Florida Geometry EOC cheat sheet

Conditional statement "if p then q": converse swaps to "if q then p"; inverse negates to "if not p then not q"; contrapositive is "if not q then not p" and is always logically equivalent to the original.

A biconditional "p if and only if q" is true only when the conditional and its converse are both true.

Law of Detachment: from p → q and p, conclude q. Law of Syllogism: from p → q and q → r, conclude p → r.

Counterexample: a single case making a conditional's hypothesis true and conclusion false disproves the statement.

Addition, subtraction, multiplication, and division properties of equality preserve equality; substitution replaces a quantity with an equal quantity.

In a proof, vertical angles are congruent, and angles supplementary to the same angle are congruent.

Segment addition: if B is between A and C, then AB + BC = AC. Angle addition: if ray OB lies inside angle AOC, then m∠AOB + m∠BOC = m∠AOC.

A definition is reversible (biconditional); a theorem must be proved before it can be used as a reason.

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

CPCTC: once two triangles are proved congruent, every pair of corresponding parts is congruent.

Isometries (reflections, rotations, translations) preserve side lengths and angle measures, so they map figures to congruent figures.

Similar polygons have congruent corresponding angles and proportional corresponding sides; the scale factor k equals the ratio of any pair of corresponding lengths.

Similarity criteria for triangles: AA, SSS similarity, and SAS similarity.

If two similar figures have scale factor k, then areas compare as k² and volumes as k³.

In a triangle, a segment parallel to one side divides the other two sides proportionally (Triangle Proportionality Theorem).

The midsegment of a triangle is parallel to the third side and half its length.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²). Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).

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

Point-slope form y − y₁ = m(x − x₁) and slope-intercept form y = mx + b describe the same line.

Reflection across the x-axis: (x, y) → (x, −y); across the y-axis: (x, y) → (−x, y); across y = x: (x, y) → (y, x).

Rotation about the origin: 90° CCW (x, y) → (−y, x); 180° (x, y) → (−x, −y); 270° CCW (x, y) → (y, −x).

Translation by vector (a, b): (x, y) → (x + a, y + b).

A dilation centered at the origin with scale factor k maps (x, y) → (kx, ky); k > 1 enlarges, 0 < k < 1 reduces.

Parallelogram tests on the coordinate plane: both pairs of opposite sides parallel, diagonals bisect each other, or one pair of opposite sides both parallel and congruent.

Pythagorean Theorem: a² + b² = c², where c is the hypotenuse; the converse holds, so a² + b² = c² implies a right triangle.

45-45-90 triangle: legs x and x, hypotenuse x√2. 30-60-90 triangle: sides x, x√3, and 2x with the shortest side opposite 30°.

Trigonometric ratios in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal, and they are equal in alternate-interior positions.

Geometric mean altitude theorem: in a right triangle the altitude to the hypotenuse creates two triangles similar to the original and to each other.

Law of Sines: a/sin A = b/sin B = c/sin C, valid for any triangle when a side and its opposite angle are known.

Law of Cosines: c² = a² + b² − 2ab·cos C, used for SAS or SSS situations.

Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) recovers an acute angle measure from a ratio of two known sides.

A central angle equals its intercepted arc; an inscribed angle equals half its intercepted arc.

An angle inscribed in a semicircle is a right angle, and inscribed angles intercepting the same arc are congruent.

Tangent-radius theorem: a tangent line is perpendicular to the radius drawn to the point of tangency.

Two tangent segments from an external point are congruent.

Chord-chord: for chords intersecting inside a circle, the products of the segments are equal (a·b = c·d).

Secant/tangent from an external point: (whole)(external) = (whole)(external); for a tangent, tangent² = (whole secant)(external part).

Arc length = (central angle/360°)·2πr; sector area = (central angle/360°)·πr².

Equation of a circle: (x − h)² + (y − k)² = r², with center (h, k) and radius r.

Perimeter and circumference: C = 2πr = πd; arc length is a fraction of the circumference.

Area formulas: rectangle A = bh; triangle A = ½bh; parallelogram A = bh; trapezoid A = ½(b₁ + b₂)h; circle A = πr².

Composite figures are decomposed into standard shapes; subtract areas of cut-out regions rather than adding them.

Surface area of a prism or cylinder = sum of all faces; for a cylinder SA = 2πr² + 2πrh.

Volume: prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³; sphere surface area SA = 4πr².

Slant height ℓ of a cone satisfies ℓ² = r² + h²; a pyramid's lateral faces are triangles with height equal to the slant height.

When every dimension is multiplied by k, surface area scales by k² and volume by k³.

Density and modeling problems use volume or area as the base quantity, e.g., mass = density × volume.

Probability of an event = (number of favorable outcomes)/(total number of equally likely outcomes).

Geometric probability = (favorable length, area, or volume)/(total length, area, or volume) for a randomly chosen point.

Complement rule: P(not A) = 1 − P(A). Addition rule: P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); for dependent events multiply by the conditional probability P(B|A).

Conditional probability: P(B|A) = P(A and B)/P(A).

Two-way frequency tables give joint, marginal, and conditional relative frequencies by dividing by the appropriate total.

Expected value = Σ(value × probability); it summarizes the long-run average of a random quantity.

In synthetic geometry data, the number of diagonals of an n-gon is n(n − 3)/2, and interior angles sum to (n − 2)·180°.

A conditional and its contrapositive are logically equivalent; the converse and inverse are equivalent to each other.

Law of Detachment applies p → q with p to conclude q; the Law of Syllogism chains p → q and q → r into p → r.

Vertical angles are congruent, and a linear pair of angles is supplementary.

Segment addition and angle addition postulates justify breaking a length or angle into parts.

A single counterexample is enough to disprove a universal conditional statement.

SSS, SAS, ASA, AAS, and HL prove triangle congruence; SSA and AAA do not.

Similar triangles satisfy AA, SSS similarity, or SAS similarity, giving proportional corresponding sides.

Scale factor k relates lengths by k, areas by k², and volumes by k³ in similar solids.

A midsegment of a triangle is parallel to the third side and half as long.

CPCTC lets you conclude corresponding parts are congruent after proving the triangles congruent.

Distance formula is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint averages the coordinates.

Perpendicular slopes multiply to −1; parallel slopes are equal.

Rotation 90° counterclockwise about the origin maps (x, y) to (−y, x).

A dilation centered at the origin multiplies each coordinate by the scale factor k.

Reflection over y = x swaps coordinates: (x, y) becomes (y, x).

Pythagorean Theorem a² + b² = c² holds only for right triangles; its converse tests for a right angle.

In a 30-60-90 triangle the sides are in ratio 1 : √3 : 2; in a 45-45-90 triangle they are 1 : 1 : √2.

sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Law of Cosines c² = a² + b² − 2ab·cos C handles SSS and SAS cases.

The angle of elevation equals the angle of depression when measured between the same horizontal and the line of sight.

An inscribed angle measures half its intercepted arc; a central angle measures the full arc.

A tangent is perpendicular to the radius at the point of tangency.

Chord products are equal for chords intersecting inside a circle.

Arc length is (θ/360°)·2πr and sector area is (θ/360°)·πr².

The standard circle equation (x − h)² + (y − k)² = r² gives center (h, k) and radius r.

Volume of a cone or pyramid is one-third of the prism or cylinder with the same base and height.

Sphere volume is (4/3)πr³ and sphere surface area is 4πr².

Composite figure area is found by adding or subtracting standard shapes.

Scaling all dimensions by k multiplies surface area by k² and volume by k³.

Cylinder surface area is 2πr² + 2πrh, counting both circular bases and the lateral rectangle.

Geometric probability compares a favorable region to a total region of the same dimension.

The complement rule states P(not A) = 1 − P(A).

For independent events, P(A and B) = P(A)·P(B).

Conditional probability is P(B|A) = P(A and B)/P(A).

Expected value is the sum of each outcome value times its probability.