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Distance Formula
(x2−x1)2+(y2−y1)2
Midpoint Formula
M = (x1+x2 / 2) , (y1+y2 /2)
Slope Formula
(y2−y1) / (x2−x1)
Area of a Triangle
A = (1/2)bh
b = base and h = height
Area of a Trapezoid
A = (1/2)(b1+b2)h
b1 and b2 are the bases
h = height.
Area of a Circle
A = πr²
r = radius.
Circumference of a Circle
C = 2πr
r = radius.
Area of a Sector
(θ/360) * πr²
θ = central angle in degrees
r = radius.
Arc Length
(θ/360) * 2πr
θ = central angle in degrees
r = radius.
Volume of a Cylinder
V = πr²h
r = radius
h = height.
Volume of a Cone
V = (1/3)πr²h
r = radius
h = height.
Volume of a Sphere
V = (4/3)πr³
r = radius.
Surface Area of a Sphere
SA = 4πr²
r = radius.
Pythagorean Theorem
a² + b² = c²
a and b = legs of a right triangle
c = hypotenuse.
ONLY WORKS ON RIGHT TRIANGLES ‼
Triangle Sum Theorem
The sum of the angles in a triangle is 180°.
Exterior Angle Theorem
An exterior angle of a triangle is equal to the sum of the two remote interior angles.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Isosceles Triangle Rule
If two sides of a triangle are equal, then the angles opposite those sides are equal.
Equilateral Triangle
All sides are equal
All angles are 60°
Right Triangle
One angle HAS to be 90°
SOH-CAH-TOA
Sine = Opposite / Hypotenuse,
Cosine = Adjacent / Hypotenuse,
Tangent = Opposite / Adjacent
Inverse Trig Functions
Used to find angles.
θ = sin⁻¹(opp/hyp),
θ = cos⁻¹(adj/hyp),
θ = tan⁻¹(opp/adj)
30-60-90 Triangle
30, 60, 90 angles,
Hypotenuse = 2x
Longer leg = x√3
Shorter leg = x
45-45-90 Triangle
45, 45, 90 angles,
Legs = x
Hypotenuse = x√2
Sine Rule
a/sinA = b/sinB = c/sinC
Used to find missing sides or angles in non-right triangles.
Cosine Rule
c² = a² + b² − 2ab cos(C)
Used to find missing sides or angles in non-right triangles.
Parallel Lines
Lines with equal slopes
Perpendicular Lines
Lines with negative reciprocal slopes.
Isosceles Triangle Proof
Prove 2 equal sides using the distance formula.
Right Triangle Proof
Prove perpendicular sides using slopes.
Parallelogram Proof
Prove 2 pairs of parallel sides.
Rhombus Proof
Prove 4 equal sides.
Rectangle Proof
Prove 4 right angles using slopes.
Square Proof
Prove rhombus + 1 right angle
Rigid Motions
Transformations that preserve shape and size (translation, rotation, reflection).
Translation
A slide that preserves distance and angle.
Rotation
A turn that preserves distance and angle.
Reflection
A flip over a line that preserves distance and angle.
Congruence
Preserved under rigid motions
Non-Rigid Motions
Transformations that do not preserve shape and size (dilation).
Dilation
Multiply coordinates from center → preserves angles, not size. Produces similar figures.
Rotation 90° CCW
(x, y) → (–y, x)
Rotation 180°
(x, y) → (–x, –y)
Rotation 270° CCW
(x, y) → (y, –x)
Reflection over x-axis
(x, y) → (x, –y)
Reflection over y-axis
(x, y) → (–x, y)
Reflection over y = x
(x, y) → (y, x)
Reflection over y = –x
(x, y) → (–y, –x)
Dilation Rule
(x, y) → (kx, ky) (k = scale factor)
CPCTC
Corresponding Parts of Congruent Triangles are Congruent
Triangle Congruence Theorems
SSS, SAS, ASA, AAS, HL (only for right triangles)
Similarity Theorems
AA, SSS~, SAS~
Parallelogram Properties
Opp sides ≅, Opp angles ≅, Diagonals bisect, Opp sides ‖
Rectangle Properties
All from parallelogram, 4 right angles, Diagonals ≅
Rhombus Properties
All sides ≅, Diagonals perpendicular, Diagonals bisect angles
Square Properties
Rhombus + Rectangle properties
Trapezoid
1 pair of ‖ sides
Isosceles Trapezoid
base angles ≅, legs ≅, diagonals ≅
Kite
2 pairs of adjacent sides ≅, 1 pair of opp angles ≅, Diagonals ⊥, One diagonal bisects the other
Central Angle
Equal to the arc it intercepts in a circle.
Inscribed Angle
½ the intercepted arc in a circle.
Angle in Circle (secant/secant or secant/tangent)
(1/2)(big arc − small arc)
Tangent Line
Perpendicular to the radius at the point of tangency.
Two Tangents from Same Point
Segments are ≅
Intersecting Chords
a⋅b = c⋅d
Two Secants from Outside Point
whole⋅outer = whole⋅outer
Secant & Tangent
tangent² = outer⋅whole
Rigid Motions
Transformations that preserve shape and size (translation, rotation, reflection).
Non-Rigid Motions
Transformations that do not preserve shape and size (dilation).
Linear pair
A pair of adjacent supplementary angles