Geometry & Trigonometry at the 700-800 Level

This is the largest hard-tier bank on the test, and it splits cleanly into two halves: right-triangle trigonometry and coordinate geometry.

Six archetypes cover nearly all of it. The two-observation height problem (a surveyor walks $d$ closer and re-sights) is solved by subtracting COTANGENTS: $h = \frac{d}{\cot\theta_{far} - \cot\theta_{near}}$. A given trig ratio hands you a triangle's shape, not its size — recognize the triple, then scale. Similar figures scale lengths by $k$ and areas by $k^{2}$, and part-to-part ratios must be converted to part-to-whole. Area as a bridge: one triangle, three bases, three altitudes. Coordinate distance, midpoint, and slope, including running the midpoint backwards as $B = 2M - A$. Perpendicular distance from a point to a line, which is never the vertical or horizontal gap.

Hard geometry rarely punishes a bad theorem. It punishes a correct calculation that stopped one step early or answered a neighbouring quantity — the intermediate leg when the perimeter was asked, the un-scaled triangle, the linear ratio where the area ratio belongs. Reread the last six words of the question before you bubble.