ACT MATH

Area of a Triangle

A \= (1/2)(b)(h)

Area of a Circle

The formula to find the area of a circle is `A = πr^2`, where `A` is the area and `r` is the radius of the circle.

Circumference of Circle

The formula for the circumference of a circle is `2πr`, where `r` is the radius of the circle.

Perimeter of a Square

P \= 4(s)

Area of a Square

The area of a square is calculated by multiplying the length of one of its sides by itself. In mathematical notation, it can be represented as A = s^2, where A is the area and s is the length of one side of the square.

Area of a Trapazoid

a and b are the two bases

Area of a Rectangle

A \= (l)(w)

Perimeter of a Rectangle

The perimeter of a rectangle is equal to the sum of all its sides. If the length and width of the rectangle are denoted by `l` and `w` respectively, then the perimeter `P` can be calculated as:

`P = 2l + 2w`

Therefore, the formula for the perimeter of a rectangle is `P = 2l + 2w`.

Properties of a Rectangle

-4 right angles; -Diagonals bisect each other and are congruent; -Opposite sides are equal

Similar Triangles

corresponding angles are equal, corresponding sides are proportionate

Pythagorean Theorem

a^2 + b^2 = c^2

Sine Ratio

SOH

Opposite leg/ hypotenuse

Cosine Ratio

CAH

adjacent leg/ hypotenuse

Tangent Ratio

TOA

opposite leg/ adjacent leg

Distance Formula

The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. It is given by:

`d = sqrt((x2 - x1)^2 + (y2 - y1)^2)`

where `d` is the distance between the two points `(x1, y1)` and `(x2, y2)`.

Midpoint Formula

The midpoint formula is a formula used to find the midpoint between two points in a coordinate plane. It is given by:

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

Where (x1, y1) and (x2, y2) are the coordinates of the two points.

Slope Formula

The slope formula is used to calculate the slope of a line given two points on the line. It is expressed as:

`m = (y2 - y1) / (x2 - x1)`

where `m` is the slope and `(x1, y1)` and `(x2, y2)` are the coordinates of the two points.

Slope-Intercept Formula

y = mx +b

Multiplying Terms with Exponents

add the exponents

Dividing Terms with Exponents

subtract the exponents

Distributing Exponents

multiply the exponents

(4y)^2 = 4^2y^2 = 16y^2

Negative Exponent

take the reciprocal and change the exponent to positive

(a/b) ^ -m = (b/a) ^ m

Positive Slope

rises from left to right

Negative Slope

falls from left to right

Volume of a Cube

V = s \* s \* s

V = s^3

Volume of a Rectangular Solid

V = LWH

Surface Area of a Cube

SA= 6s^2

Surface Area of a Rectangular Solid

SA = 2LW + 2LH + 2WH

Parallel Lines

same slope, different y intercept, no points in common

Secant Ratio - sec(A) … (1/cos)

1/cos(A) \= Hypotenuse/Adjacent

Cosecant Ratio - csc(A) (1/sin)

1/sin(A) \= Hypotenuse/Opposite

Cotangent Ratio - cot(A) … (1/tan) … (cos/sin)

1/tan(A) \= Adjacent/Opposite

Tan(A)

Sin(A)/Cos(A)

Percent Change

% change \= [(new value - old value)/ old value] x 100

what percent

x/100

is

\=

the product of

* (multiplication)

the sum of

\+ (addition)

(positive)*(negative)

\= (negative)

(negative)*(negative)

\= (positive)

Geometric Sequence

r = common ratio between terms (multiply or divide)

An example of a geometric sequence is: 2, 4, 8, 16, 32, ... where each term is obtained by multiplying the previous term by 2.

The formula for a geometric sequence is:

a_n = a_1 \* r^(n-1)

where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the number of terms.

Arithmetic Sequence

d = common difference between terms (add or subtract)

The formula for an arithmetic sequence is:

`a_n = a1 + (n-1)d`

where `a_n` is the `n`th term, `a1` is the first term, `n` is the number of terms, and `d` is the common difference between consecutive terms.

Isosceles Triangle

two angles are equal, two sides are equal

Equilateral Triangle

three angles are equal (60 degrees), three sides are equal

Hypotenuse

Longest side in a right triangle, opposite the right angle

Pythagorean Triples

3, 4, 5 ; 5, 12, 13 ; 8, 15, 17 ; 7, 24, 25

Mean (average)

\= (sum of values) / (\# of values)

Median

middle number - put all numbers in order and pick the middle number

Mode

the number that occurs the MOST

Angles in a Triangle

the three interior angles add to 180 degrees

Y Intercept

y \= mx +b (y intercept is b; when x \= 0 what is y; where a line crosses the y axis)

Slope

change in y / change in x; rise / run

Permutations & Combinations

Use the dash method (ex. \_____ * \_____ * \_____ )

Distributing

3(2x + 5) = 6x+15

Properties of an Isosceles Trapezoid

-opposite sides are equal; -top two angles are congruent; -bottom two angles are congruent; -diagonals are congruent

Triangle Side Lengths

A side length of a triangle is: (1) greater than the other two sides subtracted; (2) less than the sum of the other two sides

Perpendicular lines

1 point in common, form a right angle, slope are negative reciprocals (i.e. 2/3 and -3/2)

Radians to Degrees

Multiply by (180/pi) or (pi/180) to change the units

180 degrees, 360 degrees

pi radians, 2*pi radians

Sin^2 + Cos^2 \= 1

1 - Sin^2 \= Cos^2, 1 - Cos^2 \= Sin^2

Weighted Average

\[(Av 1)(# 1) + (Av 2)(# 2) + (Av 3)(# 3) + (Av 4)(# 4)\] / (# of items)

A weighted average is a type of average that takes into account the importance, or weight, of each value in a data set. It is calculated by multiplying each value by its weight, adding up the products, and dividing by the sum of the weights. This is commonly used in finance, accounting, and statistics to calculate averages that reflect the relative importance of different values.

Equation of a Circle

\
(h, k) - center of the circle, r = radius

Equation of an Ellipse

(h, k) - center of the ellipse, a = x radius, b = y radius

Probability Equation

(Number of desired outcomes) / (Total number of outcomes)

Probability 2 events (AND)

Probability 1 * Probability 2 ( multiply)

Probability 2 events (OR)

Probability 1 + Probability 2 (ADD)

Factors

smaller numbers that evenly go into a larger number (i.e. For the number 12 here are the factors: 1, 2, 3, 4, 6, 12)

Multiples

larger than the number (i.e. For the number 12 here are the factors: 12, 24, 36, 48, ... )

Least Common Multiple (LCM)

Greatest Common Factor (GCF)

Slope Formula

The steepness of a graph line; the ratio of the vertical change (the rise) to the horizontal change (the run).

What is the slope of the straight line passing through the points (-2,5) and (6,4)?

4-5/6\- -2\= -1/8, this graph will rise 1, and go left 8.

Slope-Intercept Formula

Ex: y\= -4/5 - 7. The graph will go up 4, and to the left 5, and the y-intercept will be -7.

Line

A straight path of points that extends forever in two directions. A line doesn't have any thickness or width. Arrows sometimes show that the line goes on forever in either direction

Line segment

The set of points on a line between any two points on that line.

Midpoint

The point halfway between two endpoints on a line segment.

Midpoint Formula

(x₁+x₂)/2, (y₁+y₂)/2

Intersect

To cross. Two lines can intersect each other much like two streets cross each other at an intersection.

Vertical Line

A line that runs straight up and down,

Horizontal Line

A line that runs straight across from left to right. *Think horizon

Parallel Lines

Lines that run in the same direction and keep the same distance apart. Parallel lines never intersect one another.

Perpendicular Lines

Two lines that intersect to form a square corner. The intersection of two perpendicular lines for a right, or 90-degree, angle

Ray

A part of a line, with one endpoint, that continues without end in one direction.

What are the angle facts?

1. No negative angle exists
2. no zero angle exists
3. its rare to see fractional angles

Right Angles

Angles that measure 90 degrees

Obtuse Angles

An angle whose measure is greater than 90 degrees and is less than 180 degrees.

Straight Angles

An angle that measures exactly 180 degrees and forms a straight line.

Complementary Angles

Two angles whose sum is 90 degrees

Supplementary Angles

Two angles whose sum is 180 degrees. Also, there is another 180 degrees below the line, with a total of 360 degrees.

Vertical Angles

Opposite angles with be equal to each other.

Reflex Angles

Angles that have measures greater than 180 degrees and less than 360 degrees

Angle Note

Angles around a point total 360 degrees.

Angle Note

The exterior angles of any figure are supplementary to the interior angles and always total 360 degrees.

Transversal

A line that intersects two or more lines

Vertical Angles

Angles that are opposite of each other have equal .

Corresponding angles

Angles in the same position around two parallel lines and a transversal and have equal measures.

Equilateral Triangle

A triangle with three congruent sides and three equal angles. Because the three angles must add up to 180 degrees, all three angles of an equilateral triangle are always equal to 60 degrees.

Isosceles Triangle

A triangle with two congruent sides. The angles opposite those sides are also equal . If angle A is 50 degrees, then angle B is also 50 degrees.

Scalene Triangle

A triangle with no congruent sides

Right Triangle

A triangle that has one inside angle that is 90 degrees.