50d ago

Geometry Brain Dump – Second Semester

Right Triangles and Trigonometry

  • Given a right triangle:

    • sin(A)=cos(B)sin(A) = cos(B)$$sin(A) = cos(B)$$

    • cos(A)=sin(B)cos(A) = sin(B)$$cos(A) = sin(B)$$

    • tan(A)=1tan(B)tan(A) = \frac{1}{tan(B)}$$tan(A) = \frac{1}{tan(B)}$$

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

    • sin(30°)=12sin(30°) = \frac{1}{2}$$sin(30°) = \frac{1}{2}$$

    • cos(30°)=32cos(30°) = \frac{\sqrt{3}}{2}$$cos(30°) = \frac{\sqrt{3}}{2}$$

    • tan(30°)=33tan(30°) = \frac{\sqrt{3}}{3}$$tan(30°) = \frac{\sqrt{3}}{3}$$

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

    • sin(45°)=22sin(45°) = \frac{\sqrt{2}}{2}$$sin(45°) = \frac{\sqrt{2}}{2}$$

    • cos(45°)=22cos(45°) = \frac{\sqrt{2}}{2}$$cos(45°) = \frac{\sqrt{2}}{2}$$

    • tan(45°)=1tan(45°) = 1$$tan(45°) = 1$$

Three-Dimensional Figures

  • Cone

    • Volume: V=13πr2hV = \frac{1}{3}πr^2h$$V = \frac{1}{3}πr^2h$$

    • Surface Area: S.A.=πrl+πr2S.A. = πrl + πr^2$$S.A. = πrl + πr^2$$, where ll$$l$$ is the slant height.

  • Rectangular Prism

    • Volume: V=lwhV = lwh$$V = lwh$$

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

  • Regular Pyramid

    • Volume: V=13lwhV = \frac{1}{3}lwh$$V = \frac{1}{3}lwh$$

  • Cylinder

    • Volume: V=πr2hV = πr^2h$$V = πr^2h$$

    • Surface Area: S.A.=2πr2+2πrhS.A. = 2πr^2 + 2πrh$$S.A. = 2πr^2 + 2πrh$$

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

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 =12= \frac{1}{2}$$= \frac{1}{2}$$(larger arc – smaller arc)

Segment Lengths

  • Tangent and Secant: A2=B(B+C)A^2 = B(B + C)$$A^2 = B(B + C)$$

  • 2 Secants: A(A+B)=C(C+D)A(A + B) = C(C + D)$$A(A + B) = C(C + D)$$

  • 2 Chords: (A)(B)=(C)(D)(A)(B) = (C)(D)$$(A)(B) = (C)(D)$$

  • 2 Tangents Formulas

Circles

  • Area of a Sector: A=(θ360°)πr2A = (\frac{θ}{360}°)πr^2$$A = (\frac{θ}{360}°)πr^2$$, where θ is the central angle in degrees.

  • Arc Length: ArcLength=2πr(θ360°)Arc Length= 2πr(\frac{θ}{360}°)$$Arc Length= 2πr(\frac{θ}{360}°)$$, where θ is the central angle in degrees.

  • Equation of circle with center at origin: x2+y2=r2x^2 + y^2 = r^2$$x^2 + y^2 = r^2$$

  • Equation of a circle with center (h,k)(h, k)$$(h, k)$$: (xh)2+(yk)2=r2(x − h)^2 + (y − k)^2 = r^2$$(x − h)^2 + (y − k)^2 = r^2$$

    • Example: Given (x+5)2+(y7)2=81(x + 5)^2 + (y − 7)^2 = 81$$(x + 5)^2 + (y − 7)^2 = 81$$, Center: (5,7)(-5, 7)$$(-5, 7)$$; Radius= 81=9\sqrt{81} = 9$$\sqrt{81} = 9$$

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

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

  • 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=DBAB = DB$$AB = DB$$ and AE=ECAE = EC$$AE = EC$$, then DEBCDE || BC$$DE || BC$$ and DE=12BCDE = \frac{1}{2} BC$$DE = \frac{1}{2} BC$$.

  • Triangle Inequality Theorem:

    • a+b>ca + b > c$$a + b > c$$

    • a+c>ba + c > b$$a + c > b$$

    • b+c>ab + c > a$$b + c > a$$


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Geometry Brain Dump – Second Semester

Right Triangles and Trigonometry

  • Given a right triangle:

    • sin(A)=cos(B)sin(A) = cos(B)

    • cos(A)=sin(B)cos(A) = sin(B)

    • tan(A)=1tan(B)tan(A) = \frac{1}{tan(B)}

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

    • sin(30°)=12sin(30°) = \frac{1}{2}

    • cos(30°)=32cos(30°) = \frac{\sqrt{3}}{2}

    • tan(30°)=33tan(30°) = \frac{\sqrt{3}}{3}

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

    • sin(45°)=22sin(45°) = \frac{\sqrt{2}}{2}

    • cos(45°)=22cos(45°) = \frac{\sqrt{2}}{2}

    • tan(45°)=1tan(45°) = 1

Three-Dimensional Figures

  • Cone

    • Volume: V=13πr2hV = \frac{1}{3}πr^2h

    • Surface Area: S.A.=πrl+πr2S.A. = πrl + πr^2, where ll is the slant height.

  • Rectangular Prism

    • Volume: V=lwhV = lwh

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

  • Regular Pyramid

    • Volume: V=13lwhV = \frac{1}{3}lwh

  • Cylinder

    • Volume: V=πr2hV = πr^2h

    • Surface Area: S.A.=2πr2+2πrhS.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: PopulationDensity=PopulationAreaPopulation 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 =12= \frac{1}{2}(larger arc – smaller arc)

Segment Lengths

  • Tangent and Secant: A2=B(B+C)A^2 = B(B + C)

  • 2 Secants: A(A+B)=C(C+D)A(A + B) = C(C + D)

  • 2 Chords: (A)(B)=(C)(D)(A)(B) = (C)(D)

  • 2 Tangents Formulas

Circles

  • Area of a Sector: A=(θ360°)πr2A = (\frac{θ}{360}°)πr^2, where θ is the central angle in degrees.

  • Arc Length: ArcLength=2πr(θ360°)Arc Length= 2πr(\frac{θ}{360}°), where θ is the central angle in degrees.

  • Equation of circle with center at origin: x2+y2=r2x^2 + y^2 = r^2

  • Equation of a circle with center (h,k)(h, k): (xh)2+(yk)2=r2(x − h)^2 + (y − k)^2 = r^2

    • Example: Given (x+5)2+(y7)2=81(x + 5)^2 + (y − 7)^2 = 81, Center: (5,7)(-5, 7); Radius= 81=9\sqrt{81} = 9

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

  • Angle Inscribed in a Semi-Circle: ABD=90°\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=DBAB = DB and AE=ECAE = EC, then DEBCDE || BC and DE=12BCDE = \frac{1}{2} BC.

  • Triangle Inequality Theorem:

    • a+b>ca + b > c

    • a+c>ba + c > b

    • b+c>ab + c > a