Geometry year summary for regents

Point

An undefined term representing a specific location with no dimensions.

Line

A one-dimensional figure that extends indefinitely in both directions.

Plane

A two-dimensional flat surface that extends infinitely.

Segment

A part of a line that has two endpoints.

Ray

A part of a line that starts at a point and extends infinitely in one direction.

Angle

Formed by two rays that share a common endpoint called the vertex.

Acute Angle

An angle that measures less than 90 degrees.

Obtuse Angle

An angle that measures greater than 90 degrees but less than 180 degrees.

Right Angle

An angle that measures exactly 90 degrees.

Straight Angle

An angle that measures exactly 180 degrees.

Complementary Angles

Two angles whose measures add up to 90 degrees.

Supplementary Angles

Two angles whose measures add up to 180 degrees.

Vertical Angles

Angles that are opposite each other when two lines intersect, and they are congruent.

Linear Pair

A pair of adjacent angles that are supplementary.

Parallel Lines

Lines in a plane that never intersect and are equidistant from each other.

Transversal

A line that intersects two or more lines at distinct points.

Perpendicular Lines

Lines that intersect at a right angle (90 degrees).

Slope

A measure of the steepness of a line; calculated as the change in y over change in x.

Distance Formula

A formula used to determine the distance between two points in a coordinate plane.

Midpoint Formula

A formula to find the midpoint of a line segment given the coordinates of its endpoints.

Isometries

Transformations that preserve size and shape, such as translations, rotations, and reflections.

Translation

A transformation that slides each point of a figure the same distance in the same direction.

Rotation

A transformation that turns a figure around a fixed point.

Reflection

A transformation that flips a figure over a line, creating a mirror image.

Dilation

A transformation that alters the size of a figure but not its shape.

Triangle Area Formula

The formula to calculate the area of a triangle, A = (1/2)bh.

Pythagorean Theorem

A statement that relates the lengths of the sides of a right triangle: a² + b² = c².

Congruence Criteria for Triangles

Criteria that determine if two triangles are congruent: SSS, SAS, ASA, AAS, HL.

Similarity Criteria for Triangles

Criteria that determine if two triangles are similar: AA, SAS, SSS.

Triangle Inequality Theorem

A theorem stating that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Centroid

The point of concurrency of the medians of a triangle.

Orthocenter

The point of concurrency of the altitudes of a triangle.

Incenter

The point of concurrency of the angle bisectors of a triangle.

Circumcenter

The point of concurrency of the perpendicular bisectors of a triangle.

Parallelogram

A quadrilateral with opposite sides that are both parallel and equal in length.

Rectangle

A parallelogram with four right angles.

Rhombus

A parallelogram with four congruent sides.

Square

A quadrilateral that is both a rectangle and a rhombus.

Trapezoid

A quadrilateral with at least one pair of parallel sides.

Circumference of a Circle

The distance around a circle; C = 2πr or C = πd.

Area of a Circle

The space contained within a circle; A = πr².

Central Angle

An angle whose vertex is at the center of a circle and whose sides are radii.

Inscribed Angle

An angle formed by two chords in a circle which has its vertex on the circle.

Intersecting Chords Theorem

The product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Tangent-Secant Theorem

For a tangent-secant pair, the square of the length of the tangent is equal to the product of the length of the secant and its external segment.

Equation of a Circle

The standard form of a circle's equation: (x-h)² + (y-k)² = r².

SOH CAH TOA

Mnemonic for the definitions of sine, cosine, and tangent in a right triangle.

Law of Sines

A relationship between the sides and angles of a triangle: a/sin A = b/sin B = c/sin C.

Law of Cosines

A formula used to relate the lengths of the sides of a triangle to the cosine of one of its angles.

Area of a Triangle

The formula for calculating the area of a triangle based on two sides and an included angle: A = (1/2)ab sin C.

Volume of a Prism

The space occupied by a prism; V = Bh, where B is the area of the base.

Surface Area of a Prism

The total area of all the surface faces of a prism: SA = 2B + Ph.

Volume of a Cylinder

The amount of space a cylinder occupies, calculated as V = πr²h.