To find the number of points with integer coordinates that lie in the interior of the triangle with vertices at (0, 0), (0, 41), and (41, 0), we can use the following steps:
### Step 1: Understand the Triangle
The triangle is formed by the points (0, 0), (0, 41), and (41, 0). The base of the triangle is along the x-axis from (0, 0) to (41, 0), and the height is along the y-axis from (0, 0) to (0, 41).
### Step 2: Calculate the Area of the Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In our case, the base and height are both 41:
\[
A = \frac{1}{2} \times 41 \times 41 = \frac{1681}{2} = 840.5
\]
### Step 3: Use Pick's Theorem
Pick's Theorem states that for a simple polygon whose vertices are lattice points (points with integer coordinates), the area \( A \) can be expressed as:
\[
A = I + \frac{B}{2} - 1
\]
where \( I \) is the number of interior lattice points, and \( B \) is the number of boundary lattice points.
### Step 4: Calculate Boundary Points
1. **On the vertical line from (0, 0) to (0, 41)**: There are 42 points (including both endpoints).
2. **On the horizontal line from (0, 0) to (41, 0)**: There are 42 points (including both endpoints).
3. **On the hypotenuse from (0, 41) to (41, 0)**: The equation of the line is \( y = 41 - x \). The integer points on this line segment can be found by checking integer values of \( x \) from 0 to 41, which gives us 41 points (excluding the endpoints).
Now, we need to account for the vertices counted multiple times:
- Total boundary points \( B = 42 + 42 + 41 - 3 = 122 \).
### Step 5: Substitute into Pick's Theorem
Now we can substitute the values into Pick's Theorem:
\[
840.5 = I + \frac{122}{2} - 1
\]
\[
840.5 = I + 61 - 1
\]
\[
840.5 = I + 60
\]
\[
I = 840.5 - 60 = 780.5
\]
Since \( I \) must be an integer, we round down to get \( I = 780 \).
### Conclusion
The number of points with integer coordinates that lie in the interior of the triangle is **780**.
