preACT math deck

x² + y² = r²

equation of a circle with a center at the origin

(x,y) is any point on the circle

R is the radius

Pythagorean Theorem

(x-h)² + (y-k)² = r²

Equation of a circle with a center other than the origin

The center is (h,k) and the radius is r.

(x,y) is any point on the circle

Area of a Parallelogram

base x height= length x width

Area of Triangles

½ x base x height

½ bh

Area of Trapezoids

½ h (sum of both bases)

½ x height x (base 1 + base 2)

Areas of Rhombuses and Kites

½ (diagonal 1 x diagonal 2)

½ ( the length of diagonal 1 x the length of diagonal 2)

Areas of Regular Polygons

½ aP

a= length of the apothem, the perpendicular distance from the center to the side

p= perimeter

a regular polygon has all congruent sides so to find the perimeter multiply the side length by the number of sides

Area of a Circle

pi x r²

Area of a Circle Sector

A= x/360° times $\pi r^2$

the ratio of the sector’s angle over 360° times the area of the circle

Rectangular prism surface area

2B + Ph

b= area of base

p= perimeter of base

h= height of prism

Surface area of a cylinder

2•area of a circle + 2•pi•r•h

Surface Area of a Regular Pyramid

b+ ½ pl

P= perimeter of base

l= slant height

Surface Area of a Cone

area of a circle + pi• r• l

l= slant height

r= radius of base

Volume of Rectangular Prism

length • width • height

Volume of a Triangular Prism

½ • base• height• lengthwise

Volume of a Cylinder

area of a circle• height

Volumes of Pyramids

1/3 • area of base • height

Volume of a Cone

1/3 • area of a circle • height

Surface Area of a Sphere

4• the area of a circle

Volume of a Sphere

4/3 • pi• r³

what is a chord?

a chord divides a circle into minor and major arcs

the minor arc= arc of a chord

Theorems about Chords

In a circle or congruent circles, congruent circles have congruent arcs

In a circle or congruent circles, congruent chords are equidistant from the center

If a diameter is perpendicular to a chord, then it bisects the chord and its arc

What is the perpendicular bisector of a chord?

The diameter

What is a tangent?

a line, segment, or ray that intersects a circle at exactly one point (called the point if tangency)

Theorems about Tangents

two circles have a common tangent if a line is tangent to both circles

a line is tangent to a circle if and only if it is perpendicular to the radius (drawn at point of tangency)

tangent segments from the same point outside the circle are congruent

what is a radian?

the measure of a central angle that has an arc length that is equal to the radius (another way of measuring angle lengths).

Common Radians

2$\pi$ radians = 360° (full circle)

$\pi$ radians= 180° (semicircle)

1 radian=57.296 degrees (180/$\pi$)

$\pi$ /2 radian= 90°

$\pi$/4 radian= 45°

$\pi$/6 radian= 30°

$\pi$/3 radian= 60°

Radian- Degree Conversions

to convert radians to degrees=multiply by 180°/ $\pi$

to convert degrees to radians=multiply by $\pi$/180°

Arc Length vs Arc Measure

2 arcs can have the same measure but different lengths

arc measure= central angle measure

arc length= measure of central angle/ 360° x d$\pi$

Congruent Arcs

Arcs with the same measure and are in the same circle or congruent circles

What type of triangle is this?

equilateral= equilangular

3 congruent sides + 3 congruent angles

What triangle is this?

Isosceles triangle

2 congruent sides + 2 congruent base angles

What is this triangle

Scalene triangle

0 congruent sides + 0 congruent angles

how many angles does an acute triangle and obtuse triangle have?

Acute = 3 acute angles

Obtuse= 1 obtuse angle

How to find exterior angles?

Sum of 2 inside angles = exterior angle

neighboring interior angle is supplementary to exterior angle

What is an included angle/side?

the angle between 2 sides

The side between two angles

All Triangle Congruence Theorems

SSS (side-side-side) Congruence Postulate- all three sides of one triangle are congruent to all three sides of another

SAS (side-angle-side) Congruence Postulate- two sides and the included angle are congruent to the corresponding 2 sides and included angle of another triangle

ASA (angle-side-angle) Congruence Postulate- two angles and an included side of one triangle are congruent to two angles and an included side of another triangle

AAS (angle-angle-side) Congruence Theorem- two angles and a non included side of one triangle are congruent to two angles and a NON included side of another triangle

HL (hypotenuse leg) Theorem- the hypotenuse and leg of two RIGHT triangles are congruent

What is a perpendicular bisector?

a line, ray, or segment that divides a line into half (bisects it) and forms right angles

If a point is on the perpendicular bisector, then it’s equidistant to the segment’s endpoints

The converse of this theorem is also true

-divided in half

-equidistance

-right angles

Collinear and Coplanar

Collinear points lie on the same line.

Coplanar points lie on the same planes.

Angle Pairs

adjacent angles- are on the same plane, and share a vertex and side, but chill alone

vertical angles- nonadjacent, opposite each other, forming two straight intersecting lines.

What is a segment bisector?

a line, ray, segment or plane that passes through a segment at its midpoint and divides the segment into two congruent segments

What is an Angle Bisector?

a ray that divides an angle into 2 congruent angles

Triangle Similarity Theorems

AA (angle-angle) Similarity Postulate- two congruent angles of two triangles

SAS (side-angle-side) Similarity Theorem- if <A is congruent to <D, and AB/DE= AC/DF, then triangle ABC is similar to triangle DEF

SSS (side-side-side) Similarity Theorem- if AB/DE= BC/EF= AC/DF (proportional sides), then the triangles are similar

What is inductive reasoning?

hypothesis formed from observations

observation: every cat Emily meets purrs

conclusion: Emily then assumes all cats purr

What are conjectures and counterexamples?

Conjecture= conclusion

Every cat Emily meets purrs- observation

Emily then assumes all cats purr- conjecture

counterexample- proving the conjecture wrong

conjecture- all cats purr

counterexample: one cat that doesn’t purr

Central Angles and Arcs

central angle- angle with a vertex at the circle center

arc- part of the circumference; name by putting a curve over the letters of the two endpoints

sector- a ‘slice’ of the circle

minor arc- <180°

major arc- >180° use three letters to name

angles add up to 360°

180° arc= semicircle

What is a secant?

a line that intersects a circle at two points (goes through it)

What happens when two secants intersect?

these formulas are also true for secant + tangent or 2 tangents

when two secants intersect inside a circle:

the angle formed= ½ (x°+ y°)

when two secants intersect outside a circle:

the angle formed= ½ (x°- y°)

Deductive Reasoning; Laws of Detachment and Syllogism

deductive reasoning- facts= conclusion

law of detachment: if p-q and p is true, q is true.

law of syllogism: if p-q and q-r are true, then p-r is true

1. If i watch a scary movie (p), then I get scared (q).

2. If i get scared (q) then I will hide under my blanket ®.

   conclusion: if i watch a scary movie (p), then I will hide under my blanket ®.

Conditional and Biconditional Statements

conditional statements: if…(p) then….(q) conjecture

regular: all fish have gills

Conditional: if it is a fish, it has gills. (p-q)

converse: if it has gills, it is a fish (q-p)

both are true statements which means this is a biconditional statement

biconditional= true conditional + true converse

Identifying Rectangles and Squares

a rectangle is a parallelogram with:

-4 right angles

-congruent diagonals

A square is a parallelogram with:

-4 right angles

-4 congruent sides

rectangle + rhombus= square

Right Triangle Trigonometric Ratios; Sine, Cosine, Tangent

SOH-CAH-TOA

sin= opposite/hypotenuse

cos= adjacent/hypotenuse

tan= opposite/adjacent

these measures are for a specified angle and opposite side, adjacent side, hypotenuse

Theorems to prove a Kite

A kite is a quadrilateral with 2 pairs of adjacent congruent sides.

-two pairs of adjacent congruent sides

-1 pair of congruent opposite angles

-perpendicular diagonals

What is a polygon?

a closed plane figure with at least 3 straight sides

Polygon Interior and Exterior Angle Measures

sum of interior angles: (n-2) • 180

{number of sides, minus 2, times 180= the sum of angle measures)

ex. 5 sides= 3 triangles= 3• 180= 540°

Polygon Exterior Angle Sum Theorem

the exterior angles of a polygon ALWAYS add up to 360°, no matter the # of sides.

1 angle per veryex

REGULAR POLYGON- all congruent angles and sides

Rotation Rules for Coordinate Points (counterclockwise)

90° (or 270° c)- (x,y) becomes (-y,x)

180°- (x,y) becomes (-x,-y)

270° (or 90° c)- (x,y) becomes (y,-x)

if the center of rotation isn’t at the origin, subtract the center from the coordinate point being rotated, add the rule, then add the center back to the point

Reflection Rules

reflections- rigid motion transformations

Preimage- triangle ABC

Image- triangle A’B’C’

‘- prime

Line of Reflection Rules:

x-axis: (x,y) becomes (x,-y)

y-axis: (x,y) becomes (-x,y)

y=x: (x,y) becomes (y,x)

Other (ex. X=2): count. The point was 2 points above the line in the pre image so in the image it’s going to be 2 points below the line

Theorems to Prove a Rhombus

same properties of a parallelogram + some more

-perpendicular diagonals

-has a diagonal that bisects a pair of opposite angles

-has one pair of consecutive congruent sides

all sides of a rhombus are congruent