Which one has the largest area?

By Pick's Theorem , the area of any lattice polygon is $A = I + \frac{1}{2}B - 1$ , where $I$ is the number of points in the interior of the polygon and $B$ is the number of points on the boundary of the polygon.

All of the triangles in the diagram are lattice polygons with $0$ interior points and $3$ boundary points, so they all have the same area of $A = 0 + \frac{1}{2} \cdot 3 - 1 = \frac{1}{2}$ .

Let the distance between each pair of horizontal and vertical dots be $1$

A triangle's area is defined by $\dfrac{\text {b} \cdot \text{h}}{2}$ where $\text{b}$ represents the triangle's base and $\text{h}$ represents the triangle's height

In $\triangle A, \triangle C$ and $\triangle E$ ,

Base = $1$ and Height = $1$

Thus their areas are the same, $\dfrac{1}{2}$

In $\triangle B$ and $\triangle D$ ,

Base = $\sqrt{1^2 + 1^2} = \sqrt{2}$ and Height = $\dfrac{\sqrt{2}}{2}$

These triangles also have the same area, $\dfrac{1}{2}$

Thus all triangles have the same area