![]() ![]() The sum of the areas of the squares built on the triangle's legs is 2 square units which is the same as the area of the square built on the triangle's hypotenuse. ![]() The coordinate grid lines divide the square on the hypotenuse into four half unit-squares, and so its area is 2 square units. The squares built on the triangle legs are unit squares in the coordinate grid so they each have an area of 1 square unit.These each measure 45 degrees so the four angles of the quadrilateral all measure 90 degrees and it is a square. Each angle of the quadrilateral is made up of two base angles of right isosceles triangles. Next we check that the four angles are right angles. All four sides of the quadrilateral built on the hypotenuse are diagonals of coordinate grid squares so they are all congruent. This means that the diagonals of the coordinate grid squares are all congruent. The squares in the coordinate grid are all congruent with side length of one unit. We will first check that all four sides of the quadrilateral are congruent and then show that it has four right angles. In this task, the result is presented for two specific triangles in geometric form representative of the ancient Greek geometers' way of thinking. 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. 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.Â.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 investigate the symmetries of the quadrilateral using geometry software or patty paper.If the teacher wishes to pursue this aspect of the task in greater depth, several options are available: 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. 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.įor 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. The goal of this task is for students to check that the Pythagorean Theorem holds for two specific examples. ![]()
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