Right Triangles
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalOne theorem and two memorized triangles cover this topic, and every trap bends one of them. The points leak at the seat, squares added when the hypotenuse was known, or sides combined without squaring at all, in the ratios, the 45-45-90 and 30-60-90 factors crossing wires or running the wrong direction, and at the end, in the finish: c-squared reported instead of c, or the hypotenuse handed in when the perimeter or area was asked.
These patterns aren't really about whether you know the Pythagorean theorem. They're about whether the hypotenuse sat alone on its side of the equation, whether each special triangle kept its own factor anchored to its short side, and whether the work continued past the root to the quantity the question named.
The work
4 ways in · any order
Lesson
Right Triangles
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Seat the hypotenuse first and let it decide add-or-subtract, recall both special-triangle ratios in both directions, scale the familiar triples, and finish to the side, perimeter, or area asked. The lesson works the method and the three patterns that derail it, and it saves the trap for last: the answer that stopped one move early.
Diagnostic
10-item topic check
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Ten items across the three patterns: legs and hypotenuse swapped in the theorem, special-triangle factors misapplied, and squared values or wrong quantities reported. A mix of ladders, diagonals, scaled triples, and perimeter-area finishes. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
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About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.