IL

Geometry year summary for regents

1. Basic Definitions and Concepts

  • Points, Lines, Planes: Undefined terms. Line (1D), Plane (2D).

  • Segments and Rays: Parts of a line.

  • Angles: Formed by two rays from a vertex.

    • Types: Acute (<90^\text{o}>), Obtuse (>90^\text{o}>), Right (=90^\text{o}>), Straight (=180^\text{o}>).

    • Pairs: Complementary (90^\text{o}>), Supplementary (180^\text{o}>), Vertical, Linear Pair.

2. Lines and Angles

  • Parallel Lines: Never intersect. Transversal creates congruent alternate interior/exterior/corresponding angles; supplementary consecutive interior angles.

  • Perpendicular Lines: Intersect at right angles.

  • Slope: m = \frac{\Delta y}{\Delta x}. Parallel lines have equal slopes; Perpendicular lines have negative reciprocal slopes (m1 = -\frac{1}{m2}>).

  • Distance Formula: d = \sqrt{(x2-x1)^2 + (y2-y1)^2}>

  • Midpoint Formula: M = \left(\frac{x1+x2}{2}, \frac{y1+y2}{2}\right)>

3. Transformations

  • Isometries (Rigid Motions): Preserve size and shape.

    • Translation: Slide. (x,y) \to (x+a, y+b)>

    • Rotation: Turn. Key rules: 90^\text{o}> CCW (x,y) \to (-y,x)>; 180^\text{o}> (x,y) \to (-x,-y)>

    • Reflection: Flip. Key rules: x-axis (x,y) \to (x,-y)>; y-axis (x,y) \to (-x,y)>; y=x (x,y) \to (y,x)>

  • Dilation: Change in size but not shape (non-isometric). (x,y) \to (kx,ky)> (k = scale factor).

4. Triangles

  • Area: A = \frac{1}{2}bh

  • Pythagorean Theorem: For right triangles, a^2 + b^2 = c^2>

  • Congruence: SSS, SAS, ASA, AAS, HL.

  • Similarity: AA, SAS, SSS.

  • Triangle Inequality: Sum of any two sides > third side.

  • Points of Concurrency: Centroid (medians), Orthocenter (altitudes), Incenter (angle bisectors), Circumcenter (perpendicular bisectors).

5. Quadrilaterals

  • Parallelogram: Opposite sides parallel/congruent; opposite angles congruent; diagonals bisect each other.

  • Rectangle: Parallelogram + 4 right angles; congruent diagonals.

  • Rhombus: Parallelogram + 4 congruent sides; perpendicular diagonals bisect angles.

  • Square: A rhombus and a rectangle.

  • Trapezoid: One pair parallel sides. Isosceles trapezoid: non-parallel sides/base angles/diagonals congruent.

6. Circles

  • Circumference: C = 2\pi r or C = \pi d

  • Area: A = \pi r^2

  • Chords, Tangents, Secants: Key lines/segments.

  • Angles: Central Angle = Arc; Inscribed Angle = \frac{1}{2} Arc.

  • Segment Relations: Intersecting Chords: a \cdot b = c \cdot d>; Tangent-Secant: (tangent)^2 = whole \cdot external>

  • Equation: (x-h)^2 + (y-k)^2 = r^2>

7. Right Triangle Trigonometry

  • SOH CAH TOA: Sine, Cosine, Tangent definitions.

  • Law of Sines: \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}>

  • Law of Cosines: c^2 = a^2 + b^2 - 2ab\cos C>

  • Area: A = \frac{1}{2}ab\sin C>

8. Three-Dimensional Geometry

  • Volume (V) & Surface Area (SA) Formulas:

    • Prism: V = Bh; SA = 2B + Ph

    • Cylinder: $$V = \pi r^2