Number of integral points (integral points means both the co-ordinates should be integer) exactly in the interior of the triangle with vertices `(0,0),(0, 21) and (21, 0)` is

Options

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

Explanation

To find the number of integral points exactly in the interior of the triangle with vertices (0,0), (0,21), and (21,0), we can follow these steps:

### Step 1: Understand the Triangle
The triangle is formed by the vertices (0,0), (0,21), and (21,0). The base of the triangle lies along the x-axis from (0,0) to (21,0), and the height extends vertically from (0,0) to (0,21).

### Step 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 is 21 and the height is also 21.
\[ A = \frac{1}{2} \times 21 \times 21 = \frac{441}{2} = 220.5 \]

### Step 3: Apply Pick's Theorem
To find the number of integral points inside the triangle, we can use Pick's Theorem, which states:
\[ A = I + \frac{B}{2} - 1 \]
where:
- \( A \) is the area of the polygon (triangle in this case),
- \( I \) is the number of interior integral points,
- \( B \) is the number of boundary integral points.

### Step 4: Calculate Boundary Points \( B \)
We need to find the number of integral points on the boundary of the triangle.

1. **From (0,0) to (0,21)**: This segment has 22 points (including both endpoints).
2. **From (0,21) to (21,0)**: The line equation is \( x + y = 21 \). The integral points on this line can be found by varying \( x \) from 0 to 21, which gives us 21 points (excluding (0,21)).
3. **From (21,0) to (0,0)**: This segment has 22 points (including both endpoints).

Now, we must be careful not to double-count the vertices:
- Points on the boundary: (0,0), (0,21), (21,0)
- Total boundary points:
\[ B = 22 + 21 + 22 - 3 = 62 \]

### Step 5: Substitute into Pick's Theorem
Now we can substitute the values into Pick's Theorem:
\[ 220.5 = I + \frac{62}{2} - 1 \]
\[ 220.5 = I + 31 - 1 \]
\[ 220.5 = I + 30 \]
\[ I = 220.5 - 30 = 190.5 \]

Since \( I \) must be an integer, we conclude that the number of integral points exactly in the interior of the triangle is:
\[ I = 190 \]

### Final Answer
The number of integral points exactly in the interior of the triangle is **190**.

---