Geometry Regents

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Flashcards for Geometry Review

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70 Terms

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Distance Formula

(x2x1)2+(y2y1)2\sqrt{(x2-x1)^2+(y2-y1)^2}

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Midpoint Formula

M = (x1+x2 / 2) , (y1+y2 /2)

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Slope Formula

(y2−y1) / (x2−x1)

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Area of a Triangle

A = (1/2)bh

b = base and h = height

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Area of a Trapezoid

A = (1/2)(b1+b2)h

b1 and b2 are the bases

h = height.

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Area of a Circle

A = πr²

r = radius.

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Circumference of a Circle

C = 2πr

r = radius.

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Area of a Sector

(θ/360) * πr²

θ = central angle in degrees

r = radius.

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Arc Length

(θ/360) * 2πr

θ = central angle in degrees

r = radius.

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Volume of a Cylinder

V = πr²h

r = radius

h = height.

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Volume of a Cone

V = (1/3)πr²h

r = radius

h = height.

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Volume of a Sphere

V = (4/3)πr³

r = radius.

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Surface Area of a Sphere

SA = 4πr²

r = radius.

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Pythagorean Theorem

a² + b² = c²

a and b = legs of a right triangle

c = hypotenuse.

ONLY WORKS ON RIGHT TRIANGLES

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Triangle Sum Theorem

The sum of the angles in a triangle is 180°.

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Exterior Angle Theorem

An exterior angle of a triangle is equal to the sum of the two remote interior angles.

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Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

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Isosceles Triangle Rule

If two sides of a triangle are equal, then the angles opposite those sides are equal.

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Equilateral Triangle

All sides are equal

All angles are 60°

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Right Triangle

One angle HAS to be 90°

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SOH-CAH-TOA

Sine = Opposite / Hypotenuse,

Cosine = Adjacent / Hypotenuse,

Tangent = Opposite / Adjacent

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Inverse Trig Functions

Used to find angles.

θ = sin⁻¹(opp/hyp),

θ = cos⁻¹(adj/hyp),

θ = tan⁻¹(opp/adj)

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30-60-90 Triangle

30, 60, 90 angles,

Hypotenuse = 2x

Longer leg = x√3

Shorter leg = x

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45-45-90 Triangle

45, 45, 90 angles,

Legs = x

Hypotenuse = x√2

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Sine Rule

a/sinA = b/sinB = c/sinC

Used to find missing sides or angles in non-right triangles.

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Cosine Rule

c² = a² + b² − 2ab cos(C)

Used to find missing sides or angles in non-right triangles.

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Parallel Lines

Lines with equal slopes

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Perpendicular Lines

Lines with negative reciprocal slopes.

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Isosceles Triangle Proof

Prove 2 equal sides using the distance formula.

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Right Triangle Proof

Prove perpendicular sides using slopes.

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Parallelogram Proof

Prove 2 pairs of parallel sides.

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Rhombus Proof

Prove 4 equal sides.

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Rectangle Proof

Prove 4 right angles using slopes.

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Square Proof

Prove rhombus + 1 right angle

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Rigid Motions

Transformations that preserve shape and size (translation, rotation, reflection).

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Translation

A slide that preserves distance and angle.

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Rotation

A turn that preserves distance and angle.

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Reflection

A flip over a line that preserves distance and angle.

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Congruence

Preserved under rigid motions

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Non-Rigid Motions

Transformations that do not preserve shape and size (dilation).

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Dilation

Multiply coordinates from center → preserves angles, not size. Produces similar figures.

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Rotation 90° CCW

(x, y) → (–y, x)

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Rotation 180°

(x, y) → (–x, –y)

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Rotation 270° CCW

(x, y) → (y, –x)

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Reflection over x-axis

(x, y) → (x, –y)

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Reflection over y-axis

(x, y) → (–x, y)

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Reflection over y = x

(x, y) → (y, x)

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Reflection over y = –x

(x, y) → (–y, –x)

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Dilation Rule

(x, y) → (kx, ky) (k = scale factor)

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CPCTC

Corresponding Parts of Congruent Triangles are Congruent

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Triangle Congruence Theorems

SSS, SAS, ASA, AAS, HL (only for right triangles)

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Similarity Theorems

AA, SSS~, SAS~

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Parallelogram Properties

Opp sides ≅, Opp angles ≅, Diagonals bisect, Opp sides ‖

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Rectangle Properties

All from parallelogram, 4 right angles, Diagonals ≅

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Rhombus Properties

All sides ≅, Diagonals perpendicular, Diagonals bisect angles

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Square Properties

Rhombus + Rectangle properties

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Trapezoid

1 pair of ‖ sides

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Isosceles Trapezoid

base angles ≅, legs ≅, diagonals ≅

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Kite

2 pairs of adjacent sides ≅, 1 pair of opp angles ≅, Diagonals ⊥, One diagonal bisects the other

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Central Angle

Equal to the arc it intercepts in a circle.

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Inscribed Angle

½ the intercepted arc in a circle.

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Angle in Circle (secant/secant or secant/tangent)

(1/2)(big arc − small arc)

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Tangent Line

Perpendicular to the radius at the point of tangency.

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Two Tangents from Same Point

Segments are ≅

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Intersecting Chords

a⋅b = c⋅d

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Two Secants from Outside Point

whole⋅outer = whole⋅outer

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Secant & Tangent

tangent² = outer⋅whole

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Rigid Motions

Transformations that preserve shape and size (translation, rotation, reflection).

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Non-Rigid Motions

Transformations that do not preserve shape and size (dilation).

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Linear pair

A pair of adjacent supplementary angles