Geometry and Analytic Geometry Key Concepts

Vocabulary flashcards covering geometry, lines, line segments, angles, function properties, line equations, and coordinate geometry concepts.

Geometry

One of the oldest branches of mathematics, named after the Greek words for "Earth" and "Measure".

Point

A fundamental geometric element used to denote a specific location in space.

Line

A geometric figure determined by two distinct points that extends indefinitely to infinity in both directions.

Line Segment

A portion of a line lying between two distinct points called endpoints; it is the only linear figure with a finite length.

Ray

A part of a line that begins at a single point called an endpoint and extends to infinity in the opposite direction.

Angle

The union of two rays that share a common endpoint called a vertex.

Protractor

A device used to measure angles in degrees by aligning its center point with the vertex of the angle.

Right Angle

An angle whose measure is exactly $90^\text{o}$.

Straight Angle

An angle whose measure is exactly $180^\text{o}$.

Complementary Angles

A pair of two angles whose sum of measures equals $90^\text{o}$.

Supplementary Angles

A pair of two angles whose sum of measures equals $180^\text{o}$.

Adjacent Angles

Two angles that share a common endpoint and a common side, but share no interior points.

Segment Addition Postulate

A postulate stating that if point $B$ lies between points $A$ and $C$ on segment $\overline{AC}$, then $AB + BC = AC$.

Angle Addition Postulate

A postulate stating that if point $D$ lies in the interior of $\angle ABC$, then $m\angle ABC = m\angle ABD + m\angle DBC$.

Axiomatic System

A consistent set of axioms from which geometric principles and theorems are logically derived without contradictions.

Euclid of Alexandria

The ancient Greek mathematician who formulated the first axiomatic geometry system in his 13-chapter work titled The Elements of Geometry.

Parallel Postulate

Euclid's Fifth Postulate, which forms the basis of Euclidean Geometry where parallel lines exist.

Playfair's Axiom

A simplified phrasing of the Parallel Postulate stating that given a line and a point not on the line, exactly one parallel line passes through the point.

Relation

A rule, relationship, or correspondence between sets of information consisting of a set of ordered pairs.

Ordered Pair

A pair of values composed of an x-coordinate (abscissa) and a y-coordinate (ordinate).

Domain

The set of all first coordinates (x-values) in a relation or function.

Range

The set of all second coordinates (y-values) in a relation or function.

Vertical Line Test

A visual method used to test if a graph represents a function; if any vertical line intersects the graph more than once, it is not a function.

Even Function

A function that is symmetrical with respect to the y-axis, satisfying $f(-x) = f(x)$.

Odd Function

A function that is symmetrical with respect to the origin, satisfying $f(-x) = -f(x)$.

Parallel Lines

Two lines $L_1$ and $L_2$ that have equal slopes ($m_{L_1} = m_{L_2}$) or coefficients satisfying $a_1 b_2 - a_2 b_1 = 0$.

Perpendicular Lines

Two lines $L_1$ and $L_2$ whose slopes are negative reciprocals ($m_{L_1} \cdot m_{L_2} = -1$) or coefficients satisfying $a_1 a_2 + b_1 b_2 = 0$.

Distance Formula between Two Points

The formula derived from the Pythagorean Theorem giving distance $d$ between points $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Shortest Distance from a Point to a Line

The directed distance $d$ from point $P(x_0, y_0)$ to line $Ax + By + C = 0$, given by $d = \frac{A x_0 + B y_0 + C}{\pm \sqrt{A^2 + B^2}}$, where the denominator's sign matches $B$.

Two-Point Form

The equation of a line passing through two known points $(x_1, y_1)$ and $(x_2, y_2)$: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$.

Point-Slope Form

The equation of a line with slope $m$ passing through point $(x_1, y_1)$: $y - y_1 = m(x - x_1)$.

Slope-Intercept Form

The equation of a line expressed as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Intercept Form

The equation of a line with x-intercept $a$ and y-intercept $b$: $\frac{x}{a} + \frac{y}{b} = 1$.

General Form of a Line

The equation of a line expressed in the standard form $ax + by + c = 0$, where $a$, $b$, and $c$ are real numbers.

Angle Between Two Intersecting Lines

The angle measure $\theta$ from line $L_1$ to line $L_2$ calculated using $\tan(\theta) = \frac{m_2 - m_1}{1 + m_2 m_1}$, provided $m_2 m_1 \neq -1$.

Division of a Line Segment

The coordinates of point $(x, y)$ dividing a line segment in ratio $r$ calculated as $x = x_1 + r(x_2 - x_1)$ and $y = y_1 + r(y_2 - y_1)$.