NY Regents Geometry cheat sheet

Congruence criteria for triangles: SSS, SAS, ASA, AAS, and HL (HL applies only to right triangles); AAA and SSA do not guarantee congruence.

A rigid motion (translation, rotation, reflection, or a composition) preserves distance and angle measure, so two figures are congruent exactly when a rigid motion maps one onto the other.

Similarity: two figures are similar when a sequence of rigid motions plus a dilation maps one onto the other; corresponding angles are equal and corresponding sides are proportional with ratio k.

Similar triangle shortcuts: AA, SSS similarity (all three side ratios equal), and SAS similarity (two proportional sides with the included angle equal).

If two similar figures have scale factor k, then lengths scale by k, areas scale by k^2, and volumes scale by k^3.

Pythagorean theorem: in a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2; the converse holds, so a^2 + b^2 = c^2 implies a right angle opposite c.

Special right triangles: 45-45-90 has sides x, x, xsqrt(2); 30-60-90 has sides x, xsqrt(3), 2x, with x opposite the 30-degree angle.

Right-triangle trigonometry: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent; also sin A = cos(90 - A) for complementary angles.

Law of Sines a/sin A = b/sin B = c/sin C and Law of Cosines c^2 = a^2 + b^2 - 2ab*cos C apply to any triangle; use the Law of Cosines for SAS or SSS data.

A central angle equals its intercepted arc; an inscribed angle equals half its intercepted arc, so an angle inscribed in a semicircle is 90 degrees.

Inscribed angles that intercept the same arc are congruent, and opposite angles of a cyclic quadrilateral are supplementary.

A tangent line is perpendicular to the radius at the point of tangency, and two tangent segments drawn from one external point are congruent.

Chord theorem: if two chords intersect inside a circle, the products of the segments are equal: ab = cd.

Secant-tangent power theorem: from an external point, (tangent)^2 = (outside secant segment)*(whole secant), and for two secants (outside1)(whole1) = (outside2)(whole2).

Arc length = (central angle/360)2pi*r, and sector area = (central angle/360)pir^2.

A line is tangent to a circle when its distance from the center equals the radius; a perpendicular from the center to a chord bisects the chord.

Equation of a circle in standard form: (x - h)^2 + (y - k)^2 = r^2, with center (h, k) and radius r.

Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).

Slope formula: m = (y2 - y1)/(x2 - x1); parallel lines have equal slopes, and perpendicular lines have slopes whose product is -1.

Point-slope form y - y1 = m(x - x1) and slope-intercept form y = mx + b both describe the same line; use slope-intercept to read slope and y-intercept directly.

Complete the square to convert x^2 + y^2 + Dx + Ey + F = 0 into center-radius form; the center is (-D/2, -E/2).

A figure is a parallelogram if one pair of opposite sides is both congruent and parallel; a parallelogram is a rectangle if its diagonals are congruent.

A quadrilateral is a rhombus if its diagonals are perpendicular bisectors of each other; a square satisfies both rectangle and rhombus conditions.

To prove a triangle is isosceles using coordinates, show two side lengths are equal with the distance formula; to prove it is right, show two slopes multiply to -1 or sides satisfy a^2 + b^2 = c^2.

The midpoint of a segment is equidistant from both endpoints, and the perpendicular bisector of a segment is the set of points equidistant from the endpoints.

Circumference C = 2pir and circle area A = pi*r^2.

Triangle area = (1/2)bh; parallelogram area = bh; trapezoid area = (1/2)(b1 + b2)*h.

Regular polygon area = (1/2)apothemperimeter, where the apothem is the perpendicular distance from center to a side.

Prism or cylinder volume V = B*h (B = base area); pyramid or cone volume V = (1/3)Bh.

Sphere volume V = (4/3)pir^3 and sphere surface area SA = 4pir^2.

Cylinder lateral area = 2pirh; cone lateral area = pir*l where l is the slant height; total surface area adds the base areas.

When a solid is dilated by factor k, surface area scales by k^2 and volume scales by k^3.

Cross-sections: a plane parallel to the base of a cone or pyramid produces a similar figure; a plane cutting a sphere produces a circle.

Density problems: density = mass/volume, so mass = density*volume; keep units consistent before computing.

Scale drawings: if a drawing uses scale 1 unit : n units, then areas scale by n^2, so convert lengths first, then square the factor for area.

Population density = population/area; to compare two regions, compute population per square unit or per square mile.

For composite figures, decompose into non-overlapping familiar shapes, add areas, and subtract any removed regions.

In real-world right-triangle problems, the angle of elevation is measured upward from horizontal and the angle of depression downward from horizontal; they are equal as alternate interior angles.

Volume flow problems: volume = rate*time; convert rates such as gallons per minute to cubic units per minute before multiplying.

Similar-figure modeling: use a proportion of corresponding lengths to find an unknown measurement, then apply k^2 or k^3 for area or volume as needed.

Rounding: carry full precision through intermediate steps and round only the final answer to the precision requested by the problem.

Congruence criteria for triangles: SSS, SAS, ASA, AAS, and HL (HL applies only to right triangles); AAA and SSA do not guarantee congruence.

A rigid motion (translation, rotation, reflection, or a composition) preserves distance and angle measure, so two figures are congruent exactly when a rigid motion maps one onto the other.

Similarity: two figures are similar when a sequence of rigid motions plus a dilation maps one onto the other; corresponding angles are equal and corresponding sides are proportional with ratio k.

Similar triangle shortcuts: AA, SSS similarity (all three side ratios equal), and SAS similarity (two proportional sides with the included angle equal).

If two similar figures have scale factor k, then lengths scale by k, areas scale by k^2, and volumes scale by k^3.

Pythagorean theorem: in a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2; the converse holds, so a^2 + b^2 = c^2 implies a right angle opposite c.

Special right triangles: 45-45-90 has sides x, x, xsqrt(2); 30-60-90 has sides x, xsqrt(3), 2x, with x opposite the 30-degree angle.

Right-triangle trigonometry: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent; also sin A = cos(90 - A) for complementary angles.

Law of Sines a/sin A = b/sin B = c/sin C and Law of Cosines c^2 = a^2 + b^2 - 2ab*cos C apply to any triangle; use the Law of Cosines for SAS or SSS data.

A central angle equals its intercepted arc; an inscribed angle equals half its intercepted arc, so an angle inscribed in a semicircle is 90 degrees.

Inscribed angles that intercept the same arc are congruent, and opposite angles of a cyclic quadrilateral are supplementary.

A tangent line is perpendicular to the radius at the point of tangency, and two tangent segments drawn from one external point are congruent.

Chord theorem: if two chords intersect inside a circle, the products of the segments are equal: ab = cd.

Secant-tangent power theorem: from an external point, (tangent)^2 = (outside secant segment)*(whole secant), and for two secants (outside1)(whole1) = (outside2)(whole2).

Arc length = (central angle/360)2pi*r, and sector area = (central angle/360)pir^2.

A line is tangent to a circle when its distance from the center equals the radius; a perpendicular from the center to a chord bisects the chord.

Equation of a circle in standard form: (x - h)^2 + (y - k)^2 = r^2, with center (h, k) and radius r.

Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).

Slope formula: m = (y2 - y1)/(x2 - x1); parallel lines have equal slopes, and perpendicular lines have slopes whose product is -1.

Point-slope form y - y1 = m(x - x1) and slope-intercept form y = mx + b both describe the same line; use slope-intercept to read slope and y-intercept directly.

Complete the square to convert x^2 + y^2 + Dx + Ey + F = 0 into center-radius form; the center is (-D/2, -E/2).

A figure is a parallelogram if one pair of opposite sides is both congruent and parallel; a parallelogram is a rectangle if its diagonals are congruent.

A quadrilateral is a rhombus if its diagonals are perpendicular bisectors of each other; a square satisfies both rectangle and rhombus conditions.

To prove a triangle is isosceles using coordinates, show two side lengths are equal with the distance formula; to prove it is right, show two slopes multiply to -1 or sides satisfy a^2 + b^2 = c^2.

The midpoint of a segment is equidistant from both endpoints, and the perpendicular bisector of a segment is the set of points equidistant from the endpoints.

Circumference C = 2pir and circle area A = pi*r^2.

Triangle area = (1/2)bh; parallelogram area = bh; trapezoid area = (1/2)(b1 + b2)*h.

Regular polygon area = (1/2)apothemperimeter, where the apothem is the perpendicular distance from center to a side.

Prism or cylinder volume V = B*h (B = base area); pyramid or cone volume V = (1/3)Bh.

Sphere volume V = (4/3)pir^3 and sphere surface area SA = 4pir^2.

Cylinder lateral area = 2pirh; cone lateral area = pir*l where l is the slant height; total surface area adds the base areas.

When a solid is dilated by factor k, surface area scales by k^2 and volume scales by k^3.

Cross-sections: a plane parallel to the base of a cone or pyramid produces a similar figure; a plane cutting a sphere produces a circle.

Density problems: density = mass/volume, so mass = density*volume; keep units consistent before computing.

Scale drawings: if a drawing uses scale 1 unit : n units, then areas scale by n^2, so convert lengths first, then square the factor for area.

Population density = population/area; to compare two regions, compute population per square unit or per square mile.

For composite figures, decompose into non-overlapping familiar shapes, add areas, and subtract any removed regions.

In real-world right-triangle problems, the angle of elevation is measured upward from horizontal and the angle of depression downward from horizontal; they are equal as alternate interior angles.

Volume flow problems: volume = rate*time; convert rates such as gallons per minute to cubic units per minute before multiplying.

Similar-figure modeling: use a proportion of corresponding lengths to find an unknown measurement, then apply k^2 or k^3 for area or volume as needed.

Rounding: carry full precision through intermediate steps and round only the final answer to the precision requested by the problem.