Overview
This lecture reviews the June 2024 NYS Geometry Regents Exam, covering core geometry concepts, problem-solving strategies, and exam-specific tips, including formula usage, similarity, transformations, and special constructions.
Exam Structure & Formula Sheet
- The new formula sheet includes only volume formulas specific to geometry: cylinder, prism, sphere, cone, and pyramid.
- Some formulas, like area of a circle, must be memorized.
Transformations & Symmetry
- Rigid motions (translations, reflections, rotations) preserve size and shape.
- Reflections change orientation; translations and rotations do not.
- Minimum rotation for a regular n-gon to map onto itself: 360°/n.
Similarity, Triangle, and Proportion
- Similar triangles are identified via angle-angle (AA) or other similarity criteria.
- In intersecting chords, corresponding angle and side proportions are equal.
- Sidesplitter theorem: a line parallel to one side of a triangle divides the other two proportionally.
Volume & Area Problems
- Volume of a sphere: ( V = \frac{4}{3}\pi r^3 ).
- Volume of a cylinder: ( V = \pi r^2 h ).
- Volume of a cone/pyramid: ( V = \frac{1}{3} ) base area × height.
- For sectors: Area of sector / Area of circle = angle / 360°.
Coordinate Geometry
- Point-slope form: (y - y_1 = m(x - x_1)).
- Slope of perpendicular lines: negative reciprocals.
- Midpoint: ( (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) ).
- Distance formula: ( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} ).
Special Triangles & Trigonometry
- Use SOHCAHTOA for right triangle trig; ensure calculators are in degree mode.
- Special triangles: 30-60-90 side ratios are ( x, x\sqrt{3}, 2x ); 45-45-90 are ( x, x, x\sqrt{2} ).
- Area of any triangle with SAS: ( \frac{1}{2}ab\sin(C) ).
Dilations & Scaling
- Dilation by scale factor k multiplies lengths by k and area by k².
- Perimeter is scaled by k; area by k².
Quadrilaterals & Proofs
- Rectangle: Parallelogram with equal diagonals.
- Rhombus: Parallelogram with consecutive sides equal or diagonals perpendicular.
Constructions & Points of Concurrency
- Circumcenter: intersection of perpendicular bisectors.
- Incenter: intersection of angle bisectors.
- Centroid: intersection of medians.
- Orthocenter: intersection of altitudes.
Key Terms & Definitions
- Rigid Motion — Transformation preserving distance and angle (size and shape).
- Sector — "Pizza slice" region of a circle, bounded by two radii and an arc.
- Dilation — Transformation enlarging or reducing a figure proportionally.
- Circumcenter — Point where the perpendicular bisectors of a triangle meet.
- Sidesplitter Theorem — A segment parallel to one triangle side divides the other two proportionally.
- Similarity — Figures with proportional sides and equal angles.
- Isosceles Triangle — Triangle with two equal sides.
- Midpoint — The point halfway between two endpoints on a segment.
Action Items / Next Steps
- Review and memorize key formulas not on the reference sheet (e.g., area of circle, distance formula).
- Practice right triangle trigonometry problems and ensure your calculator is set to degrees.
- Study geometric constructions, especially for points of concurrency.
- Complete sample Regents questions and supplementary problems as discussed in the lecture.