Reference Sheet for Algebra/Geometry

Reference sheet for Algebra/Geometryx Exponents:

a2 – b2 = (a – b) (a + b)

Copyright © Regents Exam Prep Center http://regentsprep.org

Perimeter: add the distances around the outside. Circumference: 2C r dS S

Pythagorean Theorem: Right Triangles only. 2 2 2c a b � Triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

Trig: Right triangles only

sin oA

h ( ; cos a

Ah

( ; tan oA

a (

Angle of elevation: from horizontal line of sight up. Angle of depression: from horizontal line of sight down.

Volume and Surface Area: rectangular solidV l w h < <

rectangular solid 2 2 2SA lh hw lw � � 2

cylinderV r hS 2

closed cylinder 2 2SA rh rS S �

Error in Measurement: Relative error = |measure-actual| actual % of Error = Relative • 100% Permutations: Arrangement in specific order.

!P( )!n r

nn r

�

Data: 5 Statistical Summary: minimum, maximum, median, 1st quartile, 3rd quartile Quartiles divide data into 4 equal parts. Percentiles divide data into 100 equal parts.

Percentile rank of score x 100number of scores below xn

< , where n is

the number of scores. Mean = average. Mode = most often (may be more than one answer). Median = middle. Outliers = values that are far away from the rest of the data. Median best describes data if outliers exist. Range = difference between the maximum and minimum values.

Area: 12triangleA bh

2

3

4equilateral triangles

A

rectangleA bh 2

squareA bh s

parallelogramA bh

1 2rhombus 2

d dA bh

<

trapezoid 1 21 ( )2

A h b b � 2

circleA rS 2

sector of circle 360n

A rS

2semicircle

12

A rS

2quarter circle

14

A rS

Probability: P(A’) = 1 – P(A) complement P(A and B) = P(A)•P(B) independent P(A and B) = P(A)•P(B/A) dependent P(A or B) = P(A) + P(B) mutually exclusive P(A or B) = P(A) + P(B) – P(A and B) not exclusive P(B/A) = P(A and B)/P(A) conditional probability P(B/A) means probability of B given A has occurred.

Box and Whisker Plot: 1st and 3rd quartiles are at the ends of the box, median is a vertical line in the box, and the max/min are at the ends of the whiskers. Helpful in interpreting the distribution of data.

Literal equations: a = b + cd, solve for c. a – b = cd a – b = c d Use same strategies as for solving equations.

Sets: Union - all elements in both sets. Intersection - elements where sets overlap.

' Complement - elements not in the set.

A BA BA

*�

{ } or � means null set.

Exponential Growth and Decay: Decay: where 0 and 0 1xy ab a b ! � �

Growth: where 0 and 1xy ab a b ! !


Algebra – Things to Remember! Scientific Notation: 3.2 x 1013

The first number must be 1 < n < 10 Factorial: 5! = 5•4•3•2•1 1! = 1 FYI: 0!=1

Absolute Value: | -5 | = 5 | 5 | = 5 Represents distance

Exponents: 2 2( 3) 3� z � •m n m nx x x �

02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y

Properties of Real Numbers: Commutative Property: a + b = b + a ab = ba Associative Property: a+(b+c) = (a+b)+c a(bc) = (ab)c Distributive Property: a(b+c) = ab + ac Identity: a + 0 = a a • 1 = a Inverse: a + (-a) = 0 a • (1/a) = 1 Zero Property: a • 0 = 0

Undefined: 6

7 x� is undefined when x = 7 since

the denominator = 0.

Degree: Degree of monomial = sum of exponents 4x3 is of degree 3 x2y3 is of degree 5

Polygons and sides: triangle – 3 quadrilateral – 4 pentagon – 5 hexagon – 6 septagon – 7

octagon – 8 nonagon – 9 decagon – 10 dodecagon - 12

Multiply: (distribute or FOIL) ( 3)( 2) • • 2 3• 3• 2x x x x x x� � � � � = 2 5 6x x� �

2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �

Direct Variation: y = kx where k = constant of variation k = y/x

Solving Equations: 1. Deal with any parentheses in the problem. 2. Combine similar terms on same side of = sign. 3. Get the needed variables on the same side of = sign. 4. Isolate the needed variable by add or subtract. 5. Find the needed variable by divide or multiply.

Add Fractions: Get the common denominator: 5 3 5 9 14 76 2 6 6 6 3x x x x x x� �

Factor: Look for a GCF (greatest common factor) Factor binomial or trinomial.

2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �

Interval Notation: (1,5) 1 5[1,5] 1 5

xx

l � �l d d

Inequalities: 5 3 13 Remember to

3 8 change direction4 8 of inequality when

2 mult/div by a negative.

x xx xx

x

� d �� d �� dt �

Function: Passes the vertical line test.A set of ordered pairs in which each x element has only one y element associated with it. f (x) = 3x + 4 f (3) = 3•3 + 4 = 13

Systems: y – 2x = 1 y + 2x = 9

Linear: substitute; add to eliminate one variable or graph.

y = x2 –x-6 y = 2x – 2

Linear Quadratic: substitute or graph

For inequality systems, graph. x = abscissa, y = ordinate Slope:

2 1

2 1

= = .

y yvertical change risem

horizontal change run x x�

�

Equations of Lines: m = slope

1 1

slope-intercept( ) point-slope

y mx by y m x x ��

Parallel and Perpendicular: Parallel: slopes are equal. Perpendicular: slopes are negative reciprocals (flip over and negate)

Parabola: 2y ax bx c � �

Axis of symmetry:

2

bx

a�

Roots: where the graph crosses the x-axis.






02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�







02 1 •( )n m n mx x

33

144

� m

m nn

xx

x�

( ) •n n nxy x y


Undefined: 6







2 2 2( ) 2a b a ab b� � � 2 2 2( ) 2a b a ab b� � �





2 2 ( )( )a b a b a b� � �

Quadratic Equation:

2 5 6 0 Set = 0.( 3)( 2) 0 Factor.

3; 2 Find roots

x xx x

x x

� � � �


xx

l � �l d d




x xx xx

x

� d �� d �� dt �


Systems: y – 2x = 1 y + 2x = 9


y = x2 –x-6 y = 2x – 2



2 1

2 1

= = .



�


1 1





Axis of symmetry:

2

bx

a�






sin oA

h ( ; cos a

Ah

( ; tan oA

a (




cylinderV r hS 2



!P( )!n r

nn r

�



< , where n is



2

3


A

rectangleA bh 2

squareA bh s

parallelogramA bh

1 2rhombus 2

d dA bh

<

trapezoid 1 21 ( )2

A h b b � 2

circleA rS 2


A rS

2semicircle

12

A rS

2quarter circle

14

A rS






A BA BA

*�








sin oA

h ( ; cos a

Ah

( ; tan oA

a (




cylinderV r hS 2



!P( )!n r

nn r

�



< , where n is



2

3


A

rectangleA bh 2

squareA bh s

parallelogramA bh

1 2rhombus 2

d dA bh

<

trapezoid 1 21 ( )2

A h b b � 2

circleA rS 2


A rS

2semicircle

12

A rS

2quarter circle

14

A rS






A BA BA

*�








sin oA

h ( ; cos a

Ah

( ; tan oA

a (




cylinderV r hS 2



!P( )!n r

nn r

�



< , where n is



2

3


A

rectangleA bh 2

squareA bh s

parallelogramA bh

1 2rhombus 2

d dA bh

<

trapezoid 1 21 ( )2

A h b b � 2

circleA rS 2


A rS

2semicircle

12

A rS

2quarter circle

14

A rS






A BA BA

*�




Copyright © Regents Exam Prep Center

Geometry – Things to Remember! Regular Solids: Tetrahedron – 4 faces Cube – 6 faces Octahedron – 8 faces Dodecahedron – 12 faces Icosahedron – 20 faces

3-D Figures: Prism: V = Bh

Pyramid: 13

V Bh

Cylinder: 2V r hS ; 22 2SA rh rS S �

Cone: 213

V r hS ; 2SA s r rS S �

Sphere: 343

V rS ; 2 24SA r dS S

Locus Theorems: Fixed distance from point.

Fixed distance from a line.

Equidistant from 2 points.

Equidistant 2 parallel lines.

Equidistant from 2 intersecting lines

Polygon Interior/Exterior Angles: Sum of int. angles = 180( 2)n �

Each int. angle (regular) = 180( 2)nn�

Sum of ext. angles = 360

Each ext. angle (regular) = 360n

Congruent Triangles SSS SAS ASA AAS HL (right triangles only)

NO donkey theorem (SSA or ASS)

CPCTC (use after the triangles are congruent)

Related Conditionals: Converse: switch if and then Inverse: negate if and then Contrapositive: inverse of the converse (contrapositive has the same truth value as the original statement)

Triangles: By Sides: Scalene – no congruent sides Isosceles – 2 congruent sides Equilateral – 3 congruent sides By Angles: Acute – all acute angles Right – one right angle Obtuse – one obtuse angle Equiangular – 3 congruent angles(60º) Equilateral l Equiangular Exterior angle of a triangle equals the sum of the 2 non-adjacent interior angles. Mid-segment of a triangle is parallel to the third side and half the length of the third side.

Inequalities: --Sum of the lengths of any two sides of a triangle is greater than the length of the third side. --Longest side of a triangle is opposite the largest angle. --Exterior angle of a triangle is greater than either of the two non-adjacent interior angles.

Pythagorean Theorem: 2 2 2c a b �

Converse: If the sides of a triangle satisfy 2 2 2c a b � then the triangle is a right triangle.

Similar Triangles: AA SSS for similarity SAS for similarity Corresponding sides of similar triangles are in proportion.

Mean Proportional in Right Triangle: Altitude Rule: Leg Rule:

altitupart hyp =other

deal partitude t hyp

hyp =

projele

cg

leg tion





sin oA

h ( ; cos a

Ah

( ; tan oA

a (




cylinderV r hS 2



!P( )!n r

nn r

�



< , where n is



2

3


A

rectangleA bh 2

squareA bh s

parallelogramA bh

1 2rhombus 2

d dA bh

<

trapezoid 1 21 ( )2

A h b b � 2

circleA rS 2


A rS

2semicircle

12

A rS

2quarter circle

14

A rS






A BA BA

*�




Reference Sheet for Algebra/Geometry

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