The number of points having both coordinates as integers, that lie in the interior of the triangle with vertices at (0,0), (0,41) and (41,0) isa)901b)861c)820d)780

To determine the number of points with integer coordinates lying in the interior of the given triangle, we can use Pick's theorem. Pick's theorem states that for a polygon with vertices at lattice points (points with integer coordinates), the number of lattice points strictly inside the polygon can be calculated using the formula:

A = I + B/2 - 1

where:
A is the area of the polygon,
I is the number of lattice points strictly inside the polygon, and
B is the number of lattice points on the boundary of the polygon.

In this case, the triangle has vertices at (0,0), (0,41), and (41,0), which forms a right-angled triangle.

The base of the triangle is 41 units long, and the height is 41 units as well. Therefore, the area of the triangle is:

A = (1/2) * base * height
= (1/2) * 41 * 41
= 841/2
= 420.5

Now, let's calculate the number of lattice points on the boundary of the triangle:

The base of the triangle has length 41, so there are 42 lattice points on the base.
The height of the triangle has length 41, so there are 42 lattice points on the height.
The vertex at (0,0) is a lattice point.
Therefore, the total number of lattice points on the boundary of the triangle is 42 + 42 + 1 = 85.

Now, we can substitute the values into Pick's theorem:

A = I + B/2 - 1
420.5 = I + 85/2 - 1
420.5 = I + 42.5 - 1
420.5 - 41.5 = I
379 = I

Therefore, the number of lattice points strictly inside the triangle is 379.

The correct answer choice from the given options is d) 780.