Chapter 39: Pythagorean Theorem: Situation 33 From the MACMTL–CPTM Situations Project

In both an Algebra 1 course and an Advanced Algebra course, students were given transparency cutouts of graph paper squares with side lengths from 1 unit to 25 units. Students were asked to create triangles whose sides had the side-lengths of three of the squares (see Figure 39.1). Students began to notice the squares that would create right triangles and the relationship involving the areas of those squares. A student asked, “Does this work for every right triangle?”

The Pythagorean theorem relates the squares of the lengths of the sides of a right triangle. The law of cosines establishes a more general relationship between the lengths of the sides that holds for all triangles. Using algebra and geometry together it is possible to prove both the Pythagorean theorem and the law of cosines.