ACT Math Concepts

Full PDF guide: https://bxsciencepa.org/wp-content/uploads/2016/02/50-Important-Math-Concepts-1.pdf

Rules of Exponents

x⁰

Multiplying exponents

Dividing exponents

Power of exponents

x⁰: 1

Multiplying exponents: add exponents, same base

Dividing exponents: subtract exponents, same base

Power of exponents: multiply exponents, same base

x² - y²

(x + y)(x - y)

Triangle Rules:

1. The angles of a triangle always…

2. The sum of any two sides of a triangle must…

3. Isosceles vs scalene

1. add up to 180 degrees

2. exceed the measure of the third side

3. Isosceles: at least equal two sides, scalene: no sides equal

Pythagorean triples to know:

3, 4, 5

5, 12, 13

8, 15, 17

7, 24, 25

9, 40, 41

Multiples of triples are also triples

Exterior Angle Theorem

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

Parallelogram rules:

1. adjacent angles are…

2. opposite angles are…

3. diagonals…

1. supplementary (add up to 180 degrees)

2. congruent

3. bisect each other

Parallel lines cut by a transversal:

congruent

congruent

supplementary

congruent

supplementary

circumference of a circle

2πr

arc length of a circle

arc length (degrees) = (n/360) x 2πr

area of a circle

πr²

sector area of a circle

(n/360) x πr²

How do arc length and circumference relate to each other? How do area and sector area relate to each other?

Arc length is part of the circumference (rim of circle) and sector area is part of the area of a circle (measure of full “surface”)

What happens to the area and circumference of a circle when you double its radius?

• area is quadrupled

• circumference is doubled

central angle

angle whose two sides are the radii of the circle;

equal to the number of degrees in the intercepted arc

inscribed angle

angle formed by two chords at the edge of a circle;

equal to half of the number of degrees in the intercepted arc

Angles inscribed in a semicircle are…

always 90 degrees

Area formulas

1. triangle

2. rectangle

3. trapezoid

4. rhombus

1. ½bh

2. bh

3. ½h(b₁ + b₂)

4. ½d₁d₂

area of a triangle

½bh

area of rectangle

bh

area of a trapezoid

½h(b₁ + b₂)

area of a rhombus

½d₁d₂

Volume formulas:

1. cube

2. rectangular prism

3. cylinder

4. sphere

5. cone

1. s³

2. lwh

3. πr²h

4. 4/3πr³

5. 1/3πr²h

volume of a cube

s³

^{(s = length of a side)}

volume of a rectangular prism

lwh

volume of a cylinder

πr²h

(r = radius

h = height of cylinder)

volume of a sphere

4/3πr³

volume of a cone

1/3πr²h

Surface area formulas

1. rectangular prism

2. cube

1. 2lw + 2lh + 2wh

2. 6s²

surface area of a rectangular prism formula

2lw + 2lh + 2wh

length of a diagonal of a rectangular prism formula

d² = l² + w² +h²

surface area of a cube formula

6s²

The number of degrees of any n-gon is…

180 (n - 2)

n = number of sides

The angle measurements of any n-gon with equal sides is…

(n - 2) • 180/n

The formula for the number of diagonals in any n-sided polygon:

½n(n - 3)

45-45-90 triangle ratios

If you are given the leg, the hypotenuse is the leg x (sqrt2)

If you are given the hypotenuse, the leg is the hypotenuse/ sqrt2

30-60-90 triangles

ratio is 1: cbrt3 : 2

midpoint formula

x₁+x₂ / 2 , y₁+y₂ / 2

distance formula

√(x₂ - x₁)² + (y₂ - y₁)²

equation of a circle

(x - h)² + (y - k)² = r²

(h, k) is the point the radius is centered at

when the radius is centered at the origin, h and k are set to zero, so the equation is: x² + y² = r²

parabola formula

y = a(x - h)² + k

(h, k) is the vertex

if a > 0, the parabola turns upward; if it is < 0, the parabola turns downward

Equations of a line:

1. slope intercept

2. point-slope

3. standard form

1. y = mx + b

2. y - y₁ = m(x - x₁)

3. ax + by = c

Factor:

x² + bx + c = 0

ax² + bx + c = 0

1. what two numbers multiplied together make c and added together make b

2. what two numbers multiply to make ac and add to make b

Solving absolute value equations and inequalities:

1. |ax + b| = c

   1. when will an equation have no solutions?

2. |ax + b| > c

3. | ax + b | < c

1. set up two equations; one with a positive c and and the other with a negative c

   1. if c is less than zero

2. set up two equations; it can only be one:
   ax + b > c OR ax + b < -c

3. set up one equation
   –c < ax + b < c