Overview
This lecture is a comprehensive review of the June 2024 Geometry Regents Exam, covering essential geometry concepts, problem-solving techniques, formulas, and new standards for the New York State exam.
Formula Sheet & Exam Changes
- The formula sheet now contains only basic volume formulas specific to geometry (sphere, cylinder, prism, cone, pyramid).
- Students must recall area formulas for shapes (e.g., area of a circle = πr²) since not all are provided.
- The exam aligns with updated Next Generation Learning Standards, with minor changes from Common Core.
Key Problem-Solving Techniques
Transformations & Similarity
- Translations, reflections, and rotations are "rigid motions"—they preserve size and shape.
- Dilations change size but preserve angle measures; area scales by the square of the scale factor.
- Similar triangles have proportional sides and equal corresponding angles.
2D & 3D Geometry
- Cross-section of a cylinder, cut parallel to the base, is a circle.
- Inscribed sphere in a cube: radius = ½ edge length.
- Volume formulas:
- Sphere: ( V = \frac{4}{3} \pi r^3 )
- Cone/Pyramid: ( V = \frac{1}{3} ) × base area × height
- Cylinder/Prism: ( V = ) base area × height
Angles & Triangles
- Interior angles of a triangle sum to 180°.
- Triangle inequality: sum of any two sides > third side.
- Isosceles triangle: two equal sides.
- 30-60-90 and 45-45-90 triangles have set relationships between sides.
Trigonometry
- Use sine, cosine, and tangent for solving unknowns in right triangles.
- Inverse trig functions find angle measures.
- Calculators must be in degree mode for trigonometry problems.
Coordinate & Circle Geometry
- Use the distance formula to determine lengths between points.
- Equation of a circle: ( (x-h)^2 + (y-k)^2 = r^2 ), found by completing the square.
- Sectors: Area is proportional to the central angle (( \frac{\theta}{360} \times ) area of circle).
- For chords, tangents, secants: intersecting segments have proportional relationships.
Constructions & Points of Concurrency
- Circumcenter: intersection of perpendicular bisectors of triangle sides.
- Other centers: incenter (angle bisectors), centroid (medians), orthocenter (altitudes).
Proofs & Logical Reasoning
- Use given information plus properties (reflexive, addition, parallel lines, etc.).
- Demonstrate triangle congruence by SSS, SAS, ASA, AAS, or HL as appropriate.
- Conclude with a clear statement answering the proof prompt.
Key Terms & Definitions
- Dilation — A transformation that scales figures by a factor from a center point.
- Similarity — Two shapes with equal angles and proportional sides.
- Rigid Motions — Movements (translation, reflection, rotation) preserving shape and size.
- Sector — A "slice" of a circle bounded by two radii and the included arc.
- Circumcenter — The point where a triangle's perpendicular bisectors intersect.
- Altitude — A segment from a triangle’s vertex perpendicular to the opposite side.
- Median — A segment from a vertex to the midpoint of the opposite side.
- Isosceles Triangle — A triangle with at least two equal sides.
- Triangle Inequality — The sum of any two sides of a triangle must be greater than the third.
Action Items / Next Steps
- Review the bare-bones formula sheet—memorize any missing area or angle formulas.
- Practice right triangle trigonometry with calculator in degree mode.
- Review the steps of geometric constructions, especially those related to points of concurrency.
- Complete any additional practice problems provided by your teacher on new standards.
- Bring a ruler, compass, and calculator (reset to degree mode) for the exam.