Understand and informally prove the Pythagorean theorem using an area-based argument, and verify it with known right triangles.
For any right triangle, the square of the hypotenuse (the longest side, c) equals the sum of the squares of the other two sides (the legs, a and b): a^2 + b^2 = c^2.
One classic proof uses area: a big square with side (a+b) can be split two ways -- once leaving a single tilted square of area c^2 in the middle, once leaving two squares of area a^2 and b^2 side by side. Both versions use the same big square and four identical copies of the triangle, so the leftover areas must be equal, which proves a^2 + b^2 = c^2.
Mastery looks like: They apply a^2 + b^2 = c^2 to find a missing side of a right triangle (leg or hypotenuse) and can explain the area-based proof idea in their own words.
Common stumbles: Plugging a known leg into the c^2 spot instead of a hypotenuse; forgetting to take the square root as the final step.