Mathematics
A.P. Sum Formula Choice — If *n* is odd, use Sₙ = n × middle term. If *n* is even, use Sₙ = (n/2)[2a + (n−1)d]. Odd = direct middle; even = formula.
Circle Tangent Length — From external point, two tangents to a circle are *always equal in length*. If solving for x, set tangent segments equal immediately.
Coordinate Distance Shortcut — For distance between (x₁,y₁) and (x₂,y₂): square the differences, add, then root. Remember "Diff-Square-Add-Root" (DSAR).
Heron's Formula Memory — Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. Mnemonic: "Semi-perimeter Subtracted Successively" for the three (s−side) terms.
Sector Area Quick — Area of sector = (θ/360) × πr². Length of arc = (θ/360) × 2πr. Both start with (θ/360); area adds one *r*, arc adds none.
Tangent-Chord Angle — Angle between tangent and chord equals angle in alternate segment. Look for this in circle proofs—instant angle equality.
Parallelogram Area Trap — Area = base × perpendicular height, NOT base × slant side. Always drop perpendicular; never use the sloping edge directly.
Trigonometry Complementary — sin(90−θ) = cosθ; tan(90−θ) = cotθ; sec(90−θ) = cosecθ. "Complements switch to Co-functions."
Section Formula Direction — For internal division m:n, coordinates = (mx₂+nx₁)/(m+n). Heavier weight (m) pulls toward second point (x₂). Weight goes with *opposite* point.
Construction Circle Tangent — To draw tangent from external point: join to center, bisect that line, draw semicircle—intersections give tangent contact points. Semicircle method never fails.
Physics
Ohm's Law Triangle — Draw triangle: V on top, I and R at base. Cover what you need: V = IR, I = V/R, R = V/I. "Very Important Rule."
Fleming's Left-Hand Rule — Thumb = motion/Force, First finger = Field (N→S), Middle = Current direction. "Thumb Force, First Field, Middle Current" (ThuFF-MC for motor).
Right-Hand Thumb Rule — Thumb = current direction; curled fingers = magnetic field direction around straight wire. For coils, fingers = current, thumb = North pole inside.
Lens Formula Sign — Concave lens: f is negative. Object left of lens: u is negative. Real image right of lens: v is positive. "Left and inward = negative."
Power of Lens — P (diopter) = 1/f (meters). Concave has negative power; convex has positive power. Shorter focal length = more powerful lens. P₁+P₂ = combined power.