Question number five. Six points are marked on a square grid with cells of size one by one, as shown here on the right. Conga wants to choose three of the marked points to be the vertices of a triangle. What is the smallest possible area of such a triangle? By inspection, we see that there are really just two possibilities. We can connect these three vertices, and in red, the second possibility are going to be these three vertices. And it's not exactly clear which area is bigger. It looks like the yellow triangle will have a smaller area, but let's compute both and make sure. The way we're going to do this is we note that each triangle here is contained in a rectangle, and we know the areas of those. And then we can subtract the regions that we don't need and obtain the area of the triangle we want. So let's do that. Let's look at the red area first. We have the red triangle contained in a rectangle of area six. It's a three by two. One of those diagonals is exactly the diagonal of the rectangle, so we subtract half the area. And then the other diagonal is the diagonal of a one by three rectangle, and half the area would be three halves. So we obtain that the red area is three minus three halves or three halves. And now moving on to the yellow area, we repeat similar calculation. Now that triangle is contained in a rectangle of area four, one by four. We subtract half of that as one of the diagonals is exactly the diagonal of the rectangle, and the other edge is a diagonal of a three by one rectangle. Half of the area would be three halves, so we obtain two minus three halves or one halves. And so like we suspected, the yellow triangle has a smaller area. Turns out that C, one half, is the answer to the problem.

triangle

smallest area

square grid

enclosing rectangle

yellow triangle