Geometry Brain Dump Review

Comprehensive vocabulary and formula flashcards covering geometry logic, transformations, triangles, polygons, circles, and quadrilateral properties based on the student's brain dump notes.

$$\wedge$$

The logical operator meaning 'and'.

$$\vee$$

The logical operator meaning 'or'.

$$\sim$$

The logical symbol for 'inverse' or negation.

Converse

A logical statement written as $q \rightarrow p$.

Contrapositive

A logical statement written as $\sim q \rightarrow \sim p$.

Syllogism

A logical rule stating that if $p \rightarrow q$ and $q \rightarrow r$, then $p \rightarrow r$.

Law of Detachment

If $p \rightarrow q$ is true and $p$ is true, then $q$ is true.

$90^{\circ}$ Rotation

The transformation rule $(x, y) \rightarrow (-y, x)$.

$180^{\circ}$ Rotation

The transformation rule $(x, y) \rightarrow (-x, -y)$.

$270^{\circ}$ Rotation

The transformation rule $(x, y) \rightarrow (y, -x)$.

Reflection over $y=x$

The transformation rule $(x, y) \rightarrow (y, x)$.

Reflection over $y=-x$

The transformation rule $(x, y) \rightarrow (-y, -x)$.

$45-45-90$ Triangle

A special right triangle with side ratios of $x : x : x\sqrt{2}$.

$30-60-90$ Triangle

A special right triangle with side ratios of $x : x\sqrt{3} : 2x$.

SOH CAH TOA

$\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$, $\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$

Law of Sines

$$\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$$

Law of Cosines

$$a^2 = b^2 + c^2 - 2bc \times \cos(A)$$

Sum of Interior Angles of a Polygon

$$(n-2) \times 180$$

Regular Polygon Interior Angle

$$\frac{(n-2) \times 180}{n}$$

Regular Polygon Exterior Angle

$$\frac{360}{n}$$

Distance Formula

$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

Midpoint Formula

$$(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$$

Intersection of Chords Angle

$\angle = \frac{1}{2}(a+b)$ where $a$ and $b$ are intercepted arcs.

Exterior Circle Angle (Secant/Tangent)

$\angle = \frac{1}{2}(a-b)$ where $a$ and $b$ are intercepted arcs.

Circle Segment Rules (Chords)

$$ab = cd$$

Circle Segment Rules (Secants)

$$a(a+b) = c(c+d)$$

Area of a Sector

$$\frac{n}{360} \times \pi r^2$$

Arc Length (Degrees)

$$L = \frac{n}{360} \times 2 \pi r$$

Arc Length (Radians)

$L = r \theta$ where $\theta$ is the central angle in radians.

Scale Factor Ratios

$a:b$ for scale factor, $a^2:b^2$ for surface area, and $a^3:b^3$ for volume.

Parallelogram Properties

Opposite sides are parallel and equal, opposite angles are equal, consecutive angles are supplementary, and diagonals bisect each other.

Rhombi Properties

Four equal sides, perpendicular diagonals, and diagonals that bisect the angles.

Rectangle Properties

Four $90^{\circ}$ angles and diagonals that are equal.

Isosceles Trapezoid Properties

Diagonals are equal, base angles are equal, and opposite angles are supplementary (add to $180^{\circ}$).