Geometry Brain Dump – Second Semester

Right Triangles and Trigonometry

• Given a right triangle:

  • $sin(A) = cos(B)$

  • $cos(A) = sin(B)$

  • $tan(A) = \frac{1}{tan(B)}$

• 30° - 60° - 90° Trigonometric Ratios:

  • $sin(30°) = \frac{1}{2}$

  • $cos(30°) = \frac{\sqrt{3}}{2}$

  • $tan(30°) = \frac{\sqrt{3}}{3}$

• 45° - 45° - 90° Trigonometric Ratios:

  • $sin(45°) = \frac{\sqrt{2}}{2}$

  • $cos(45°) = \frac{\sqrt{2}}{2}$

  • $tan(45°) = 1$

Three-Dimensional Figures

• Cone

  • Volume: $V = \frac{1}{3}πr^2h$

  • Surface Area: $S.A. = πrl + πr^2$, where $l$ is the slant height.

• Rectangular Prism

  • Volume: $V = lwh$

  • Surface Area: $S.A. = 2lw + 2wh + 2lh$

• Regular Pyramid

  • Volume: $V = \frac{1}{3}lwh$

• Cylinder

  • Volume: $V = πr^2h$

  • Surface Area: $S.A. = 2πr^2 + 2πrh$

• Cavalieri's Principle: If two solids have congruent altitudes and every plane parallel to the bases results in cross-sections of equal area, then the solids have equal volumes.

• Population Density: $Population Density = \frac{Population}{Area}$

Shape - Three-Dimensional Shape (360° Rotation)

• Rectangle → Cylinder

• Triangle → Cone

• Semi-Circle → Sphere

Angle/Arc Measures

• Secant and Tangent

• 2 Tangents

• 2 Secants

  • Measure of angle formed $= \frac{1}{2}$(larger arc – smaller arc)

Segment Lengths

• Tangent and Secant: $A^2 = B(B + C)$

• 2 Secants: $A(A + B) = C(C + D)$

• 2 Chords: $(A)(B) = (C)(D)$

• 2 Tangents Formulas

Circles

• Area of a Sector: $A = (\frac{θ}{360}°)πr^2$, where θ is the central angle in degrees.

• Arc Length: $Arc Length= 2πr(\frac{θ}{360}°)$, where θ is the central angle in degrees.

• Equation of circle with center at origin: $x^2 + y^2 = r^2$

• Equation of a circle with center $(h, k)$: $(x − h)^2 + (y − k)^2 = r^2$

  • Example: Given $(x + 5)^2 + (y − 7)^2 = 81$, Center: $(-5, 7)$; Radius= $\sqrt{81} = 9$

• Inscribed Angle: $\measuredangle BCD = 2\measuredangle BAC$

• Angle Inscribed in a Semi-Circle: $\measuredangle ABD = 90°$

• Inscribed Quadrilateral Theorem: Opposite angles of an inscribed quadrilateral are supplementary.

Cross Section Shape

• Cylinder

  • Horizontal Plane: Circle

  • Vertical Plane: Rectangle

• Cone

  • Horizontal Plane: Circle

  • Vertical Plane: Triangle

• Rectangular Prism

  • Horizontal Plane: Rectangle

  • Vertical Plane: Rectangle

• Triangular Prism

  • Horizontal Plane: Triangle

  • Vertical Plane: Rectangle

• Sphere

  • Horizontal Plane: Circle

  • Vertical Plane: Circle

Triangles

• Triangle Midsegment Theorem: If $AB = DB$ and $AE = EC$, then $DE || BC$ and $DE = \frac{1}{2} BC$.

• Triangle Inequality Theorem:

  • a + b > c

  • a + c > b

  • b + c > a