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