Geometry and Trigonometry: Shapes, Angles, and Right Triangles
Core ideas
Geometry questions require careful use of angle relationships, triangle properties, similarity, the Pythagorean theorem, area and volume formulas, and circle facts. In coordinate geometry, know distance, midpoint, slope, and equations of lines and circles. For right-triangle trigonometry, use SOH-CAH-TOA and recognize special triangles such as 30-60-90 and 45-45-90. Radian and degree conversion is also fair game.
Traps
Do not confuse radius with diameter, or central angle with inscribed angle. When figures are similar, side lengths scale by a factor k, areas by k^2, and volumes by k^3. Verify whether an angle is complementary (sum 90 degrees) or supplementary (sum 180 degrees). In trigonometry, confirm that your calculator is in the correct mode and that you use the side opposite or adjacent to the angle you are given.
Worked example
A right triangle has legs 6 and 8. The hypotenuse is sqrt(6^2 + 8^2) = sqrt(100) = 10. The sine of the angle opposite the side of length 6 is 6/10 = 0.6, and the cosine of that same angle is 8/10 = 0.8.
Rule of thumb
Draw and label a diagram, then mark every known length and angle. Use similarity or trigonometric ratios before resorting to lengthy coordinate calculations.