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

• $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1} = \frac{\Delta y}{\Delta x}$
• Slope-intercept form: $y = mx + b$
• Point-slope form: $y - y<em>1 = m(x - x</em>1)$

Midpoint

• $M = (\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2})$

Distance

• $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$

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 ($m<em>1 \cdot m</em>2 = -1$).

Angles

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

Partitioning a Line Segment

• $(x, y) = (x<em>1 + \frac{a}{a+b}(x</em>2 - x<em>1), y</em>1 + \frac{a}{a+b}(y<em>2 - y</em>1))$

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: $V = \frac{4}{3} \pi r^3$