NY Regents Geometry key terms

A transformation that changes a figure's position or orientation but never its size or shape. Translations, rotations, and reflections are the three rigid motions, and each preserves distance and angle measure.

A rigid motion that slides every point of a figure the same distance in the same direction. It preserves lengths, angle measures, and orientation, and can be described by a vector or coordinate rule.

A rigid motion that flips a figure across a line, called the line of reflection, so each point and its image are equally distant from that line. It preserves size and shape but reverses orientation.

A rigid motion that turns a figure about a fixed center point through a given angle, either clockwise or counterclockwise. Distances from the center and all angle measures are preserved.

Two figures are congruent when a sequence of rigid motions maps one exactly onto the other. Congruent figures have equal corresponding side lengths and equal corresponding angle measures.

The sides and angles of one figure that match, in position and order, the sides and angles of another figure under a transformation. In congruent figures corresponding parts are always equal in measure.

Shortcut conditions proving two triangles congruent without checking all six parts: SSS, SAS, ASA, AAS, and HL for right triangles. SSA and AAA do not guarantee congruence.

If two sides of a triangle are congruent, the angles opposite those sides are congruent, and conversely. The bisector of the vertex angle also bisects the base and is perpendicular to it.

A line that reflects a figure onto itself, dividing it into two mirror-image halves. A regular polygon with n sides has exactly n lines of symmetry.

A figure has rotational symmetry when some rotation of less than 360 degrees about its center maps it onto itself. The smallest such angle is the angle of rotational symmetry.

A transformation that enlarges or reduces a figure by a scale factor from a fixed center point. It preserves angle measures and shape but multiplies all lengths by the scale factor.

Figures related by a sequence of rigid motions and dilations, so corresponding angles are equal and corresponding side lengths are proportional. Similar figures share shape but not necessarily size.

The ratio comparing a length in an image to the corresponding length in the original figure. In a dilation it equals the image length divided by the original length.

Corresponding side lengths of similar figures that form equal ratios. Setting these ratios equal creates a proportion that can be solved for an unknown length.

An equation stating that two ratios are equal, such as a/b = c/d. Cross multiplication converts it to ad = bc, which allows solving for an unknown value.

Conditions proving two triangles similar: AA (two pairs of congruent angles), SSS similarity (all three side pairs proportional), and SAS similarity (two proportional sides with congruent included angles).

An angle bisector of a triangle divides the opposite side into segments proportional to the two adjacent sides. This gives a proportion useful for finding unknown segment lengths.

For positive numbers a and b, the geometric mean is the square root of their product. In a right triangle the altitude to the hypotenuse creates two geometric-mean relationships involving the legs.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a squared plus b squared equals c squared. Its converse tests whether a triangle is right.

A set of three positive integers that satisfy the Pythagorean theorem, such as 3-4-5 or 5-12-13. Any integer multiple of a triple is also a triple.

The 45-45-90 triangle, whose legs are equal and hypotenuse is leg times the square root of 2, and the 30-60-90 triangle, whose sides are in ratio 1 to square root of 3 to 2.

For an acute angle in a right triangle: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent. These ratios depend only on the angle's measure.

The angle measured upward from a horizontal line to a line of sight toward an object above. It appears in right-triangle word problems solved with trigonometric ratios.

The angle measured downward from a horizontal line to a line of sight toward an object below. It equals the angle of elevation from the object back to the observer, by alternate interior angles.

In any triangle, the ratio of a side to the sine of its opposite angle is constant for all three sides. It solves for missing parts when given two angles and a side or two sides and a non-included angle.

In any triangle, the square of a side equals the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. It generalizes the Pythagorean theorem.

An angle whose vertex is the center of a circle and whose sides are radii. Its measure equals the measure of the intercepted arc.

An angle whose vertex lies on a circle and whose sides are chords. Its measure is half the measure of its intercepted arc.

The measure of an arc in degrees, equal to the measure of its central angle. Arc length, by contrast, is a distance measured along the circle.

The distance along a circle between two points, computed as the circumference multiplied by the central angle divided by 360 degrees. It is a portion of the full circumference.

The region of a circle bounded by two radii and the arc between them, shaped like a pie slice. Its area equals the circle's area times the central angle divided by 360 degrees.

An inscribed angle measures half its intercepted arc. Consequences include: angles inscribed in a semicircle are right angles, and inscribed angles intercepting the same arc are congruent.

If a triangle is inscribed in a circle and one side is a diameter, the angle opposite that diameter is a right angle. The converse also holds: a right angle subtends a diameter.

A line intersecting a circle in exactly one point, the point of tangency. A tangent is perpendicular to the radius drawn to that point.

A line or segment that intersects a circle at two points, extending through the interior. Secant segments from an external point satisfy product relationships with tangent segments.

A quadrilateral whose four vertices lie on one circle. Its opposite angles are supplementary, and its converse identifies when four points are concyclic.

The standard form (x - h) squared plus (y - k) squared equals r squared describes a circle with center (h, k) and radius r. Completing the square converts general form to standard form.

Rewriting a quadratic expression as a perfect square plus a constant, by adding and subtracting the square of half the linear coefficient. It reveals the center and radius of a circle from general form.

The ratio of vertical change to horizontal change between two points on a line, computed as rise over run. It measures steepness and direction.

The equation y = mx + b, where m is the slope and b is the y-intercept. It lets you graph a line quickly and identify parallel lines by equal slopes.

Lines in the same plane that never intersect. In the coordinate plane they have equal slopes, and when a transversal cuts them, alternate interior and corresponding angles are congruent.

Lines that intersect to form right angles. In the coordinate plane their slopes are negative reciprocals, so the product of the slopes equals negative one.

The line perpendicular to a segment at its midpoint. Every point on it is equidistant from the segment's endpoints, and it is the locus of such points.

The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is the average of the x-coordinates and the average of the y-coordinates. It locates the exact center of a segment.

The distance between points (x1, y1) and (x2, y2) is the square root of the sum of the squared differences of the coordinates. It follows from the Pythagorean theorem.

The amount of three-dimensional space a solid occupies, measured in cubic units. Prisms and cylinders use base area times height; pyramids and cones use one third of that.

If two solids have equal heights and equal cross-sectional areas at every level, they have equal volumes. It justifies the one-third factor in pyramid and cone volume formulas.

The total area of all faces or curved surfaces of a solid, measured in square units. Lateral area excludes the bases, while total surface area includes them.

A two-dimensional pattern that folds into a three-dimensional solid. A net shows every face of the solid laid flat, making surface area easy to compute.

The two-dimensional figure formed where a plane slices through a solid. Slicing a cylinder or prism parallel to its base gives a congruent copy of the base.

Mass per unit volume of a substance, often expressed in grams per cubic centimeter. Density problems use volume to find mass, or mass to find volume.

Using geometry to represent a real-world situation, choosing shapes and formulas that approximate it, then interpreting the result in context with appropriate units and reasonableness checks.

The number of people per unit area, computed by dividing population by land area. It is an area-based model often used with circular or rectangular regions.