math final!!

ratio

the quotient of two quantities

written as:

x to y

x/y

x:y

the order matters! always put the first term as the numerator

extended ratio

compares 3 or more numbers

proportion

an equation stating that two ratios are equal

extremes are the first and last numbers in a proportion

means are the middle two

cross products are the product of the means and also the product of the extremes

cross products property

in a proportion, the products of the extremes and means equal each other

properties of proportions

you can write the reciprocal of each ratio

you can switch the means

you can add the denominator to the numerator and divide by the denominator

similar figures

figures that have the same shape but not necessarily the same size

corresponding angles are congruent, and corresponding segments are proportional

order matters in congruency statements!

Angle-Angle Similarity Postulate

if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar

Side-Angle-Side Similarity Theorem

if the angle of one triangle is congruent with the angle of another triangle and the included sides are proportional, the triangles are similar

Side-Side-Side Similarity Theorem

if corresponding sides of triangles are proportional, the triangles are similar

arithmetic sequence

pattern of numbers where any term in the sequence is found by adding or subtracting the previous term by the *common difference*

geometric sequence

pattern of numbers where any term is found by multiplying the previous term by a *common factor*

geometric mean

a term between two terms of a geometric sequence

consecutive terms of a geometric sequence are proportional

formula: a/x = x/b or x=√ab

altitude

segment drawn from a vertex that is perpendicular to the opp. of a triangle

right triangle similarity theorem

if the altitude is drawn to the hyp. of a right triangle the two triangles made + the orginial are similar to each other

Heartbeat (altitude theorem)

the altitude from the right angle to the hyp. divides the hyp. into two segments. the length of the altitude is the GM of the two segments

Boomerang (leg theorem)

the length of each leg in a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hyp. that is adjacent to the leg

side-splitter theorem

if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally

part two of side-splitter

if 3 or more parallel lines intersect two transversals then the segments intercepted on the transversals are proportional

triangle-angle-bisector theorem

if a ray bisects an angle of a triangle it divides the opposite side into 2 segments that are proportional to the other two sides of the triangle

transformation

when you change the position, shape, or size of a figure

the original figure is the preimage, the result is the image

isometry

when the preimage and image are congruent, also called a %%rigid transformation%%

the three types are: @@translations@@, reflections, and ==rotations==

naming images

arrow notation (→)

prime notation (‘) is used to identity image points

example: K→K’

translation rule

ex: (x,y) → (x-3,y+2)

or:

reflections

when a figure flips across a line, the preimage and image are congruent with opposite orientations

line symmetry/reflectional symmetry

figures are reflections on either sides of the line, and are congruent

the line of symmetry divides these parts

rotations

the center of rotation is what an object rotates from

always counter clockwise

center of a regular polygon

point that is equidistant from its vertices

the amount of vertices determine the number of congruent triangles

rotational symetry

if a figure rotates ≤180° and becomes its own image

angle of rotation is the smallest angle needed for this to happen

dilation

scale factor makes a smaller or larger copy of a figure that is similar

enlargements are if the scale factor is greater than 1

reductions are if the scale factor is b/t 0 and 1

pythagorean theorem

a²+b²=c²

%%c= hypotenuse%%

pythagorean triples are integers that work

if c² > a²+b² the triangle is obtuse

if c²< a²+b² the triangle is acute

45-45-90 Triangles

isosceles right triangles

hyp = leg x √2

30-60-90 Triangles

\-can seperate equilateral triangles into 2 of these

long leg = short leg x √3

hyp. = short leg x 2

trigonometry!!

\-calc in degrees mode

\-round to the nearest thousandth

tangent ratio

tan = opposite leg/adjacent leg

tangent ratios cannot be 0 or negative

sine

sin = opposite leg/hypotenuse

always b/t 0 and 1

cosine

cos = adjacent/hyp.

always b/t 0 and 1

inverse ratios

use to find missing angles

(tan, sin, cos) to the negative first power on ur calc

elevation and depression angles

A of E - the angle formed between the horizontal and diagonal when looking up

A of D - the angle formed between the horizontal and diagonal when looking %%down%%

the line of sight is what you see

DONT USE THE VERTICAL SIDE!!!

skipping unit 10!

refer back to those other flashcards for that :)

3 dimensional nets

A 3D net is a 2D pattern that can be folded to create a 3D shape. It shows all the faces of the shape and how they connect.

Surface area

the area of the surface the solid

LA + area of the bases

SA = 2B + Ph

B = area of the base

prism

polyhedron with two congruent parallel faces (bases),

the other face are lateral faces

name a prism using the base shapes

Lateral area

the sum of the areas of the lateral faces

LA = Ph

P = perimeter of one base

h = height of the prism

cylinder

Surface area of a cylinder

2B + 2πrh or 2πr²+2πrh

pyramid

a polyhedron where the base it a polygon and the lateral faces are triangles that meet

Sa=B+½Pl

l = fancy squiggly l, aka the slant height

cone

the base of a cone is always a circle

SA = B + πrl

or SA = πr²+πrl

l= fancy l aka slant height

volume of a prism

volume is the amount of space a figure occupies

V = Bh

B= area of the base

Cavalieri’s Principle

if two solids have the same height and same area at every cross section they have the same volume

volume of a cylinder

V=Bh

V=πr²h

composite solid

combination of two simple solids

volume of a pyramid

V=1/3Bh

it is one third the volume of a prism

volume of a cone

V=1/3Bh

V=1/3πr²h