geometry final 6-11

circumference

mesure around the circle

2 pie r

arc length

a portion of the circumference in units

ratio of length of given arc to circumference is equal to ration of measure of arc to 360

how to find arc length

arc length over 2 pie r equals measure of arc over 360

sector of a circle

region bounded by two radii of the circle and their intercepted arc

area of a circle

pie r squared

radius

a segment whose endpoints are the center and any point on a circle

diameter

a chord (segment) the contains the center of the circlewhich is twice the length of the radius.

how to find the area of a sector

area= measure of arc over 360 times pie r squared

population density

the measure of how many people live within a given area

number of people over area of line in square miles

chord

segment whose endpoints are on a circle

secant

a line that INTERSECTS a circe in 2 points

tangent

a line in the plane of a circle that intersecta the circle in exactly 1 point

point of tangency

the point where a tangent intersects a circle

tangent circles

coplanar circles that intersect at one point

concentric circles

coplanar circles that have a common center

comeon tangent

a line of segment that is tangent to 2 coplanar circles

common internal tangent

intersects the segments that joins the centers of the 2 circles

common external tangent

DOES NOT intersect the segment that joins the center of the 2 circles

tangent line to circle theorem

a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle

external tangent congruence theorem

tangent segments from a common external point are congruent

standard equation of a circle

(x-h)² +(y-k)²=r² center (h,k)

central angle

an angle whose vertex is the center of the circle

the measure of a circular arc is the measure of its central angle

minor arc

less than 180

major arc

more than 180

semicircle

arc with endpoints that are the endpoints of the diamenter

adjacent arcs

two arcs of the same circle that intersect at exactly 1 point

arc addition postulate

the measure of an arc formed by 2 adjacent arcs is the sum measure of the 2 arcs

congruent circles theorem

two circles are congruent if and only if they have the same radius

congruent arcs

two arcs are congruent if and only if they have the same measure and they are arcs of the same circle or of congruent circles

congruent central angles theorem

in the same circle of in congruent circles two minor arcs are congruent if and only if their corresponding central angles are congruent

similar circles theorem

all circles are similar

similar arcs

two arcs are similar if and only if they have the same measure. all congruent ares are similar but NOT all similar arcs are congruent

inscribed right triangle theorem

the measure of an inscribed angle is 90 if and only if the segment across from It is the diameter

inscribed quadrilateral theorem

a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary

measure of inscribed angles

is one HALF the measure of its intercepted arc→add up to 360

(angle whose vertex is on a circle)

how to find the cent and radius from this equation

x²+y²-8x+4y-16=0

isolate constants→x²-8x+y²+4y=16

complete squares(half it squared)→x²-8x+16+y²+4y+4=16+16+4

factor left, simplify right→(x-4)²+(y+2)²=36

rewrite equation →(x-4)²+(y+2)²=6²

how to find radius for standard equation for circle

A)use distance formula with given center and point

B) count if you have a graph

C) plug into formula with given point and center

polyhedron

solid that is bounded by polygons called faces

vertex of polyhedron

where 3 or more faces meet

prism

can cut it in the same way and get the same shape

pyramid

comes to a point

to name a prism or pyramid you use the

base

a plane cuts through a solid, the intersection is called the

cross section

solid of revolution

3 dimensional figure formed by rotating a 2 dimensional shape around an axis

axis of revolution

the line around the shape is rotated

area of rhombus and kite

½ diagonal1 times diagonal2

center of regular polygon

center of its circumscribed circle

apothem

distance from the center to any SIDE

area of a regular polygon

½ aPwhere a is the apothem and P is the perimeter.

how to find area of regular polygon

1) finder central angle →360 over number of sides

2)find apothem→ cos half central angle=apothem over radius

3) find side length→tan half central angle=side length/apothem x 2

4) find perimeter→side lengthxside number

5)find area→1/2 aP, where a is the apothem and P is the perimeter.

right triangle similarity theorem

if the altitude(height) is drawn to the hypoteneus of a right triangle then the two triangles formed are similar to the original

geometric mean of 2 positive numbers

a/x=x/b

geometric mean(altitude) theoremn

in a right triangle the altitude from the right angle to the hypotenuse divides the hypotneus into 2 segments. the length of the altitude is the geometric mean of the lengthes of the 2 segments of the hypotenuse

CD²(altitude)=AD+BD(2segments of hypotenuse)

how to find geometric mean

a and b

square root of a times square root of bis equal to }\sqrt{a \cdot b}.

area of a triangle

½ the product of the lengths of the 2 sides times the sine of their INCLUDED angle

law of sines

used to solve a triangle when 2 angles and the length of any side are known(AAS or ASA) of when the lengths of 2 sides and an angle opposite on of the 2 sides are known(SSA)

Sine A/a=sine B/b=sine C/c

when to use negative trig functions

when solving for an angle measure

EX. solving for sin B with given sine= 115,side a -20 and side b-11

sin 115/20=sin B/11→20(sinB)-9.97/20→sinB=0.49→sin-1 0.49=29.9

inverse tangent

if tan A=x than tan -1 x=m angle A

inverse sine

if sine A=x sine -1 x=m angle A

inverse cosine

if cos A=x than cos -1 x=m angle A

solving a right triangle

to find all unknown side lengths and angle measures. you can do this when you know either 2 side lengths of 1 side length and the measure of 1 acute angle

SOH

sine(x)=opposite side over hypotonuse

CAH

cosine(x)=adjacent side over hypotenuse

TOA

tangent(x)=opposite side over adjacent side

sine and cosine of complementary angles

the sine of an acute angle is equal to the cosine of its compliment(opposite angle)

sin A=cos(90-A)=cos B →all functions interchangable

pythagreon triples

3x,4x,5x _5x,12x,13x_ 8x,15x,17x_ 7x,24x,25x

pythagreon inequality theorem

for any triangle ABC where c is the length of the longest side the following statements are true

if c² is less than a² +b² triangle ABC is acute

if c² is greater than a² +b² triangle ABC is obtuse

to determine if it is a triangle 2 smaller legs must be

larger than the longest side added together

45,45,90 triangle theorem

hypotnuse is root 2 times as long as each leg

30,60,90 triangle theorem

hypotenuse is twice as long as the shorter leg and the longer leg is root 3 times as long as the shorter leg

triganomic ratio

ration of the lengths of the 2 sides in a right triangle