Lesson objective: Apply the Pythagorean Theorem to solve for missing side lengths in right triangles.

Students bring prior knowledge of estimating decimal equivalents of square roots from 8.NS.A.2. This prior knowledge is extended to work with the Pythagorean Theorem as students solve for unknown distances. A conceptual challenge students may encounter is that diagonal distance will be derived through writing and solving an equation including square numbers and square roots.

The concept is developed through work with an emphasis on comparing the distance of legs to the distance of a hypotenuse, which clarifies the concept that the legs of a right triangle hold a consistent relationship.

This work helps students deepen their understanding of operations and equivalence because they practice the operation of taking a square root of both sides of an equation, while understanding that both sides of the equation will remain equal to one another.

Students engage in Mathematical Practice 8 (look for and express regularity in repeated reasoning) as they use and reuse the Pythagorean Theorem to solve a variety of right triangle problems, and understand that \(a^2 + b^2 = c^2\) holds true for all of the problems. Students also engage in Mathematical Practice 6 (attend to precision) as they calculate squares, approximate square roots, and compare lengths.

**Key vocabulary:**

- diagonal
- distance
- horizontal
- vertical