Overview
This lecture reviews the June 21, 2024 Geometry Regents Exam, covering major topics, strategies, and key problem-solving techniques relevant to New York State's Next Generation Learning Standards.
Exam & Formula Sheet Changes
- Next Generation standards are now in effect, similar to Common Core with some differences.
- The formula sheet for this exam includes only volume formulas specific to geometry (e.g., cylinder, prism, sphere, cone, pyramid).
Volume & Area Problems
- Volume formulas provided: cylinder (V = Bh), cone/pyramid (V = 1/3Bh), sphere (V = 4/3πr³), where B = area of base.
- When solving volume questions, identify radius/diameter carefully and use correct substitution.
- For area of sectors, use the proportion of the angle over 360° multiplied by total circle area.
- For non-right triangles with two sides and included angle, use Area = 1/2 ab sin(C).
Transformations & Symmetry
- Translation, reflection, and rotation are rigid motions which preserve angle and length.
- For regular polygons, minimum angle of rotation = 360°/number of sides.
- Congruence and similarity are preserved under rigid motions; dilations multiply lengths, areas by scale factor squared.
Triangles & Similarity
- Use proportions when right triangles share an angle or are divided by an altitude.
- The triangle inequality: sum of any two side lengths > third side.
- Special triangle relationships: 30-60-90 triangle side ratio is 1 : √3 : 2; 45-45-90 triangle is 1 : 1 : √2.
Coordinate Geometry & Slope
- For line equations, use point-slope form: y - y₁ = m(x - x₁).
- Perpendicular slopes are negative reciprocals.
- Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Distance: sqrt((x₂-x₁)² + (y₂-y₁)²).
- To prove segment is perpendicular bisector, show negative reciprocal slopes and midpoint.
Parallel Lines, Parallelograms & Special Quadrilaterals
- Prove rectangles by showing parallelogram with equal diagonals; prove rhombus by all sides equal.
- Sides splitter theorem: a line parallel to one side of a triangle divides the other two sides proportionally.
Trigonometry
- Right triangle trig: use sine, cosine, tangent ratios based on given sides/angles.
- When given sin(a) = cos(b), then a + b = 90°.
Circle Theorems
- Inscribed angles intercepting the same arc are congruent.
- Tangent-secant and chord-chord theorems relate segment lengths.
Proofs & Constructions
- Proof strategies: use given info, mark diagrams, and cite properties (e.g., alternate interior angles, reflexive property, ASA).
- Know constructions for equilateral triangle, perpendicular bisector, and points of concurrency (circumcenter, centroid, incentive, orthocenter).
Key Terms & Definitions
- Rigid Motion — A transformation preserving size and shape (translation, reflection, rotation).
- Sector — Portion of a circle bounded by two radii and the included arc.
- Triangle Inequality — In any triangle, the sum of two side lengths must exceed the third.
- Circumcenter — Intersection of perpendicular bisectors of a triangle's sides.
- Similarity — Figures with equal corresponding angles and proportional side lengths.
- Point-Slope Form — Equation of a line: y - y₁ = m(x - x₁).
Action Items / Next Steps
- Review and practice new formula sheet content, especially volume.
- Memorize special triangle relationships (30-60-90, 45-45-90).
- Practice right triangle trigonometry and constructions for all points of concurrency.
- Set calculator to degree mode for trigonometry problems.
- Complete any assigned supplementary problems or readings on new standards.