Geometry
Core ideas
Geometry questions test transformations, congruence and similarity, angle relationships, the Pythagorean theorem, and volume. Rigid motions-translations, rotations, and reflections-preserve side lengths and angle measures, so they produce congruent figures. Dilations change size by a scale factor and produce similar figures with equal corresponding angles and proportional sides. For right triangles, a^2+b^2=c^2, where c is the hypotenuse. The converse also matters: if the three sides satisfy the equation, the triangle is right. Know angle pairs: complementary angles sum to 90°, supplementary angles sum to 180°, vertical angles are equal, and parallel lines cut by a transversal create equal corresponding and alternate interior angles.
Traps
Do not apply a scale factor directly to area or volume. If lengths scale by k, areas scale by k^2 and volumes by k^3. The hypotenuse is always the longest side and is opposite the right angle; substituting a leg for c is a common error. In similar figures, match corresponding vertices before writing proportions. Do not assume a figure is drawn to scale, and do not use side lengths to prove similarity without checking angle equality or proportional sides.
Worked example
A rectangle 4 cm by 6 cm is dilated by a scale factor of 3. Its new dimensions are 12 cm by 18 cm. The area changes from 24 cm² to 216 cm², which is 3^2 = 9 times larger. Rule of thumb: length ratios scale by k, area ratios by k^2, and volume ratios by k^3.