{and} \quad 65^2 = 4225 \]
Thus,
\[ OA = \sqrt{2601 + 4225} = \sqrt{6826} \]

Now, calculate \( OB \):
\[ OB = \sqrt{(x_B - x_O)^2 + (y_B - y_O)^2} \]
Given \( B(75, 81) \):
\[ OB = \sqrt{(75 - 0)^2 + (81 - 0)^2} = \sqrt{75^2 + 81^2} \]

Calculating \( 75^2 \) and \( 81^2 \):
\[ 75^2 = 5625 \quad \text{and} \quad 81^2 = 6561 \]
Thus,
\[ OB = \sqrt{5625 + 6561} = \sqrt{12186} \]

### Step 2: Calculate the distance \( AB \)

The distance \( AB \) can be calculated as:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
\[ AB = \sqrt{(75 - 51)^2 + (81 - 65)^2} = \sqrt{24^2 + 16^2} \]

Calculating \( 24^2 \) and \( 16^2 \):
\[ 24^2 = 576 \quad \text{and} \quad 16^2 = 256 \]
Thus,
\[ AB = \sqrt{576 + 256} = \sqrt{832} \]

### Step 3: Use the cosine rule to find \( \cos(\angle AOB) \)

According to the cosine rule:
\[ AB^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cdot \cos(\angle AOB) \]
Rearranging gives:
\[ \cos(\angle AOB) = \frac{OA^2 + OB^2 - AB^2}{2 \cdot OA \cdot OB} \]

### Step 4: Substitute the values

Now we substitute the values we calculated:
- \( OA^2 = 6826 \)
- \( OB^2 = 12186 \)
- \( AB^2 = 832 \)

Substituting into the cosine formula:
\[ \cos(\angle AOB) = \frac{6826 + 12186 - 832}{2 \cdot \sqrt{6826} \cdot \sqrt{12186}} \]
Calculating the numerator:
\[ 6826 + 12186 - 832 = 18280 \]

### Step 5: Calculate \( OA \times OB \times \cos(\angle AOB) \)

Now we can find \( OA \times OB \times \cos(\angle AOB) \):
\[ OA \times OB \times \cos(\angle AOB) = \frac{18280}{2} \]
Thus,
\[ OA \times OB \times \cos(\angle AOB) = 9140 \]

### Final Answer
The value of \( OA \times OB \times \cos(\angle AOB) \) is \( 9140 \).