Half a Square
This lesson is optional. Many standardized assessments provide the 45-45-90 special triangle relationship as a reference and ask students to reason about similar triangles. 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.
Special right triangle relationships give students the experience of knowing an angle measurement and a single side length and being able to figure out the other sides, which previews trigonometry. In addition, knowing some ratios of side lengths for certain angle measures provides a reference point as students begin filling in tables of trigonometric ratios.
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 that the diagonal of a rectangle is always the scale factor times the diagonal of a similar unit rectangle. 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 one and a diagonal length of \(\sqrt2\), and all squares are similar to the unit square with their side length as the scale factor. Therefore, all squares with side length \(s\) have a diagonal of length \(s\sqrt2\).
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.
- Calculate side lengths in 45-45-90 triangles.
- Generalize the properties of 45-45-90 triangles including side lengths and ratios (using words and other representations).
- Let’s investigate the properties of diagonals of squares.
- I can determine the side lengths of triangles with 45, 45, and 90 degree angles.
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