## IM Commentary

The goal of this task is for students to check that the Pythagorean Theorem holds for two specific examples. Although the work of this task does not provide a proof for the full Pythagorean Theorem, it prepares students for the area calculations they will need to make as well as the difficulty of showing that a quadrilateral in the plane is a square.

For the right isosceles triangle, students can find the area of the three squares built on the sides of the triangle and verify that that the sum of the areas of the smaller squares is equal to the area of the largest square. One special feature of this right triangle is that students can produce a good argument for why the quadrilateral with the hypotenuse as one edge is a square. In the second example, this verification would be challenging and so students are allowed to assume that this quadrilateral is in fact a square. If the teacher wishes to pursue this aspect of the task in greater depth, several options are available:

- Students can investigate the symmetries of the quadrilateral using geometry software or patty paper.
- Students can see via rigid motions that the sides of the quadrilateral are congruent: this can be done via successive reflections of the coordinate grid or via rotations if students can identify the center of a 90 degree rotation which preserves the quadrilateral.
- Students can use the fact that the sum of the angles in a triangle are 180 degrees to show (using either of the pictures in the solution to (b)) that the four angles in the quadrilateral are right angles.

In modern times, the Pythagorean theorem is often stated in algebraic form: $a^2 + b^2 = c^2$ where $a$, $b$, and $c$ are the lengths of the sides of a right triangle, with $c$ being the length of the hypotenuse. In this task, the result is presented for two specific triangles in geometric form representative of the ancient Greek geometers' way of thinking.