This lesson is optional because it goes beyond grade-level expectations. This lesson gives students an opportunity to apply their reasoning about similar figures and their skill with the Pythagorean Theorem to make sense of the relationship and understand how they might use a reference diagram to solve problems. Many standardized assessments provide the 45-45-90 special triangle relationship as a reference and ask students to reason about similar triangles. 

As students solve multiple problems involving finding the lengths of diagonals in similar rectangles, and then squares, they have an opportunity to use repeated reasoning (MP8) to generalize a process. The strategy of making a table can help students to recognize and make use of their repeated reasoning. In the case of the square, a unit square has side lengths of 1 unit and a diagonal length of  units. All squares are similar to the unit square: their side length is the scale factor. Therefore, all squares with side length have a diagonal of length .

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.