Free NY Regents Geometry Exam Flashcards

A transformation that preserves distance and angle measure, so size and shape do not change. Translations, reflections, and rotations are rigid motions; a dilation is NOT, because it changes size.

Add the shift to each coordinate: (x, y) -> (x + a, y + b) for a shift of a right and b up. A translation slides a figure without rotating or flipping it, so the image is congruent to the preimage.

(x, y) -> (x, -y). A reflection flips a figure over a line of reflection; points on the line stay fixed, and the line is the perpendicular bisector of every segment joining a point and its image.

(x, y) -> (-x, y). The y-axis is the perpendicular bisector of each segment from a point to its image, so the x-coordinate changes sign and the y-coordinate stays the same.

For a counterclockwise rotation, (x, y) -> (-y, x). A rotation turns a figure about a fixed point by a given angle; positive angles are counterclockwise and the distance from the center of rotation is preserved.

A mapping sends the preimage to the image; the inverse sends each image point back to its preimage. For a rigid motion the inverse is also a rigid motion (for example, the inverse of 'translate right 3' is 'translate left 3').

Corresponding Parts of Congruent Triangles are Congruent. Use it only AFTER you have established that two triangles are congruent; it lets you conclude that matching sides and angles are equal.

SSS, SAS, ASA, AAS, and HL (hypotenuse-leg, right triangles only). Each is a minimal set of corresponding parts that guarantees congruence; SSA and AAA do NOT.

Two different triangles can have the same two sides and a non-included angle (the 'ambiguous case'). The third side can swing to two positions, giving an acute or obtuse triangle, so the parts do not determine a unique triangle.

For two right triangles, if the hypotenuse and one leg are congruent to the corresponding parts of another, the triangles are congruent. It is the right-triangle-only replacement for SSA, made valid because the right angle fixes the ambiguous swing.

If two angles of a triangle are congruent, then the sides opposite them are congruent. The original theorem goes the other way: congruent sides imply congruent opposite angles.

Identify a pair of triangles that share the parts you want to prove equal, establish a shortcut (SSS, SAS, ASA, AAS, or HL) using given info, shared sides, vertical angles, or parallel-line angle pairs, then apply CPCTC to conclude the target parts are equal.

Both pairs of opposite sides parallel AND congruent, opposite angles congruent, consecutive angles supplementary, and diagonals that bisect each other. Any one of these conditions is enough to prove a quadrilateral is a parallelogram.

A rectangle is a parallelogram with four right angles, which forces the diagonals to be congruent. A general parallelogram has congruent opposite sides and angles but its diagonals need not be equal.

All four sides are congruent and the diagonals are perpendicular bisectors of each other, with each diagonal bisecting a pair of opposite angles. A square has every property of both a rectangle and a rhombus.

Show both pairs of opposite sides are parallel (equal slopes) OR that one pair of opposite sides is both parallel and congruent (same slope AND equal length). Diagonals with the same midpoint also works because parallelogram diagonals bisect each other.

A drawing made using only a compass (to copy lengths and draw arcs) and an unmarked straightedge (to draw lines through two known points). No markings or measurements are allowed on the straightedge, so constructions are exact, not approximate.

From each endpoint, swing arcs of equal radius that intersect on both sides of the segment; the line through the two intersection points is the perpendicular bisector. Every point on it is equidistant from the endpoints.

Swing an arc on the original angle to mark where it crosses both sides, copy that chord length onto a ray from a new vertex, then draw the second ray through the new intersection. Equal chord lengths force equal angles.

A dilation stretches or shrinks a figure from a center point by a constant scale factor k; the image is similar to the preimage (same shape, possibly different size). If k > 1 the image enlarges, if 0 < k < 1 it shrinks, and if k = 1 it is the identity.

(x, y) -> (kx, ky) for scale factor k. The center of dilation maps to itself, and every image point lies on the line through the center and the preimage point, so orientation and angle measures are preserved.

Congruent figures have the same size AND shape (corresponding sides and angles equal). Similar figures have the same shape but possibly different size: corresponding angles are equal and corresponding side lengths are proportional by a constant scale factor.

AA (two pairs of congruent angles), SSS similarity (all three pairs of sides proportional), and SAS similarity (two pairs of sides proportional with the included angles congruent). AA alone is enough because the third angle is forced.

Set up a proportion of corresponding sides, e.g. (image side)/(preimage side) = k, and cross-multiply to solve. Remember that perimeter scales by k, area scales by k^2, and volume (for similar solids) scales by k^3.

Show two pairs of congruent angles (AA) using given parallel lines, shared angles, vertical angles, or right angles, then state the similarity statement in matching vertex order (e.g. triangle ABC ~ triangle DEF) so corresponding parts are unambiguous.

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. They are ratios of side lengths in a right triangle, so they depend only on the angle, not on the triangle's size.

In a right triangle, a^2 + b^2 = c^2 where c is the hypotenuse. The converse says if the sides satisfy this equation, the triangle IS a right triangle, which lets you check for right angles from side lengths alone.

45-45-90: legs are equal and hypotenuse = leg * sqrt(2). 30-60-90: short leg : long leg : hypotenuse = 1 : sqrt(3) : 2, where the short leg is opposite the 30-degree angle. Memorize these to skip re-deriving them on timed questions.

Use the inverse function (sin^-1, cos^-1, tan^-1) on the calculator with the ratio as input. For a right triangle, if sin A = 0.5 then A = sin^-1(0.5) = 30 degrees; make sure the calculator is in degree mode.

Angle of elevation is measured upward from the horizontal line of sight; angle of depression is measured downward from the horizontal. They are alternate interior angles, so when two observers look at each other across a horizontal the two angles are equal.

The altitude to the hypotenuse creates three similar right triangles. The altitude is the geometric mean of the two hypotenuse segments, and each leg is the geometric mean of the whole hypotenuse and its adjacent segment.

Use the Law of Sines (a/sin A = b/sin B = c/sin C) when you know a side and its opposite angle plus one more part, or the Law of Cosines (c^2 = a^2 + b^2 - 2ab*cos C) when you have two sides and the included angle or all three sides.

In a circle, the central angle has the same degree measure as its intercepted arc. An inscribed angle intercepting the same arc measures half the central angle, because it subtends the arc from the circumference.

An inscribed angle has its vertex on the circle and sides that are chords. It measures half of its intercepted arc; an inscribed angle intercepting a diameter is always 90 degrees.

A tangent is perpendicular to the radius drawn to the point of tangency. This gives a right angle you can use with the Pythagorean theorem to find the distance from an external point to the circle's center.

Congruent chords intercept congruent arcs and are equidistant from the center. The converse also holds: chords the same distance from the center are congruent, so chord length and arc measure are tied to the central angle.

Arc length = (theta/360) * 2 * pi * r, where theta is the central angle in degrees. Sector area = (theta/360) * pi * r^2. Both are a fraction of the full circumference or circle area determined by the central angle's proportion of 360 degrees.

Distance between (x1, y1) and (x2, y2) is sqrt((x2 - x1)^2 + (y2 - y1)^2). It is the Pythagorean theorem applied to the horizontal and vertical legs of the right triangle formed by the two points.

Midpoint = ((x1 + x2)/2, (y1 + y2)/2), the average of the coordinates. It lies on the segment exactly halfway between the endpoints, so it is the same distance from each endpoint.

Parallel lines have equal slopes (m1 = m2). Perpendicular lines have slopes that are negative reciprocals: m1 * m2 = -1 (e.g. slope 2 is perpendicular to slope -1/2). A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope).

To partition a segment from A to B in ratio m:n, the point P = A + (m/(m + n)) * (B - A), applied to each coordinate. It places P so that AP:PB = m:n along the segment, which is the coordinate version of a weighted average.

(x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. To graph, plot the center and go right, left, up, and down r units to find four points on the circle.

Group x and y terms, add the square of half the linear coefficient to each group (and add the same amount to the other side), then factor each trinomial. The result reveals the center (h, k) and radius r.

Prism/cylinder: V = Bh (base area times height). Pyramid/cone: V = (1/3)Bh. Sphere: V = (4/3) * pi * r^3. For a cylinder you can also use V = pi * r^2 * h since the base is a circle.

A cross-section is the intersection of a plane with a solid. A plane perpendicular to the axis of a cylinder gives a circle; a plane parallel to a cone's slant side gives a parabola; rotating a right triangle about a leg gives a cone.

Under a dilation with scale factor k, every linear measurement (edge, radius, height) multiplies by k, area (including surface area) multiplies by k^2, and volume multiplies by k^3. Doubling all dimensions gives 4x the surface area and 8x the volume.

Density = mass/volume (or population/area for population density). On Regents modeling questions, compute volume from the geometry, then multiply by the density to get total mass or population.

A scale drawing preserves angles and multiplies all lengths by the scale factor; area is multiplied by the scale factor squared. Read the units carefully (1 inch = 50 feet means linear scale 1:600, area scale 1:360000).

Four parts, 35 questions: Part I has 24 multiple-choice (2 credits each); Part II has 7 two-credit constructed-response; Part III has 3 four-credit constructed-response; Part IV has 1 six-credit question. Max raw score is 80 credits and the time limit is three hours.

Required tools are a graphing calculator, compass, and straightedge; use the calculator in degree mode for trig and the pi button for exact values unless told to approximate. For constructed-response items, write each reason (name the postulate or theorem) because partial credit is awarded for valid work even if the final answer is wrong.