Squares on a coordinate grid
Task
In the picture below a square is outlined whose vertices lie on the coordinate grid points:

The area of this particular square is 16 square units. For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.
IM Commentary
The purpose of this task is to use the Pythagorean Theorem and knowledge about quadrilaterals in order to construct squares of different sizes on a coordinate grid. In addition to the Pythagorean Theorem, students will need to show that the shapes they have constructed are squares. This can be done by using the Pythagorean Theorem a second time to check that, in addition to having four congruent sides, the quadrilaterals also have congruent diagonals. Alternatively, as is done in the solution, they can check that the lines containing intersecting sides of the square are perpendicular. Addressing all of these issues will require patience and skill, making the task ideally suited for group work. In addition, the task builds upon many different strands of knowledge and so it is recommended for use with experienced students.
This task leads to many interesting questions:
- In general, for which numbers n can we find a square on the grid whose area is n square units? For which n is there no such square? This line of reasoning leads to some very interesting questions in number theory.
- What other polygons can we find with vertices on the coordinate grid? Can we find an equilateral triangle? A regular hexagon? A parallelogram with specified angles? These are all challenging and interesting questions.
The idea for this task came from an Albuquerque Math Teachers' Circle meeting on October 1, 2013.
Solution
For n = 1, 2, 4, 5, 8, 9, 10 there are squares of area n units^2. The easiest squares to visualize are those having areas of 1, 4, and 9 units^2:

The next diagram shows examples of squares whose areas are 2, 5, and 8 units^2:

The last diagram shows a square of area 10 units^2:

We will first explain why the oblique squares have the desired areas and then work on showing that they are squares. The different cases are similar so we focus on the diagram above with the square of area 10 units^2. We first choose a pair of adjacent vertices (2,2) and (3,5). According to the Pythagorean Theorem the length of this side is \sqrt{(2-3)^2 + (2-5)^2} = \sqrt{10} \text{ units}.
For the angles, the side joining (2,2) to (5,1) has slope \frac{2-1}{2-5} = -\frac{1}{3}.
A second way to show that these are squares is to use the fact that a square is a quadrilateral which is both a rectangle and a rhombus and then use properties of rectangles and rhombi. For example, a square is a rhombus whose diagonals are congruent. Working with the last quadrilateral again (which we have shown to be a rhombus by calculating its side lengths), the diagonal joining (2,2) to (6,4) has length \sqrt{(6-2)^2+(4-2)^2} = \sqrt{20} \text{ units}.
Lastly, we need to explain why there are no squares of area 3, 6, and 7 units^{2} whose vertices lie at grid points. Suppose (a_1,b_1) and (a_2,b_2) are vertices of a side of a square. Then the length of that side is \sqrt{(a_2-a_1)^2 + (b_2-b_1)^2} \text{ units}^2
Squares on a coordinate grid
In the picture below a square is outlined whose vertices lie on the coordinate grid points:

The area of this particular square is 16 square units. For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.