Key Fact SAT Ch. 1-6

Key Fact A28

For any number a: 1 * a = a and a/1 = a. For any integer n: 1^n = 1 1 is a divisor of every integer. 1 is the smallest positive integer. 1 is an odd integer. 1 is the only integer with only one divisor. It is not a prime.

Key Fact J1

In any triangle, the sum of the measure of the three angles is 180 degrees.

Key Fact J12 (Triangle Inequality)

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

Key Fact A27

0 is the only number that is neither positive nor negatives. O is smaller than every positive number and greater than every negative number. 0 is an even integer. 0 is a multiple of every integer. For every number a: a + 0 = a - 0 = a For every number a: a * 0 = 0. For every integer n: 0^n + 0 For every number a ( including 0): a/0 and a/0 are meaningless expressions. (They are undefined.) For every number a other than 0: 0/a = 0/a = 0 0 is the only number that is equal to its opposite: 0 = -0. If the product of two or more numbers is o, at least one of the numbers is 0.

Key Fact C8

A decrease of a% followed by a decrease of b% always results in a smaller decrease than a single decrease of (a + b)%. Similarly, an increase of a% followed by an increase of b% always results in a larger increase than a single increase of (a + b)%. In particular, an increase (or decrease) of a% followed by another increase (or decrease) of a% is never the same as a single increase (or decrease) of 2a%.

Key Fact A25

Adding a number to an inequality or subtracting a number form the inequality preserves the inequality. If a < b, then a + c < b +c and a - c < b-c Adding inequalities in the same direction preserves them. If a < b, then c < d , then a + c < b + d Multiplying or dividing an inequality by a positive number preserves the inequality. If a< b and c is positive, then ac < bc and a/c < b/c. Multiplying or dividing an inequality by a negative number reverses the inequality. If a < b, and c is negative, then ac> bc and a/c > b/c. Taking negatives reverses and inequality . If a < b, then -a > -b, and if a > b, then -a < -b. If two numbers are each positive or each negative, taking reciprocals reverses an inequality. If a and b are both positive or both negative and a < b, the 1/a > 1/b.

Key Fact N1

All the points on a horizontal line have the same y-coordinate. To find the distance between them, subtract their x-coordinates. All the points on a vertical line have the same x- coordinate. To find the distance between them, subtract their y- coordinates.

Key Fact J10

An altitude divides an equilateral triangle into two 30-60-90 triangles.

Key Fact C7

An increase of k% followed by a decrease of k% is equal to a decrease of k% followed by an increase of k%, and is always less than the original value. The original value is never regained.

Key Fact I1

Angles are classified according to their degree measures. An acute angle measures less than 90 degrees The right angle measure 90 degrees An obtuse angle measures more than 90 degrees but less than 180 degrees. A straight angle measures 180 degrees.

Key Fact E4

Assume that the average of a set of numbers is A. If a number, x, is added tot he set and a new average is calculated, then the new average will be less than, equal to, or greater than A, depending on whether x is less than, equal to, or greater the A, respectively.

Key Fact A11

Every integer greater than 1 that is not a prime can be written as a product of primes. The least common multiple of two or more integers is the small positive integer that is a multiple of each of them. The greatest common factor or greatest common divisor of two or more integers is the largest integer that is a factor of each of them.

Key Fact A10

Every integer has a finite set factors (or divisors) and an infinite set of multiples. Positive integers, such as 7, that have exactly two positive divisors are called prime number or primes.

Key Fact A15

For any b : b ^ 1 = b For any number b and integer n > 1 : b ^ n = b * b * ..... * b, where b is used as a factor n times.

Key Fact A3

For any number a : a * 0 = 0. Conversely, if the product of two or more numbers is 0, at least one of them must be 0. If ab = 0, then a = 0 or b = 0 If xyz = 0, then x = 0 or y = 0 or z = 0

Key Fact A2

For any number a and positive number b: / a / = b _____ a = b or a = -b / a / < b ______ -b < a < b / a / > b ______ a < -b or a > b

Key Fact A1

For any number a, exactly one of the following is true: a is negative a = 0 a is positive The absolute value of a number a, denoted as /a/, is the distance between a and 0 on the number line. Since 3 is 3 units to the right of 0 on the number line and -3 is 3 units to the left of 0, both have ------ of 3:

Key Fact A24

For any numbers a and b, exactly one of the following is true: a> b or a = b or a < b.

Key Fact A23

For any numbers a and b: a > b means that a - b is positive. For any numbers a and b: a < b means that a-b is negative

Key Fact A16

For any numbers b and c and positive integers m and n: b^mb^n = b^m+n b^m/b^n = b^m-n (b^m)^n = b ^mn b^mc^m = (bc)^m

Key Fact A17

For any positive integer n: 0^n = 0 If a is positive, a^n is positive. If a is positive, a^n is positive. If a is negative, a^n is positive if n is even, and negative if n is odd.

Key Facts A18

For any positive number a, there is a positive number b that satisfies the equation b^2 = a. That number is called the square root of a, and we write b^2 = a. That number is called the square root of a , and we write b = √a. Therefore, for any positive number a : √a * √a = (√a)^2 = a.

Key Fact C3

For any positive number a: a% of 100 is a. In any problem involving percents, use the number 100.

Key Fact J7

For any positive number x, there is a right triangle whose sides are 3x, 4x, 5x.

Key Fact A19

For any positive numbers a and b: √ab = √a * √b and √a/b = √a/√b.

Key Fact C4

For any positive numbers a and b: a% of b = b% of a.

Key Fact A20

For any real number a is not = 0: a^0 = 1 For any real number a is not = 0: a^-n = 1/a^n

Key Fact N7

For any real number a: x = a is the equation of the vertical line that crosses the x- axis at ( a, 0). For any real number b: y = b is the equation of the horizontal line that crosses the y-axis at ( 0, b ). For any real numbers b and m: y = mx + b is the equation of the line that crosses the y-axis at ( 0, b ) and whose slope is m.

Key Fact A22 (the distributive law)

For any real numbers a, b, and c: a(b+c) = ab+ac a(b-c) = ab-ac and if a is not = 0, b+c/a = b/a + c/a b-c/a = b/a - c/a

Key Fact A26

If 0 < x < 1, and a is positive, then xa < a. If 0 < x < 1, and m and n are integers with m > n > 1, then x^m < x^n < x. If 0 < x < 1, then √x > x. If 0 < x < 1, then 1/x > x. In fact, 1/x > 1.

Key Fact J15

If A represents the are of an equilateral triangle with side s, then A = s^2√3 / 4.

Key Fact C6

If a < b, the percent increase in going from a to b is always greater than percent decrease in going from b to a.

Key Fact I8

If a line is perpendicular to each of a pair of lines, then these lines are parallel.

Key Fact I7

If a pair of lines that are not parallel is cut by a transversal, none of the statements listed in KEY FACT I6 is true.

Key Fact I6

If a pair of parallel lines is cut by a transversal that is not perpendicular to the parallel lines: Four of the angles are acute, and four are obtuse. All four acute angles ate congruent. All four obtuse angles are congruent. The sum of any acute angle and any obtuse angles is 180 degrees.

Key Fact I5

If a pair of parallel lines is cut by a transversal that is perpendicular to the parallel lines, all eight angles are right angles.

Key Fact D1

If a set of objects is divided into two groups in the ratio of a : b, then the first group contains a / a + b of the objects and the second group contains b / a + b of the objects. In any ration problem, write the letter x after each number and use some given information to solve for x.

Key Fact E1

If all the numbers in a set are the same, then that number is the average. If you know the average, A, of a set of n numbers, multiply A by n to get their sum.

Key Fact N3

If any two points , the midpoint, M, of segment PQ is the point whose coordinates are ( x + x /2, y + y / 2 )

Key Fact G5

If one of the equations in the system of equations consists of a single variable equal to some expression, substitute that expression for the variable in the other equation.

Key Fact J17

If the measures of two angles of one triangle are equal to the measures of two angles of a second triangle, the triangles are similar.

Key Fact E2

If the numbers in a set are not all the same, then the average must be greater than the smallest number and less than the largest number. Equivalently, at least one of the number is less than the average and at least one is greater.

Key Fact A14

If two integers are both even or both odd, their sum and difference are even. If one integers is even and the other odd, their sum and difference are odd. The product of two integers is even unless both of them are odd.

Key Facts N6

If two non vertical lines are parallel, their slopes are equal. If two non vertical lines are perpendicular, the product of their slopes is -1.

Key Fact D2

If two numbers are in the ration a : b, then, for some number x, the first number is ax and the second number is bx. If the ratio is in lowest terms, and if the quantities must be integers, then x is also an integer. Solve proportions by cross-multiplying: if a / b = c /d, then ad = bc.

Key Fact I2

If two or more angles form a straight angle, the sum of their measures is 180 degrees.

Key Fact J18

If two triangles are similar, and if k is the ratio of similitude, then: The ratio of all the linear measurements of the triangles is k. The ratio of the areas of the triangles is k^2

Key Fact J11

In a 30-60-90 right triangle the sides are x, x√3, and 2x.

Key Fact J8

In a 45-45-90 right triangle, the sides are x, x, and x√2 Therefore: By multiplying the length of a leg by √2, you get the hypotenuse. By dividing the hypotenuse by √2, you get the length of each leg.

Key Fact K3

In any polygon, the sum of the measures of the exterior angles, taking one at each vertex, is 360 degrees.

Key Fact K1

In any quadrilateral, the sum of the measures of the four angles is 360 degrees.

Key Facts J4

In any right triangle, the sum of the measures of the two acute angles is 90 degrees.

Key Fact J3

In any triangle: The longest side is opposite the largest angle The shortest side is opposite the smallest angle Sides with the same length are opposite angles with the same measure.

Key Fact J5

Let a, b, and c be the sides of triangle ABC, with a < or equal to b < or equal to c. If triangle ABC is a right triangle, a^2 + b^2 = c^2 If a^2 + b^2 = c^2, then triangle ABC is a right triangle.

Key Fact J6

Let a, b, and c be the sides of triangle ABC, with a < or equal to b < or equal to c. a^2 + b^2 = c^2 if and only if angle C is a right angle. a^2 + b^2 < c^2 if and only if angle C is obtuse. a^2 + b^2 > c^2 if and only if angle C is acute.

Key Fact G1

Memorize the six steps in order, and use this method whenever you have to solve this type of equation or inequality.

Key Fact G3

Occasionally on the SAT, you will have to solve an equation such as 3 √x - 1 = 5, which involves a radical. Proceed normally, treating the radical as the variable and using whichever of the six steps are necessary until you have a radical equal to a number. Then raise each side to the same power.

Key Fact D4

On the SAT, if one quantity increases while another decreases, multiply them; their product is a constant. Problems like this one are examples of inverse variation. We say that one variable varies inversely with a second variable if their product is constant. If y varies inversely with x, there is a constant k such that xy = k. On the SAT, the statement "y varies inversely with x" is oftern expressed as "y is inversely proportional to x."

Key Fact K4

Parallelograms have the following properties: Opposite sides are congruent: AB = CD and AD = BC. Opposite angles are congruent: a = c and b = d. Consecutive angles add up to 180 degrees: a + b = 180, b + c = 180, c + d = 180, and a + d = 180. Two diagonals bisect each other: AE = EC and BE = ED. A diagonal divides the parallelogram into two triangles that have exactly the same size and shape. (The triangles are congruent.)

Key Fact K5

Since a rectangle is a parallelogram, all of the properties listed in KEY FACT K4 hold for rectangles. In addition: The measure of each angle in a rectangle is 90 degrees The diagonals of a rectangle are congruent: AC = BD.

Key Fact K6

Since a square is a rectangle, all of the properties listed in KEY FACT K4 and K5 hold for squares. In addition: All four sides have the same length Each diagonal divides the square into two 45-45-90 right triangles. The diagonals are perpendicular to each other: AC is perpendicular to BD.

Key Fact J14

The area of a triangle is given by A = 1/2bh, where b = base and h = height.

Key Fact J9

The diagonal of a square divides the square into two isosceles right triangles.

Key Fact J13

The difference between the lengths of any two sides of a triangle is less than the length of the third side.

Key Fact N2

The distance between two points, can be calculated using the distance formula

Key Fact A21

The laws of exponents given in KEY FACT A16 are true for any exponents, not just positive integers.

Key Fact J2

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

Key Fact F1

The only term of a polynomial that can be combined are like terms.

Key Fact C5

The percent increase of a quantity is actual increase / original amount * 100% The percent decrease of a quantity is actual decrease / original amount * 100%

Key Fact A4

The product and the quotient of two positive numbers or two negative numbers are positive; the product and the quotient of a positive number and a negative number are negative.

Key Fact A5

The product of an even number of negative factors is positive. The product of an odd number of negative factors is negative.

Key Fact A12

The product of the GCF and LMC of two numbers is equal to the product of the two numbers.

Key Fact A6

The reciprocal of any nonzero number a is 1 / a. The product of any number and its reciprocal is 1: a (1 / a) = 1.

Key Fact N5

The slope of any horizontal line is 0: slope of RS = 1 - 1 / 4 - (-2) = 0/6 = 0 The slope of any line that goes up as you move from left to right is positive: slope of RT = 4 - 1 / 0 - (-2) = 3/2 The slope of any line that goes down as you move from left to right is negative: slope of ST = 1 - 4 / 4 - 0 = 3 / 4.

Key Fact A8

The sum of any number and its opposite is 0: a + (-a) = 0.

Key Fact I3

The sum of the measures of all the angles around a point of 360 degrees.

Key Fact K2

The sum of the measures of the n angles in a polygon with n sides is ( n-2) * 180 degrees.

Key Fact A7

The sum of two positive number is positive. The sum of two negative numbers is negative. To find the sum of a positive and negative number, find the difference of their absolute values and use the sign of the number with the larger absolute value.

Key Fact F7

The three most important binomial products on SAT are these: ( x - y )( x + y ) = x^2 + xy - yx - y^2 = x^2 - y^2 ( x - y)^2 ( x - y )( x - y ) = x^2 - xy - yx + y^2 = x^2 -2xy + y^2 ( x + y ) ^2 = ( x + y )( x + y ) = x^2 + xy + yx + y^2 = x^2 + 2xy + y^2

Key Fact E3

The total deviation below the average is equal to the total deviation above the average.

Key Fact F2

To add two polynomials, first enclose each one in parentheses and put a plus sign between them; then erase the parentheses and combine like terms.

Key Fact E6

To calculate the weighted average of a set of numbers, multiply each number in the set by the number of times it appears, add all the products, and divide by the total number of numbers in the set.

Key Fact C2

To convert a decimal to a percent, or a fraction to a percent, follow these rules: 1. To convert a decimal to a percent, move the decimal point two places to the right, adding 0's if necessary, and add the % symbol. 2. To convert a fraction to a percent, first convert the fraction to a decimal, then do step 1.

Key Fact C1

To convert a percent to a decimal, or a percent to a fraction, follow these rules: 1. To convert a percent to a decimal, drop the % symbol and move the decimal point two places to the left, adding 0's if necessary. (Remember: it is assumed that there is a decimal point to the right of any whole number.) 2. To convert a percent to a fraction, drop the % symbol, write the number over 100, and reduce.

Key Fact F8

To divide a polynomial by a monomial, use the distributive law. Then simplify each term by reducing the fraction formed by the coefficients to lowest terms and applying the laws of exponents.

Key Fact F9

To factor a polynomial, the first step is always to use the distributive property to remove the greatest common factor of all the terms.

Key Fact F10

To factor a trinomial use trial and error to find the binomials whose product is the given trinomial.

Key Fact A13

To find the GCF or Lcm of two or more integers, first get their prime factorizations. The even numbers are all the multiples of 2. The odd numbers are all the integers not divisible by 2.

Key Fact F5

To multiply a monomial by any polynomial, just multiply each term of the polynomial by the monomial.

Key Fact F4

To multiply monomials, first multiply their coefficients, and then multiply their variables by adding the exponents.

Key Fact F6

To multiply two binomials, use the so-called FOIL method, which is really nothing more than the distributive law. Multiply each term in the first parentheses by each term in the second parentheses and simplify by combining terms, if possible.

Key Fact G4

To solve a system of equation, add or subtract them. If there are more than two equations at them.

Key Fact A9

To subtract signed numbers, change the problem to an addition problem by changing the sign of what is is being subtracted, and then use KEY FACT A7. The integers are {... , -4, -3, -2, -1, 0, 1, 2, 3, 4, ...} The positive number are { 1, 2, 3, 4, 5, ....} The negative integers are {..., -5, -4, -3, -2, -1} Consecutive integers are two or more integers, written in a sequence, each of which is 1 more than the preceding integer.

Key Fact F3

To subtract two polynomials, enclose each one in parentheses, change the minus sign between them to a plus sign and change the sign of every term in the second parentheses. Then use KEY FACT F2 to add them: erase the parentheses and combine like terms.

Key Fact J16

Two triangles are similar provided that the following two conditions are satisfied. 1. The three angles in the first triangle are congruent to the three angles in the second triangle. 2. The lengths of the corresponding sides of the two triangles are in proportion: AB/ DE = BC / EF = AC/ DF

Key Fact I4

Vertical angles are congruent.

Key Facts N4

Vertical lines do not have slopes. To find the slope of any other line, proceed as follows: 1. Choose any two points on the line. 2. Take the differences of the y-coordinates, y - y, and the x- coordinates, x - x. 3. Divide: slope = y - y / x - x

Key Fact G2

When you have to solve for one variable in terms of the others, treat all of the others as if they were numbers, and apply the six-step method.

Key Fact E5

Whenever n numbers form an arithmetic sequence (one in which the difference between any two consecutive terms is the same): (i) if n is odd, the average of the numbers is the middle term in the sequence; and (ii) if n is even, the average of the numbers is the average of the two middle terms.

Key Facts D3

applies to extended rations, as well. If a set of objects is divided into three groups in the ration a : b : c, then the first group contains a / a + b + c of the objects, the second b / a + b + c, and the third c / a + b + c. Set rate problems up just like ratio problems. Then, solve the proportions by cross-multiplying.