Ohio OST Geometry cheat sheet

A rigid motion (translation, rotation, reflection) preserves distance and angle measure, so any figure and its image under a sequence of rigid motions are congruent.

Translation by vector (a, b): (x, y) -> (x + a, y + b). Rotation 90 degrees counterclockwise about the origin: (x, y) -> (-y, x). Rotation 180 degrees: (x, y) -> (-x, -y).

Reflection rules about the origin's axes: across the x-axis (x, y) -> (x, -y); across the y-axis (x, y) -> (-x, y); across y = x (x, y) -> (y, x); across y = -x (x, y) -> (-y, -x).

Two figures are similar if one can be obtained from the other by a sequence of rigid motions and dilations. Similarity preserves angle measure and ratios of corresponding lengths; scale factor k multiplies lengths by k and areas by k squared.

Triangle similarity criteria: AA (two pairs of congruent angles), SSS similarity (all three pairs of corresponding sides proportional), and SAS similarity (two proportional sides with congruent included angles).

Triangle congruence criteria: SSS, SAS, ASA, AAS, and HL (right triangles, hypotenuse and one leg). SSA and AAA do not guarantee congruence.

In a triangle, the segment joining midpoints of two sides is parallel to the third side and half its length (midsegment theorem).

A line parallel to one side of a triangle intersecting the other two sides divides them proportionally (Triangle Proportionality Theorem); its converse also holds.

Coordinate proof strategy: place a vertex at the origin and align sides with axes, assign variables to coordinates, then verify properties using the distance, midpoint, and slope formulas.

Circle equation in standard form: (x - h)^2 + (y - h)^2 = r^2 with center (h, k) and radius r; completing the square converts general form x^2 + y^2 + Dx + Ey + F = 0 to standard form.

A tangent line is perpendicular to the radius at the point of tangency. Two tangent segments from an external point to the same circle are congruent.

Inscribed angle measure equals half its intercepted arc; a central angle equals its intercepted arc. An angle inscribed in a semicircle is a right angle.

Arc length = (central angle in degrees / 360) * 2(pi)r; sector area = (central angle in degrees / 360) * (pi)r^2.

Intersecting chords: if two chords intersect inside a circle, the products of the segments of each chord are equal (ab = cd).

Volume formulas: cylinder V = (pi)r^2 h; cone V = (1/3)(pi)r^2 h; sphere V = (4/3)(pi)r^3; prism/cylinder cross-section area times height for prisms.

Surface area of a sphere = 4(pi)r^2; lateral surface area of a right cylinder = 2(pi)rh; lateral area of a right cone = (pi)rl where l is slant height.

Density and modeling: density = mass / volume; population density = count / area. Scaling a solid by factor k multiplies surface area by k squared and volume by k cubed.

Slope of parallel lines is equal; slope of perpendicular lines is the negative reciprocal (product = -1). Distance formula: d = square root of ((x2 - x1)^2 + (y2 - y1)^2).