Geometry Unit 9 & 10 Circle/Area OL IHS

circle

set of points that are equidistant from a given point called a center

-shaded area of circle is ellipse

radius

one point in them middle and one at the end

r= d/2

diameter

goes through the center of the circle

-a diameter is always a chord

chord

segment with endpoints on a circle

-chord is not always diameter

secant

line that intersects a circle in two points

tangent

intersects circle in exactly 1 point

intersecting chords theorem

if two chords intersect, the product of the lengths of segments of one chord is equal to the product of the lengths of the segments of another

intersecting secants length theorem

when two sacant lines intersect, PA(PB) = PC(PD)

-multiply outside of segment with the entire segment by the other outside of segment and entire segment

Intersecting secant and tangent theorem

PC^2 = PA(PB)

-pc is tangent, pa is the outside part of segment, pb is the entire part of the segment

formula for circumference

C=dπ

-circumference increases/decreases when diameter increases/decreases

-distance around the edge of circle

area for circle

πr²

-amount of space inside

central angle

vertex at center of circle, enclosed by two radii

-major central angle greater than 180

-minor central angle less than 180

arc

part of a circle with endpoints that lie on the circle

-defined by central angle

-major or minor

-if two chords are congruent, then so are the arcs

arc addition postulate

if an arc is formed by its adjacent arcs, then the measure of that arc is the sum of adjacent arcs

sector

portion of one circle that is enclosed by 2 radii and an arc of the circle

-major arcs --> major sector, vise versa

semicircle

sector enclosed by diameter of a circle and arc of the circle

-central angle is 180

quad circle

sector with a 90 degree central angle

how does the angle formed by the radii determine the sector type

-major/minor central angle = sector is major/minor

-180 degrees = semi circle

-90 degrees = quad circle

formula for arc length

mAB/360 times 2pi(radius)

measure of arc vs length of arc

-incline created by two sides that forms the angle of the arc

-distance between 2 end points (length)

--> ratio of degrees of arc to circle (360)

tangent line

touches circle at only 1 point

-point of tangency

two tangent theorems

-If a line is tangent to a circle, it is perpendicular to the radius drawn in the point of tangency

-tangent segments to a circle from the same external point are congruent

Radian

Unit for measuring angles

-angle measure for 1 radian produces arc length equal to circle's radius

- 1 rad = 57.3 degrees

- 360 = 2pi rad

- 180 = pi rad

How to convert from degrees to radians

multiply the degree by pi/180

-reduce to lowest fraction, keep the pi

How to convert from radians to degrees

Multiply given value by 180/pi rad

How to tell if a point is inside/outside, or on circle

if the distance b/w a point and a circle's center is less than the radius, then it is inside

-if it is greater, then it is outside

-if it is the exact same, then it is on the circle

inscribed angle

formed by two chords of a circle with a common endpoint that forms a vertex

circumscribed angle

angle formed by two tangents to a circle with a common endpoint (vertex)

inscribed angle theorem and intercepted arcs

measure of an inscribed angle is half of the measure of the intercepted arc

properties of inscribed figures

-figure inside a circle

-All regular polygons can be inscribed in a circle

-All vertices of an inscribed polygon are on the circle.

-The center of an inscribed regular polygon is the center of the circle

-The radius of a regular polygon is also the radius of the circle.

inscribed quadrilateral theorem

a quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary

how to use a perpendicular bisector to construct a square in a circle

-always divides an segment into 2

-forms right angle

-to divide circle into equal angles, diving by number of vertices 360/n

tangent radius theorem

if a line is perpendicular to the raidus of a circle at its endpoint on the circle, then it is a tangent

-converse is also true

-forms right angles

-may need to use pythagorean theorem

tangent tangent angle theorem

If two segments from the same exterior point are tangent to a circle, then they are congruent

-forms a tangent tangent angle

-m

equation of a circle

(x-h)^2 + (y-k)^2 = r^2

- (h,k) is center, r is radius

-take opposites of h and k

-if given points instead of radius, use distance formula from the equation and the point to find the radius

how to plot a circle

plot center, then at least 4 points that are r units from center

review how to find equations of circle from different forms using algebra

1

perimeter of regular polygons

-side lengths are the same

how to tell if its regular polygon

all sides and angles are congruent

how to find perimeter irregular polygon

-add up all sides to find perimeter

-can find it using area formula, pythagorean theorem, trig ratios

finding perimeter of irregular polygons with area

use area formula and substitute known values

finding perimeter of irregular polygons with pythagorean theorem

-u will know 2 out of 3 sides

finding perimeter of irregular polygons with trig ratios

-sin = opp/hyp

-cos A/ adj/hyp

-tan = opp/adj

hypotenuse (trig ratios)

-longest side of a right triangle and the side across from right angle

opposite (trig rtios)

side of an angle that is across the angle

adjacnet

side that touches the angle, not hypotenuse

-multiply by what isolates the variable

how to find perimeter of a polygon on a graph

use pythagorean theorem or distance formula

obtuse triangle

A triangle with one angle that is greater than 90 degrees.

-height is always outside of the triangle

acute triangle

A triangle that contains only angles that are less than 90 degrees.

-base has to be perpendicular to height

equation to calculate area of regular polygons

A = perimeter(apothem/2)

-triangle and rectangles must not overlap and you must be able to find area of them

apothem

a segment from the center of a regular polygon that is perpendicular to any side

area to find a sector

n/360pir^2

-n is measure of central angle

segment of a circle

a region of a circle between a chord and the minor arc formed by that chord

area to find segment of a circle

Asegmnet = Asector - A lower triangle

unknown figure

any shape that does not fit a geometric definition

using sine to find unknown height of a triangle

a = 1/2 a times b times sin0

-sin0 is included angle of sides and b

density

how packed something is

-number of things/total space

solid

object with length, width, depth, height

-3 dimensions

-has faces, edges, and vertices

polyhedron

not curved solid

volume

The amount of space an object takes up

surface area

all faces

eulers theorem

F+V=E+2

= F + V - E = 12

-true for all polyhedrons

types of solids

prism, pyramid, cone, sphere, cube, cylinder

area for sphere

4πr^2

area for cube

6s^2

pyramid

-polyhedron

-all lateral faces intersect at vertex

-vertical height called altitude

-slant height (l)

-base must be regular polygon

-1/2Pl + B

-P is perimeter of base, B is area of base

cone

-has altitude, slant height, base is circle

-not polyhedron

- piR(l + r)

regular pyramid

a pyramid whose base is a regular polygon and is a right pyramid

properties of a prism

1. sides of prism are parallelograms

2. cross section of prism is same across all lengths

3. prism is a polyhedron

4. bases are congruent

5. can find SA using nets

types of prisms

rectangular, cubes, triangular, pentagonol, hexagonol

formula for surface area of a prism

SA=2B+Ph

surface area for a cylinder

2πr²+2πrh

st eps for unit conversions

1. cnfirm units can be converted

2. find conversion factor

3. convert

- 100 cm = 1 m

- 3.28084 ft = 1 m

-if using area, square conversion factor first

-12 in = 1 ft