The number of integral points that lie exactly in the interior of the triangle with vertices O (0,0) A (21,0) B (0,21) is

Options

A. 105
B. 133
C. 190
D. 233

Answer

To find the number of integral points that lie exactly in the interior of the triangle with vertices O(0,0), A(21,0), and B(0,21), we can use the formula derived from Pick's Theorem. However, since we are looking for the interior points specifically, we can derive the solution step by step as follows:

### Step-by-Step Solution:

1. **Identify the vertices of the triangle**:
The vertices of the triangle are O(0,0), A(21,0), and B(0,21).

2. **Determine the area of the triangle**:
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base OA is 21 and the height OB is also 21. Therefore,
\[ A = \frac{1}{2} \times 21 \times 21 = \frac{441}{2} = 220.5 \]

3. **Calculate the number of boundary points**:
The number of integral points on the boundary of the triangle can be calculated as follows:
- From O to A (along the x-axis): There are 22 points (including both endpoints).
- From O to B (along the y-axis): There are 22 points (including both endpoints).
- From A to B (the hypotenuse): The equation of the line AB is \( x + y = 21 \). The integral points on this line can be found by setting \( x = k \) where \( k \) ranges from 1 to 20 (since both endpoints O and B are already counted). This gives us 20 points.

Thus, the total number of boundary points \( B \) is:
\[ B = 22 + 22 + 20 - 3 = 61 \]
(We subtract 3 to avoid double counting the vertices O, A, and B.)

4. **Apply Pick's Theorem**:
Pick's Theorem states that:
\[ A = I + \frac{B}{2} - 1 \]
where \( I \) is the number of interior points, \( A \) is the area, and \( B \) is the number of boundary points. Rearranging the formula to solve for \( I \):
\[ I = A - \frac{B}{2} + 1 \]
Substituting the values we found:
\[ I = 220.5 - \frac{61}{2} + 1 \]
\[ I = 220.5 - 30.5 + 1 = 191 \]

5. **Conclusion**:
The number of integral points that lie exactly in the interior of the triangle is **191**.