Geometry Brain Dump – First Semester

Formulas

  • Pythagorean Theorem: a2+b2=c2a^2 + b^2 = c^2
  • Slope-Intercept Form: y=mx+by = mx + b
  • Point-Slope Form: y−y<em>1=m(x−x</em>1)y − y<em>1 = m(x − x</em>1)
  • Side-Splitter Theorem:
    • APPB=AQQC\frac{AP}{PB} = \frac{AQ}{QC}
    • PBAB=QCAC\frac{PB}{AB} = \frac{QC}{AC}
  • Geometric Mean Altitude Theorem: HCHA=HAHB  ⟹  HC⋅HB=(HA)2\frac{HC}{HA} = \frac{HA}{HB} \implies HC \cdot HB = (HA)^2
  • Geometric Mean Leg Theorem: HBBA=BABC  ⟹  HB⋅BC=(BA)2\frac{HB}{BA} = \frac{BA}{BC} \implies HB \cdot BC = (BA)^2

Angle Relationships

  • Linear Pair: Sum of 180∘180^\circ, forms straight line
  • Supplementary: Sum of 180∘180^\circ
  • Complementary: Sum of 90∘90^\circ
  • Acute: Less than 90∘90^\circ
  • Right: 90∘90^\circ
  • Obtuse: In between 90∘90^\circ and 180∘180^\circ
  • Straight: 180∘180^\circ
  • Vertical: Congruent, equal measures
  • Alternate Interior: Congruent, equal measures
  • Alternate Exterior: Congruent, equal measures
  • Corresponding: Congruent, equal measures
  • Same Side (Consecutive) Interior: Sum of 180∘180^\circ
  • Same Side (Consecutive) Exterior: Sum of 180∘180^\circ

Rotation Rules

  • 90∘90^\circ Clockwise or 270∘270^\circ Counterclockwise: (x,y)→(y,−x)(x, y) \rightarrow (y, -x)
  • 90∘90^\circ Counterclockwise or 270∘270^\circ Clockwise: (x,y)→(−y,x)(x, y) \rightarrow (-y, x)
  • 180∘180^\circ: (x,y)→(−x,−y)(x, y) \rightarrow (-x, -y)

Reflection Rules

  • Over x-axis: (x,y)→(x,−y)(x, y) \rightarrow (x, -y)
  • Over y-axis: (x,y)→(−x,y)(x, y) \rightarrow (-x, y)
  • Over y=xy = x: (x,y)→(y,x)(x, y) \rightarrow (y, x)
  • Over y=−xy = -x: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x)

Dilation Rule

  • Dilation with respect to the origin and scale factor of k: (x,y)→(kx,ky)(x, y) \rightarrow (kx, ky)

Triangle Types

  • Right Triangle: One right angle
  • Acute Triangle: Three acute angles
  • Obtuse Triangle: One obtuse angle
  • Equiangular Triangle: All 60∘60^\circ angles
  • Isosceles Triangle: Two congruent sides/angles
  • Scalene Triangle: No congruent sides/angles
  • Equilateral Triangle: Three congruent sides

Triangle Congruence

  • Side-Side-Side (SSS)
  • Side-Angle-Side (SAS)
  • Angle-Side-Angle (ASA)
  • Angle-Angle-Side (AAS)
  • Hypotenuse-Leg (HL)

Triangle Similarity

  • Angle-Angle (AA~)
  • Side-Side-Side (SSS~)
  • Side-Angle-Side (SAS~)

Transformations

  • Rigid Transformations (Preserves Distance): Translations, Reflections, Rotations
  • Non-Rigid Transformation (Does Not Preserve Distance if scale factor is not 1): Dilations

Conditional Statements (Given p→qp \rightarrow q)

  • Converse: q→pq \rightarrow p
  • Inverse: ∼p→∼q\sim p \rightarrow \sim q
  • Contrapositive: ∼q→∼p\sim q \rightarrow \sim p
  • Biconditional: p↔qp \leftrightarrow q