Polygon with five sides

Geometry Level pending

$A$ , $B$ , $C$ , $D$ and $E$ are lattice points on the grid so that $ABCDE$ is a convex pentagon.

What is the minimum value of the area of $ABCDE$ ?

3,5 2,25 2,5 3 1 1,75 1,5 2

2 solutions

Marta Reece
Jun 10, 2017

But how do you know if this is the minimum possible area?

Pi Han Goh - 3 years, 12 months ago

Áron Bán-Szabó
Jun 10, 2017

If there is a point in the polygon (or on its perimeter), then we can find an other polygon which has a smaller area and its vertices are on the grid. Since there are finite number of points in the polygon. Let's look at that polygon with 5 sides, which perimeter doesn't include any point except the vertices. The parities of the coordinates have 4 ways: EE, EO, OE, OO (O:odd, E:even). There will be two vertices which are in the same way. The midpoint of these points will be in the polygon. If we link this point with the vertices, we get five triangles on the grid. Since the minimum value of a triangle on the grid is $0.5$ , the minimum value of the polygon's area is $5(0.5)=2.5$ .

This can be easily made.

Please note that your answer at the end of the solution is in the notation 2.5 while your choices of answers are 2,5 etc. This is somewhat confusing to people not used to that notation, as they may think that you are listing two numbers there, separated by a comma.

Marta Reece - 4 years ago

Thank you! How can I correct it?

Áron Bán-Szabó - 4 years ago

I do not believe you can, without deleting the problem and starting over. What you can do is be aware of it in the future. Someone on Brilliant staff could make the changes, however.

Marta Reece - 4 years ago

Polygon with five sides

2 solutions

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