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 ($m<em>1 = -\frac{1}{m</em>2}>$).

• Distance Formula: $d = \sqrt{(x<em>2-x</em>1)^2 + (y<em>2-y</em>1)^2}>$

• Midpoint Formula: $M = \left(\frac{x<em>1+x</em>2}{2}, \frac{y<em>1+y</em>2}{2}\right)>$

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