# SAT Facts and Formulas

#### Contents

## Numbers and Operations

### Numbers

- Even numbers: $\ldots, -6, -4, -2, 0, 2, 4, 6, \ldots , 2n, \ldots$, where $n$ is an integer.
- Odd numbers: $\ldots, -7, -5, -3, -1, 1, 3, 5, 7, \ldots, 2n+ 1, \ldots$, where $n$ is an integer.
- Consecutive integers: $\ldots, n-1, n, n+1, n+2, n+3, \ldots$, where $n$ is an integer.

### Number Line

- Consecutive integers: $\ldots, n-1, n, n+1, n+2, n+3, \ldots$, where $n$ is an integer.

### Factors, Divisibility, and Remainders

### Fractions and Decimals

- When dealing with fractions, one whole unit = 1.

### Ratios, Proportions, and Percents

A

**ratio**is is a comparison between two quantities of the same kind. It can be expressed as $a$ to $b$, $a:b,$ $a/b$, or $\frac{a}{b}.$A

**proportion**is an equation stating that two ratios are equivalent.A

**percent**is a fraction of a hundred. It can be expressed as $\frac{a}{100},$ $a\%,$ or $a$ percent.The properties of proportion:

If $a, b, c,$ and $d$ are nonzero and $\frac{a}{b}=\frac{c}{d},$ then $\begin{array}{r c l} \frac{b}{a} &=& \frac{d}{c}\\ \\ \frac{a}{c} &=& \frac{b}{d}\\ \\ ad&=&bc \\ \\ \frac{a}{a+b} &=& \frac{c}{c+d} \\ \\ \frac{a+c}{b+d}&=&\frac{a}{b}=\frac{c}{d}. \end{array}$

### Sequences and Series

- Find the $n^\text{th}$ term in an arithmetic sequence: $a_n = a_1 + (n-1)d$.
- Find the $n^\text{th}$ term in a geometric sequence: $a_n = a_1 \times r^{n-1}$.
- Find the sum of $n$ consecutive terms in an arithmetic sequence: $S_n = \frac{n(a_1+a_n)}{2}$.
- Find the sum of $n$ consecutive terms in a geometric sequence: $S_{n}= a_1 \times \frac{r^n -1} {r-1}, \text{where}\ r\neq 1$.

## Algebra and Functions

- If $a$ and $b$ are integers, and $a, a+1, a+2,\ldots, b-1, b$ is a sequence of consecutive integers, then the number of terms between $a$ and $b$ is $b-a+1$.

### Polynomials

- $a^{2}-b^{2}=(a-b)(a+b)$
- $(a \pm b)^{2} = a^{2} \pm 2ab + b^{2}$

### Exponents

- Rules of Exponents:

$\begin{array} { c | c } \text{ Rule name} & \text{ Rule } \\ \hline \text{Product Rule} & a^m \times a^n = a^{ m + n } \\ & a ^n \times b^n = (a \times b)^ n \\ \hline \text{Quotient Rule} & a^n / a^m = a^ { n - m } \\ & a^n / b^n = (a/b) ^ n \\ \hline \text{Negative Exponent} & a^ {-n} = \frac{1}{a^n} \\ \hline \text{Power Rule} & (a^n)^m = a^ { n \times m } \\ & a ^ { n^ m } = a ^ { \left ( n^ m \right) } \\ \hline \text{Fraction Rule} & a ^ { 1/n} = \sqrt[n]{a } \\ & \sqrt[m]{ a^n} = a^ { n/m} \\ \hline \text{Zero Rule} & a^0 = 1 \\ & 0^ a = 0 \text{ for } a > 0 \\ \hline \text{One Rule} & a^1 = a \\ & 1^a = 1 \\ \end{array}$

The first few perfect squares are $(1, 4, 9, \ldots, 400),$ and cubes $(1, 8, 27, \ldots, 1000).$

$\sqrt{x^{2}} = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$

### Change the Subject

- Know the rules of exponents.

### Inequalities

$x^{2} \geq 0.$

The properties of inequality:

- If $a<b$, then $a+c<b+c$ and $a-c<b-c.$
- If $a<b$ and $c<d$, then $a+c<b+d.$
- If $a<b$ and $c>0$, then $ac<bc$ and $\frac{a}{c}<\frac{b}{c}.$
- If $a<b$ and $c<0$, then $ac>bc$ and $\frac{a}{c} > \frac{b}{c}.$

Reversing the inequality:

- If $a<b$, then $-a>-b.$
- If $a>b$, then $-a<-b.$
- If $a$ and $b$ are both positive or both negative and $a<b$, then $\frac{1}{a}>\frac{1}{b}.$

The properties of numbers between $0$ and $1$ $(\text{for } 0<a<1)$:

- If $0 < b < 1$, then $ba<a$.
- If $m$ and $n$ are integers such that $1<n<m$, then $x<x^{n}<x^{m}$.
- $a<\sqrt{a}$.
- $a<1<\frac{1}{a}$.

### Absolute Value

- $|x| = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$
- $x^{2} \geq 0.$

### Functions

- The domain of $f(g(x))$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
- $\sqrt{x^{2}} = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$
- For $f(x)=\sqrt{x},$ the domain is $x\geq 0$ and the range is $f(x) \geq 0.$

### Linear Functions

- The slope of a line is defined as $\frac{(\text{change in}\ y)}{(\text{change in}\ x)}.$
- A line with a positive slope rises from left to right.
- A line with a negative slope falls from left to right.
- The slope-intercept form is $y=mx+b,$ where $m$ is the line's slope and $b$ is its $y$-intercept.
- The point-slope form is $y-y_{1}=m(x-x_{1}),$ where $m$ is the line's slope and $(x_{1}, y_{1})$ is a point on the line.
- The line $x = a$ is a vertical line that crosses the $x$-axis at $(a,0).$
- The line $y = b$ is a horizontal line that crosses the $y$-axis at $(0,b).$
- If two lines are parallel, their slopes are equal.
- If two lines are perpendicular, their slopes are negative reciprocals of each other.
- If two functions intersect at point $(x,y),$ then $f(x) = y = g(x)$.

### Quadratic Functions

- The parabola $y=a(x-h)^{2}+k$ opens up if $a>0.$
- The parabola $y=a(x-h)^{2}+k$ opens down if $a<0.$
- The parabola $y=ax^{2}+bx+c$ opens up if $a>0.$
- The parabola $y=ax^{2}+bx+c$ opens down if $a<0.$
- The parabola $y=ax^{2}+bx+c$ has a $y$-intercept at $y=c.$
- The parabola $y=a(x-h)^{2}+k$ has a vertex at $(h, k).$
- The parabola $y=a(x-h)^{2}+k$ has an axis of symmetry at $x=h$.
- The parabola $y=ax^{2}+bx+c$ has a vertex at $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right).$
- The parabola $y=ax^{2}+bx+c$ has an axis of symmetry at $x=-\frac{b}{2a}$.

### Coordinate Geometry

- The line $x = a$ is a vertical line that crosses the $x$-axis at $(a,0).$
- The line $y = b$ is a horizontal line that crosses the $y$-axis at $(0,b).$
- If two lines are parallel, their slopes are equal.
- If two lines are perpendicular, their slopes are negative reciprocals of each other.
- If two functions intersect at point $(x,y),$ then $f(x) = y = g(x)$.
- Distance formula: $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}.$
- Midpoint formula: $M=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right).$
- When transforming graphs, trace what happens to each point.

### Functions as Models

- Exponential growth: $y=ba^{x}$, where $a>1$ and $b > 0.$
- Exponential decay: $y=ba^{x}$, where $0<a<1$ and $b>0.$

### Newly Defined Functions

### Direct and Inverse Variation

- Direct variation: $y=k\cdot x, \quad k\neq0.$
- Inverse variation: $y=k\cdot\frac{1}{x}, \quad k\neq0.$

### Translating Word Problems

### Word Problemes

- (Distance) = (Rate) $\times$ (Time).

### Student-Produced Response

## Geometry and Measurement

### Lines and Angles

*Acute:*angles with measure $< 90^\circ$.*Right:*angles with measure $= 90^\circ$.*Obtuse:*angles with measure $> 90^\circ$ and $< 180 ^ \circ$.*Straight:*angles with measure $= 180^ \circ$.Angles at a point sum to $360 ^ \circ$.

- Angles on a line sum to $180 ^ \circ$.
Vertical angles are equal.

$\angle A$ and $\angle B$ are

*complementary*if $m\angle A + m\angle B=90^\circ.$$\angle A$ and $\angle B$ are

*supplementary*if $m\angle A + m\angle B=180^\circ.$*Segment Addition Postulate:*If point $B$ lies between points $A$ and $C,$ then $AB + BC = AC.$*Angle Addition Postulate:*If point $B$ lies inside $\angle AOC,$ then $\angle AOB + \angle BOC = \angle AOC.$The

*midpoint*divides a segment into two congruent segments. If $M$ is the midpoint of $\overline{AB},$ then $\overline{AM} \cong \overline{MB}, AM = MB = \frac{1}{2}AB.$The

*angle bisector*is a ray or a line that divides an angle into two congruent angles. If $\overrightarrow{BD}$ is the bisector of $\angle ABC,$ then $\angle ABD = \angle DBC = \frac{1}{2} \angle ABC.$Angles in a triangle sum to $180 ^ \circ$.

### Parallel Lines

The Properties of Parallel Lines:

Consider the two parallel lines $m$ and $n,$ and the transversal, $p.$

*Interior angles*are angles located on the inside of the parallel lines: $\angle 3, \angle 4, \angle 5,\ \text {and}\ \angle 6.$*Exterior angles*are angles located on the outside of the parallel lines: $\angle 1, \angle 2, \angle 7,\ \text{and}\ \angle 8.$*Corresponding angles*are two angles located in the same position relative to the parallel lines. Corresponding angles are congruent: $\angle 1 \cong \angle 5, \quad \angle 2 \cong \angle 6, \quad \angle 3 \cong \angle 7, \quad \angle 4 \cong \angle 8.$*Alternate-interior angles*are two nonadjacent interior angles located on opposite sides of the transversal. Alternate-interior angles are congruent: $\angle 3 \cong \angle 5 \quad \text{and} \quad \angle 4 \cong \angle 6.$*Alternate-exterior angles*are two nonadjacent exterior angles located on opposite sides of the transversal. Alternate-exterior angles are congruent: $\angle 1 \cong \angle 7 \quad \text{and} \quad \angle 2 \cong \angle 8.$*Same-side interior angles*are two nonadjacent interior angles located on the same side of the transversal. Same-side interior angles are supplementary: $\angle 3 + \angle 6 = 180^\circ \quad \text{and} \quad \angle 4 + \angle 5 = 180^\circ.$*Same-side exterior angles*are two nonadjacent exterior angles located on the same side of the transversal: $\angle 1 + \angle 8 = 180^\circ \quad\text{and} \quad \angle 2 + \angle 7 = 180^\circ.$

### Triangles

*Scalene triangle:*none of its sides are congruent.*Isosceles triangle:*at least two sides are congruent.*Equilateral triangle:*all three sides are congruent.*Acute triangle:*All angles measure less than $90^\circ$.*Obtuse triangle:*One angle measures more than $90^\circ$ and less than $180^\circ$.*Right triangle:*One angle measures $90^\circ.$*Isosceles Triangle Theorem:*The angles opposite the two congruent sides are congruent.The three angles of an equilateral triangle are all equal to $60^\circ$.

The measures of the angles in a triangle add to $180^\circ.$

The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles.

**Inequalities:**

If one side of a triangle is longer than another side, then the angle opposite the first side is bigger than the angle opposite the second side.

If one angle in a triangle is bigger than another angle, then the side opposite the first angle is longer than the side opposite the second angle.

*Triangle Inequality:*The sum of the lengths of any two sides in a triangle is greater than the length of its third side.

**Perimeter and Area:**

Perimeter of a polygon is defined as the sum of the lengths of its sides.

Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$

Area Addition Postulate: The area of a region is the sum of the areas of its non-overlapping parts.

If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.

If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights.

If two figures are similar, and their scale factor is $a:b,$ then the ratio of their perimeters is $a:b$ and the ratio of their areas is $a^2:b^2.$

### Right Triangles

*Pythagorean Theorem*

In a right triangle, the square of the hypotenuse equals the sum of the square of the legs: $c^2 = a^2 + b^2.$- If $c^2 = a^2 + b^2,$ then $\angle C = 90^\circ$ and $\triangle ABC$ is right.
- If $c^2 < a^2 + b^2,$ then $\angle C < 90^\circ$ and $\triangle ABC$ is acute.
If $c^2 > a^2 + b^2,$ then $\angle C > 90^\circ$ and $\triangle ABC$ is obtuse.

*$30^\circ\text{-}60^\circ\text{-}90^\circ$ triangle*

In a $30^\circ\text{-}60^\circ\text{-}90^\circ$ triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is $\sqrt{3}$ times as long as the shorter leg.*$45^\circ\text{-}45^\circ\text{-}90^\circ$ Triangle*

In a $45^\circ\text{-}45^\circ\text{-}90^\circ$ triangle, the hypotenuse is $\sqrt{2}$ times longer than either of the legs.

### Polygons

- $A_\text{parallelogram} = bh,$ where $b$ is the length of the base, and $h$ is the height.
- Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
- Area of a square with side length $s: A_{\square} = s^2.$
- The sum of the measures of the interior angles of a convex polygon with $n$ sides is $180(n-2).$
- The sum of the measures of the exterior angles, one per vertex, of any convex polygon is $360^\circ.$

### Circles

- The circumference of a circle with radius $r$ and diameter $d: C = 2\pi r = \pi d.$
- Area of a circle with radius $r: A_{\bigodot} = \pi r^2.$
- The measure of an arc equals the measure of its central angle.
- The length of an arc with measure $x$ is $\frac{x}{360}\cdot 2 \pi r.$
- The area of the sector formed by an arc measuring $x$ and two radii is $\frac{x}{360} \cdot \pi r^2.$

### Congruence and Similarity

- If two figures are similar, and their scale factor is $a:b,$ then the ratio of their perimeters is $a:b$ and the ratio of their areas is $a^2:b^2$.

### Composite Figures

### Solid Geometry

- Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
- Know the $30^\circ\text{-}60^\circ\text{-}90^\circ$ and the $45^\circ\text{-}45^\circ\text{-}90^\circ$ theorems.
- Area of a circle with radius $r: A_{\bigodot} = \pi r^2.$
- The perimeter of a square with side length $s$: $P_{\square} = 4s.$
- The volume of a cube with edge length $s$: $V = s^3.$
- The volume of a rectangular solid with length $l,$ width $w,$ and height $h: V = l \cdot w \cdot h.$
- The surface area of a cube with edge length $s$: $SA = 6s^2.$
- Volume of a cylinder with base radius $r$ and height $h: V = \pi r^2 h.$

### Conceptual Geometry

## Data Analysis, Statistics, and Probability

### Mean, Median, and Mode

- The average of $n$ numbers is the sum of the numbers divided by $n.$
- If the average of a set of numbers is $A$ and a new number $x=A$ is introduced to the set, the new average will also equal $A.$
- If $n$ numbers are arranged in increasing order, the median is the middle value if $n$ is odd, and it is the average of the two middle values if $n$ is even.
- In a set of numbers, the mode is the number that appears most frequently.
- To find the weighted mean of some numbers, find the product of each number and its weight, then divide the sum of these products by the sum of the weights.

### Data - Tables

### Data - Graphs and Charts

### Sets and Venn Diagrams

- The union of two sets, $A$ and $B,$ is that collection of elements that are in $A,$ or in $B,$ or in both $A$ and $B.$
- The intersection of two sets, $A$ and $B,$ is that collection of elements that are only in both $A$ and $B.$
- If every element in set $A$ is an element in set $B,$ then $A$ is a subset of $B.$

### Counting and Probability

- If $a<b$ are two integers, the number of integers between $a$ and $b$ when one endpoint is included is $b-a.$
- If $a<b$ are two integers, the number of integers between $a$ and $b$ when both endpoints are included is $b-a+1.$
- If $a<b$ are two integers, the number of integers between $a$ and $b,$ endpoints NOT included is $b-a-1.$
- If there are $n$ ways for an event to happen and $m$ ways for another event to happen, then the number of ways for both events to happen is $m\cdot n.$
- If $P(A)$ is the probability that event $A$ will occur, then $0 \leq P(A) \leq 1.$
- If $P(B)$ is the probability that event $A$ does NOT occur, then $P(A) = 1- P(B).$
- Assuming that all the possible outcomes of an event $A$ are equally likely, the probability that $A$ will occur is $P(A) = \frac{(\text{\# of favorable outcomes})}{(\text{total \# of outcomes})}.$
- Two events are independent if the outcome of one does not affect the outcome of the other.
- If events $A$ and $B$ are independent, then $P(A\ \text{and}\ B) = P(A) \cdot P(B).$
- The probability that two events, $A$ and $B,$ happen together is $P(A\ \text{and}\ B) = P(A)\cdot P(B\ \text{given}\ A).$
- If events $A$ and $B$ are mutually exclusive, then $P(A\ \text{or}\ B) = P(A) + P(B).$
- If a point is chosen at random in a geometric figure, the probability that the point lies in a particular region is: $\frac{(\text{area of region})}{(\text{area of whole figure})}.$

**Cite as:**SAT Facts and Formulas.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/sat-facts-and-formulas/