Geometry EOC Review Notes

Rigid Motions

  • Preserve length and angle measure.
  • Include translation, reflection, and rotation.

Translation

  • (x, y) \rightarrow (x + a, y + b)

Reflection

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

Rotation Rules

  • Around the origin.
  • 90° counterclockwise: (x, y) \rightarrow (-y, x)
  • 180°: (x, y) \rightarrow (-x, -y)
  • 270° counterclockwise: (x, y) \rightarrow (y, -x)

Non-Rigid Motion

  • Dilation: Preserves only angles; result is similar, not congruent.
  • (x, y) \rightarrow (kx, ky), where k is the scale factor.
    • k < 1: Reduction (shrink).
    • k = 1: Same size (congruent).
    • k > 1: Enlargement (stretch).
  • k = \frac{image}{preimage}

Pythagorean Theorem

  • a^2 + b^2 = c^2

Slope

  • m = \frac{y2 - y1}{x2 - x1} = \frac{\Delta y}{\Delta x}
  • Slope-intercept form: y = mx + b
  • Point-slope form: y - y1 = m(x - x1)

Midpoint

  • M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})

Distance

  • d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Equation of a Circle

  • (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.

Trigonometry

  • sin(A) = \frac{opposite}{hypotenuse}
  • cos(A) = \frac{adjacent}{hypotenuse}
  • tan(A) = \frac{opposite}{adjacent}
  • sin(A) = cos(90 - A)
  • cos(A) = sin(90 - A)

Parallel and Perpendicular Lines

  • Parallel lines: Same slope.
  • Perpendicular lines: Negative reciprocal slopes (m1 \cdot m2 = -1).

Angles

  • Complementary angles: Sum to 90°.
  • Supplementary angles: Sum to 180°.
  • Vertical angles: Congruent.

Partitioning a Line Segment

  • (x, y) = (x1 + \frac{a}{a+b}(x2 - x1), y1 + \frac{a}{a+b}(y2 - y1))

Triangle Properties

  • Sum of angles in a triangle: 180°.
  • Equilateral triangle: All angles are 60°.
  • Isosceles triangle: Two congruent sides and two congruent angles.
  • Triangle Inequality Theorem: The sum of any two sides of a triangle is greater than the third side.

Proving Triangle Congruence

  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • AAS (Angle-Angle-Side)
  • RHS (Right Angle-Hypotenuse-Side)

Triangle Centers

  • Median: Connects a vertex to the midpoint of the opposite side.
  • Centroid: Point of concurrency of the medians; divides each median in a 2:1 ratio.
  • Midsegment: Connects the midpoints of two sides; parallel to and half the length of the third side.
  • Circumcenter: Point of concurrency of the perpendicular bisectors; equidistant from the vertices.
  • Incenter: Point of concurrency of the angle bisectors; center of the incircle.

Circle Theorems

  • Area = \pi r^2
  • Circumference = 2 \pi r
  • Central angle = Intercepted arc
  • Inscribed angle = 1/2 Intercepted arc
  • Tangent-Radius Theorem: Angle=90^{\circ}
  • Arc Length: (\frac{\theta}{360}) * 2* \pi *r
  • Area of Sector: (\frac{\theta}{360}) * \pi *r^2

Density

  • Density = \frac{Mass}{Volume}

Volume

  • Cylinder: V = Bh
  • Sphere: V = \frac{4}{3} \pi r^3