Geometry
Geometry on NYS mathematics exams tests congruence and similarity, triangle and circle relationships, coordinate geometry, and measurement. Many items require a short justification, so name the theorem or property you use.
Core rules and concepts
- Rigid motions preserve congruence; dilations produce similarity. If the scale factor is k, lengths scale by k, areas by k^2, and volumes by k^3.
- Triangle congruence criteria include SSS, SAS, ASA, AAS, and HL for right triangles. CPCTC follows after congruence is established.
- Right triangles: Pythagorean theorem, special 45-45-90 and 30-60-90 ratios, and trig ratios sin, cos, tan.
- Circles: an inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency; central angles relate to arc measures.
- Coordinate geometry uses slope, midpoint, distance, and completing the square for circle equations.
Worked example
Find the center and radius of x^2+y^2-6x+4y-12=0. Group: (x^2-6x)+(y^2+4y)=12. Complete each square by adding (6/2)^2=9 and (4/2)^2=4 to both sides: (x-3)^2+(y+2)^2=25. The center is (3,-2) and the radius is sqrt(25)=5. Reasoning: the equation must be in standard form (x-h)^2+(y-k)^2=r^2 before reading center and radius.
Traps that cost marks
- Reporting r^2 as the radius.
- Using a similarity scale factor directly for area or volume.
- Confusing an inscribed angle with a central angle.
- Assuming a diagram is to scale or that marked-looking angles are given.