Chords and Tangents - Full List

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Circumference

C = 2 π r = π d

Rotation (tranformation)

CIRCULAR MOVEMENT AROUND AN AXIS

Describe a circle defined by (x+3)^2 +(y-2)^2=16

Center (-3,2) Radius=4

Central Angle

Central angles that intercept congruent arcs are congruent. Congruent central angles intercept congruent arcs.

Central Angles

Central angles that intercept congruent arcs are congruent/congruent central angles intercept congruent arcs

Find the centre and radius of the circle (x+2)²+(y-3)²=25

Centre (-2,3), radius 5

Find the centre and radius of the circle (x+4)²+(y+1)²=7

Centre (-4,-1), radius √7

Find the centre and radius of the circle (x+5)²+(y-2)²=4

Centre (-5,2), radius 2

Find the centre and radius of the circle x²+y²+12x−8y+48=0

Centre (-6,4), radius 2

Find the centre and radius of the circle x²+18x+y²+10y+6=0

Centre (-9,-5), radius 10

Find the centre and radius of the circle (x-3)²+(y-2)²=16

Centre (3,2), radius 4

Find the centre and radius of the circle x²-8x+y²-4y+11=0

Centre (4,2), radius 3

Find the centre and radius of the circle (x-8)²+(y-3)²=49

Centre (8,3), radius 7

A segment whose endpoints lie on a circle

Chord

Diameter

Chord that contains the center

If two central angles are congruent, then the _____ they form are congruent

Chords

Concentric Circles

Circles that share the same center.

Congruent Circles

Circles with the same radius.

A chord is a segment connecting two points on the _______ of a circle

Circumference

Congruent chords intercept ______ arcs

Congruent

If two tangent lines of the same intercept a point outside the circle, then they are _____

Congruent

Inscribed angles that intercept the same/congruent arc are ______

Congruent

Parallel chords intercept ______ arcs

Congruent

Two tangents drawn from the same external point are _____

Congruent

Congruent Chords

Congruent chords form congruent arcs

When angles are ON the circle

Cut in HALF

Which equation represents a circle with a center at (-3, -5) and a radius of 6 units? (x - 3)2 + (y - 5)2 = 6 (x - 3)2 + (y - 5)2 = 36 (x + 3)2 + (y + 5)2 = 6 (x + 3)2 + (y + 5)2 = 36

D) (x + 3)^2 + (y + 5)^2 = 36

Angle-Side-Angle (ASA)

DOES prove triangles congruent when two adjacent angles and the included side are congruent

If a diameter is perpendicular to a chord, then the ______ bisects the chord

Diameter

In a circle, a diameter drawn through the midpoint of a chord that is not a diameter is perpendicular to the chord.

Diameter bisects chord → ⊥ to chord

In a circle, a diameter drawn perpendicular to a chord bisects the chord and is major and minor arcs.

Diameter ⊥ to chord → Bisects chord and arcs

construct

Display information in a diagrammatic or logical form.

The measure of a central angle is ____ to the measure of its intercepted arc

Equal

Congruent chord are _____ form the center of the circle

Equidistant

Theorem #4

If a tangent and a secant (or two secants or two tangents) intersect at a point in the exterior of a circle, then the measure of the angle formed is equal to one-half the absolute value of the difference of the measures of the intercepted arcs. angle formed = bigger intercepted arc - smaller intercepted arc / 2

One Secant & One Tangent/Angles Outside the Circle

If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Two Secants/Angles Outside the Circle

If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Two Tangents/Angles Outside the Circle

If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Tangent Radius

If a tangent intersects a radius, then they are perpendicular

Tangent Radius

If a tangent intersects a radius, they are perpendicular

Inscribed Angle in a Semi-Circle

If an angle inscribed to a semi-circle then it is a right angle

Inscribed angle in a semi-circle

If an angle is inscribed to a semi-circle, then it is a right angle

SAS Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

90°

If line TR is tangent to circle P, what is the measure of angle PTR?

exterior

If secants and/or tangents intersect on the __________________ of a circle, then the measure of the angle formed is equal to half the difference of the intercepted arcs.

exterior

If secants intersect on the __________________ of a circle, then the measure of the angle formed is equal to half the difference of the intercepted arcs.

Tangent Chord

If the angle formed by a tangent and a chord intercept the same arc as an inscribed angle then they are congruent

tangent chord

If the angle formed by a tangent and a chord intercept the same arc as an inscribed angle, then they are congruent

SSS Similarity Theorem

If the sides of two triangles are in proportion, then the triangles are similar.

Side-Side-Side (SSS)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS)

If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent.

AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Corollary #4

If two arcs of a circle are included between parallel segments, then the arcs are congruent.

Chord Arcs Conjecture

If two chords in a circle are congruent, then their intercepted arcs are congruent

Chord Arcs Conjecture

If two chords in a circle are congruent, then their intercepted arcs are congruent.

Chord Central Angles Conjecture

If two chords in a circle are congruent, then they determine two central angles that are congruent

Chord Central Angles Conjecture

If two chords in a circle are congruent, then they determine two central angles that are congruent.

Chord Central Angles Conjecture

If two chords in a circle are congruent, there central angles are also congruent.

Chord-chord

If two chords intersect in a circle, then the product of the pieces of one chord equals the product of the pieces of the other chord

Theorem #3

If two chords intersect in the interior of a circle, then the measure of one of the angles formed is equal to one-half the sum of the measures of its intercepted are and the arc intercepted by its vertical pair. one of the angles formed = intercepted arc + intercepted arc by vertical pair / 2

Chord-Chord Product Theorem

If two chords intersect in the interior of a circle, then the products of the lengths of those chord's segments are equal.

Chords Intersect Inside the Circle/Angles Inside the Circle

If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Corollary #2

If two inscribed angles intercept the same arc/congruent arcs, then the angles are congruent.

Inscribed Angles of a Circle Theorem

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Parallel Lines Intercepted Arc Conjecture

If two lines are parallel then they intercept congruent arcs on a circle

Secant-secant

If two secant lines intersect outside a circle, then the product of the outside piece of one secant line and the entire secant line, is equal to the product of the outside piece and entire other secant line (out)(whole)=(out)(whole)

Secant-Secant Product Theorem

If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

intersect

If two secants or chords ______________ inside a circle, then the measure of the angle formed is equal to HALF the sum of the measures of the intercepted arcs.

congruent

If two segments from the same external point are tangent to a circle, then they are __________________________.

Theorem (External Point Tangent)

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

Side-Angle-Side (SAS)

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Line

It has no endpoint.

Ray

It has one endpoint.

Line Segment

It has two endpoints.

What size is the inscribed angle of a semicircle?

Its a right angle - 90 degrees.

JK = ?

Total Area Cone

Lateral Area + 2πr

Chord

Segment whose endpoints lie on the circle

SOH CAH TOA stands for?

Sine= Opposite/Hypotenuse Cosine= Adjacent/Hypotenuse Tangent= Opposite/Adjacent

x= -3

Solve for x.

x=8

Solve for x.

How do you find the radius of a circle?

Square root the number at the end of the equation.

What do you do if its angles>

Subtract

If a quadrilateral is inscribed in a circle, the opposite angles are _____

Supplementary

The measure of an angle formed by a secant and a chord is ____ to the measure if the angle adjacent to it

Supplementary

Theorem

The measure of an angel formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs m<1= 1/2 (outside # x - inside number y)

Intersecting Chords Theorem

The measure of an angle formed by 2 intersecting chords is half the sum of the measure of the 2 intercepted arcs.

Intersecting Secants Theorem

The measure of an angle formed by 2 secants that intersect outside a circle is half the difference of the larger arc measure and the smaller arc measure.

Theorem

The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc

Tangent-Secant Theorem

The measure of an angle formed by an intersecting tangent and secant to a circle is half the difference of the larger intercepted arc measure and the smaller intercepted arc measure.

Intersecting Tangents Theorem

The measure of an angle formed by intersecting tangents to a circle is 180° minus the smaller intercepted arc measure.

Theorem

The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs m< 1= 1/2 [m(arc AC) + m (BD)]

Inscribed Angle Conjecture

The measure of an angle inscribed in a circle is half the measure of the central angle

Theorem #1

The measure of an inscribed angle is equal to one-half the measure of its intercepted arc. inscribed angle = intercepted arc / 2

Inscribed Angle Conjecture

The measure of an inscribed angle is half the measure of its intercepted arc.

Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is one-half the measure of its intercepted arc.

Tangent Chord Theorem

The measure of the angle formed by the intersection of a tangent and a chord at the point of tangency is half the measure of the intercepted arc.

Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the sum of the measure of these two arcs.

What do the opposite angles of quadrilateral inscribed in circle add up to?

The opposite angles of a cyclic quadrilateral add up to 180 degrees. They are supplementary

Cyclic Quadrilateral Conjecture

The opposite angles of a cyclic quadrilateral are supplementary

Cyclic Quadrilateral Conjecture

The opposite angles of a cyclic quadrilateral are supplementary.

Perpendicular Bisector Of A Chord Conjecture

The perpendicular bisector of a chord passes through the center of the circle

Perpendicular Bisector of a Chord Conjecture

The perpendicular bisector of a chord passes through the center of the circle.

Perpendicular to a Chord Conjecture

The perpendicular from the center of a circle to a chord is the bisectie of the chord.

Perpendicular To A Chord Conjecture

The perpendicular from the center of a circle to a chord is the bisector of the chord

Perpendicular to a Chord Conjecture

The perpendicular from the center of a circle to a chord is the bisector of the chord.

What do we know about a perpendicular line from the center that intersects with a chord?

The perpendicular line from the center of the circle to a chord also bisects the chord.

point of tangency

The point where the tangent line and the radius intersect

Means-Extremes Property of Proportions

The product of the extremes equals the product of the means

How do you find the vertical angle that two intersecting chords make?

The vertical angle is the average of the two arcs that are created

What do I know about two congruent chords in a circle?

They are the same distance from the center

What do we know about two chords that are the same distance from the center?

They must be congruent

linear pair

Two angles are said to be linear if they are adjacent angles formed by two intersecting lines (opposite rays). The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.

Inscribed Angles Intercepting Arcs Conjecture

Two angles that intercept the same arc will be congruent

complementary angles

Two angles whose sum is 90 degrees

arcs

Two chords are congruent if their corresponding ____________________ are congruent.

equidistant

Two chords are congruent if they are ____________________ from the center.

Chord Distance To Center Conjecture

Two congruent chords in a circle are equidistant from the center of the circle

Chord Distance to Center Conjecture

Two congruent chords in a circle are equidistant from the center of the circle.

perpendicular

Two lines that intersect to form right angles

perpendicular lines

Two lines that intersect to form right angles

Two Tangent Segments

Two tangent segments from the same external point are congruent

Two tangents drawn to a circle from an external point are congruent.

Equiangular

all angles are congruent

equilateral

all sides are congruent

Fixed Point

also called the center

radius of a regular polygon

an angle formed by two radii drawn from consecutive vertices

central angle

an angle formed by two radii with the vertex at the center of the circle

central angle

an angle made at the center of the circle by 2 radii

inscribed angle

an angle made by 2 chords that intersect on the circle ( divide central angle by 2)

right angle (triangles)

an angle that measures 90 degrees

acute angle (triangles)

an angle that measures less than 90 degrees

inscribed angle

an angle whose vertex is on a circle and whose sides contain chords of the circle

central angle

an angle with its vertex at the center of the circle

central angle (of a circle)

an angle with its vertex at the center of the circle

Central angle

an angle with the vertex is at the center of the circle

How can you identify a central angle?

an angle with the vertex is at the center of the circle

What is a inscribed angle?

an angle with the vertex on the circle

major arc

an arc larger than 180 degrees

minor arc

an arc smaller than 180 degrees

major arc

an arc that is larger than a semicircle

minor arc

an arc that is smaller than a semicircle

point

an exact location in space

How does an inscribed angle compare to its arc?

an inscribed angle is half as big as its arc

exterior intersection formula

angle = (far arc - near arc)/2

interior intersection formula

angle = (pizza crust arc + kissing fish arc)/2

Diameter

any chord of the circle that passes through the center of a line.

a radius

any segment that joins the center to a point of the circle

A Radius

any segment that joins the center to a point of the circle, all are congruent

two inscribed angles that intercept the same _______ are _______________

arc, congruent

within a circle or in congruent circles, congruent chords have congruent

arcs

congruent arcs

arcs in the same circle or in congruent circles have equal meaures

adjacent angles (of a circle)

arcs that have exactly one point in common

within a circle or in congruent circles, congruent central angles have congruent ____________ and congruent ________________

arcs, chords

Segment of a circle formula

area of sector - area of triangle area/360 • ¶r^2 - 1/2bh = Segment

sector of a circle formula

area/ 360 • ¶r^2 = Sector

a° + x° + 90° + 90° = 360°

a°+x° = 180°

Volume Prism

base area x height

major arc

bigger arc (rest of pie slice)

A diameter that is perpendicular to a chord...

bisects the chord and its arc.

If a diameter of a circle is perpendicular to a chord, then the diameter _________________________ the chord and its arc.

bisects. CE = ED

Circumference

c = 2πr or c = πd

sphere

center o and radius r is the set of all points in space at a distance r from point o

in a circle, the perpendicular bisector of a chord contains the _____________________ ____ ____ _______________

center of the circle

Central angle

central angle and intercepted arc have the SAME MEASURE

Angle between - tangent or chord - tangent on circle

central angle ÷ 2 (exactly like inscribed angle)

central angle

central angle=measure of intercepted arc

within a circle or in congruent circles, congruent arcs and congruent chords have congruent __________________ ____________

central angles

diameter

chord that contains the center of a circle

In a circle, if a diameter bisects a _____________ that is not a diameter, then it is ______________ to the chord

chord, perpendicular

within a circle or in congruent circles, congruent arcs have congruent

chords

In the same circle or in congruent circles...

chords equally distant from the center (or centers) are congruent; congruent chords are equally distant from the center (or centers)

an inscribed angle is an angle whose vertex is on the _____________ and is made up of ______ _____________

circle, two, chords

Congruent Circles (or spheres)

circles (or spheres) that have congruent radii

congruent circles (or spheres)

circles (or spheres) that have congruent radii

Congruent or Equal Circles

circles having congruent radii.

congruent circles

circles that have congruent radii

Concentric Circles

circles that lie in the same plane and have the same center

concentric circles

circles that lie in the same plane and have the same center

tangent circles

co-planar lines that are tangent to the same line at the same point

In the same circle, two minor arcs are __________________________ if their corresponding chords are congruent.

congruent

Tangent segments from a common external point are _____________________.

congruent

Tangent-Tangent relationships are ____

congruent

Two inscribed angles that intercept to same arc are

congruent

in the same circle or in congruent circles, chords equally distant from the center are...

congruent

radii

congruent

tangents to a circle from a point are...

congruent

In the same circle or in congruent circles...

congruent arcs have congruent chords; congruent chords have congruent arcs.

congruent chords

congruent chords form congruent arcs

Tangents to a circle from a point are...

congruent.

copy an angle

construct a congruent angle

copy a segments

construct a congruent segment

construct a hexagon inscribed in a circle

bisect an angle

construct a line segment the divides an angle into two congruent angles

construct a parallel line thru a point not on a line

construct a perpendicular from a point not on a line segment

construct a perpendicular from a point on a line segment

construct a perpendicular bisector

construct a segment the is perpendicular to and bisects another segment

construct a square inscribed in a circle

construct an equilateral triangle

construct a triangle with three congruent sides

construct an equilateral triangle inscribed in a circle

Diameter

d = 2 x radius

scale factor can be used to...

determine measures for the rest of the shape

A chord that is a perpendicular bisector of another chord is a _______________ of the circle.

diameter

Circumference of a Circle Formula

diameter times pi (or 2 times pi times radius)

diameter

distance across a circle through the center; twice the radius of a circle

circumference

distance around the edge of a circle

radius

distance from the center to any point on the circle; ½ diameter of a circle

Common external tangent

does not intersect the segment joining the centers of two coplanar planes.

(postulate 18) Area Congruence Postulate

if 2 figures are congruent, then they have the same area

chord-chord product thm

if two chords intersect in the interior of circle, then the products of lengths are equal

Intersecting chords

length of line segments are proportional (ab=cd); chord-chord relationship

tangent

line in the plane of the circle that intersects the circle in exactly one point

Tangent

line that intersects the circle at 1 point

Secant

line that intersects the circle at 2 points

parallel lines

lines in the same plane that never intersect

Formula for Interior Angles

m(of angle)= arc1 + arc2 /2

Angles Formed by Intersecting Tangents

m<P = (arcSXT - arcST)/2

If vertex is outside circle

measure is 1/2 the difference of the measures of its intercepted arcs

If vertex is inside circle

measure is 1/2 the sum of the measures of its intercepted arcs

109

m∠1

128

m∠1

133

m∠1

a chord's two endpoints are

on the circle

a tangent intersects at how many points

one

the __________ angles of quadrilateral inscribed in a circle are _________________________ (add up to _______)

opposite, supplementary, 180

parallel chords

parallel chords form congruent arcs

Lateral Area Prism

perimeter of base x height

A tangent line to a circle is___________________ to the radius/diameter at the point of tangency

perpendicular

If a chord is a ________________ of another chord, the first chord is a diameter.

perpendicular bisector

if a line is tangent to a circle, then the line is...

perpendicular to the radius drawn to the point of tangency

if a line is _____________________ to a radius at its endpoint on the circle then the line is ________________________ to the circle

perpendicular, tangent

Area of a Circle Formula

pi times radius squared

minor arc

points on sides of central angle (interior) equal to measure of central angle named by 2 points

major arc

points surround the central angle (exterior) equal to 360 - measure of central angle named by 3 points

Collinear

points that lie on the same line

Coplanar

points that lie on the same plane

Radius

r = 1/2 diameter

D = 2 * r so r=

r= D ÷ 2

A line is tangent to a circle if and only if the line is perpendicular to a radius at a point of the circle.

radius ⊥ tangent

if a line is tangent to a circle at the _____________ then the line is __________________ to the circle at the point of tangency

radius, perpendicular

ratio of a circle's circumference to its diameter

An angle inscribed in a semicircle is a

right angle

an angle inscribed in a semi-circle is a ____________ ______________. (when there is a triangle drawn in the circle the top angle will be ____)

right angle, 90

tangent-chord, tangent segant angle

same as intercepted angle: 1/2 its intercepted arc

Measure of a Arc

same as the central angle.

Chord

segment whose endpoints lie on a circle

chord

segment whose endpoints lie on a circle

circle

set of points in a plane at a given distance from a given point in that plane

circle

set of points in a plane which are the same distance from the center

(Theorem 8.1) If the altitude is drawn from the hypotenuse of a right triangle, then the two triangles formed are...

similar to the original triangle and to eachother

minor arc

smaller arc

Concentric Spheres

spheres that have the same center

radius

sq rt (x-h)^2 + (y-k)^2

The opposite angles of a quadrilateral inscribed in a circle are

supplementary (add to 180)

Formula for Tangent-Secant

tan^2= OW

Tangent Line to Circle Theorem

tangent lines create a perpendicular angle to a radius at the point of tangency.

if a line in the plane of a circle in perpendicular to a radius at its outer endpoint, then the line is...

tangent to the circle

angle of elevation

the angle formed by a horizontal line and a line of sight to a point above

angle of depression

the angle formed by a horizontal line and a line of sight to a point below

How can you find the arc of any central angle?

the arc of a central angle is equal to that angle

in the same circle or in congruent circles, congruent chords are equally distant from...

the center

center of a regular polygon

the center of the circumscribed circle

a diameter that is perpendicular to a chord bisects...

the chord and it's arc

Inscribed

the circle is circumscribed about the polygon

arc measure

the degree of an arc

radius

the distance from the point of a circle to it's outside line

Point of Tangency

the exact point in which a tangent intersects the circle

When the altitude is drawn from the hypotenuse of a right triangle, each leg is...

the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg

When the altitude is drawn from the hypotenuse of a right triangle, the length of the altitude is...

the geometric mean between the segments of the hypotenuse

The Radius

the given distance

Center

the given point

30-60-90 triangle

the hypotenuse equals 2x the shorter leg; the longer leg equals the square root of 3 times the shorter leg

measure of semi circle

the measure is 180 degrees

Theorem

the measure of an inscribed angle is equal to half the measure of its intercepted arc

If a tangent and secant/chord intersect on a circle at the point of tangency

the measure of angle formed is 1/2 the measure of its intercepted arc

If two secants/chords intersect in the circle's interior

the measure of each angle formed is 1/2 the sum of its intercepted measures

measure of minor arc

the measure of its central arc

arc addition postulate

the measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs

If a tangent + secant , 2 tangents, or 2 secants intersect in a circle's exterior

the measure of the formed angle is 1/2 the difference of the measures of its intercepted arc's

45-45-90 triangle

the measure of the hypotenuse is (√2) times the measure of a leg; the legs equal eachother

center

the middle of a circle

means

the middle terms of a proportion

Segment of a Circle

the part of a circle bounded by an arc and the chord joining its endpoints

circumference

the perimeter of a circle

apothem of a regular polygon

the perpendicular distance from the center of the polygon to a side

(Theorem 11.2) the area of a parallelogram equals

the product of a base and the height to that base (A=bh)

(Theroem 11.1) the area of a rectangle equals

the product of its base and height (A=bh)

When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals...

the product of the other secant segment and its external segment.

When two chords intersect inside a circle, the product of the segment of one chord equals...

the product of the segments of the other chord.

What do we know about two chords that intersect in a circle?

the product of their segments (pieces) must equal each other

ratio (of one number to another)

the quotient when the first number is divided by the second

Tangent ratio

the ratio of the lengths of the legs in a right triangle

scale factor

the ratio of the lengths of two corresponding sides of two similar polygons

If two triangles have equal heights, then

the ratio of their areas equals the ratio of their bases

If two triangles have equal bases, then

the ratio of their areas equals the ratio of their heights

If two triangles are similar, then

the ratio of their areas equals the square of their scale factor

Annulus of a circle

the region between two concentric circles

Sector of a Circle

the region bounded by two radii and the intercepted arc

sector of a circle

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

Common internal tangent

the segment joining the centers of two coplanar circles.

radius

the segment that connects the center of a circle to any point on the circle

diameter

the segment that connects two points on a circle and passes through the center of the circle

circle

the set of points equidistant from a center point

Circle

the set of points in a plane at a given distance from a given point in that plane

circle

the set of points in plane at a given distance from a given point

(Postulate 17) The area of a square is

the square of the length of a side (A= 2s)

When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to...

the square of the tangent segment.

semicircle

½ of a circle

annulus of a circle formula

πR^2 - πr^2 (area of large circle - area of small circle)

Circumference (C) of a circle with a diameter (d) equals

πd

Lateral Area Cone

πr x slant height

Area of a circle with a radius (r) equals

πr^2

Volume Cylinder

πr²x height

Intercepted Arc

• The endpoints of the arc lie on the angle • All the other points of the arc are in the interior of the angle • Each side of the angle contains at least one endpoint of the arc

Inscribed Angle

• Vertex is on the circle • Each side contains a chord of the circle • Intercepts an arc of the circle

In a 45: 45: 90 triangle, the hypotenuse is

√2 times as long as the leg

In a 30: 60: 90 triangle, the longer leg is

√3 times as long as the shorter leg

In a circle, two chords are congruent if and only if they are equidistant to the center

≅ chords ↔ equidistant to center

In a circle, the perpendicular bisector of a chord contains the center of the circle.

⊥ bisector of chord contains center.

point of tangency

"exactly one point" the tangent intersects

The center of a circle represented by the equation (x + 9)2 + (y − 6)2 = 102 is .

(-9, 6)

Tangent and Secant Segments

(PT)^2 = PQ (PQ + QR)

Write the equation Center (-3,5) Radius=9

(X+3)^2 + (y-5)^2 =81

Arc length =

(arc measure/360) x 2πr

Area of a sector of a circle =

(arc measure/360) x πr^2

polygon is circumscribed in a circle

(circle is in polygon) each side of polygon is tangent to a circle

circle is inscribed in a polygon

(circle is still in polygon) each side of polygon is tangent to a circle

Volume of Sphere

4/3πr³

What is a Chord?

A chord is a straight line joining two points on the circumference of a circle. A diameter is a chord which passes through the centre of the circle.

Diameter

A chord that contains the center point.

Sphere

A circle

polygon

A closed plane figure made up of line segments

Theorem

A diameter that is perpendicular to a chord bisects the chord and its arc

conjecture

A guess, often one based on inadequate or faulty evidence

Radius

A segment from center to point on the circle

chord

A segment whose endpoints lie on a circle

Diagonal lines

A slanting line between horizontal and vertical lines

Radius

A straight line from the center to the circumference of a circle or sphere.

diameter

A straight line passing from side to side through the center of a circle or sphere.

What is a tangent?

A tangent is a straight line which touches a circle at one point only.

A tangent line intersects the circle at exactly ______ point, called the point of tangency.

perpendicular

A tangent line is always ____________________________ to the radius

A tangent of a circle is perpendicular to the radius drawn to the point of tangency.

Tangent Conjecture

A tangent to a circle is perpendicular to the radius drawn to the point of tangency

Tangent Conjecture

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

Reflection (tranformation)

A transformation that "flips" a figure over a mirror or reflection line.

Dilation (tranformation)

A transformation that changes the size of an object, but not the shape.

Tangent Line

Intersect with Radius at a Right Angle

Measure of an arc

Is equal to the measure of the central angle

measure of intercepted arc

Is equal to the measure of the central angle

A tangent is a line that intersects the circle ______

Once

Cyclic Quadrilaterals Conjecture

Opposite angles of a cyclic quadrilateral are supplementary

The measure of an inscribed angle is

half the measure of its intercepted arc

Parallel Chords

Parallel chords form congruent arcs

Parallel Lines Intercepted Arcs Conjecture

Parallel lines intercept congruent arcs on a circle

Parallel Lines Intercepted Arcs Conjecture

Parallel lines intercept congruent arcs on a circle.

Arc

Part of a circle connecting two points on the circle.

Arc

Part of the circle

Arc

Part of the edge (circumference) of a circle or arc.

If the diameter bisects a chord, then the diameter is _______ to the chord

Perpendicular

If the radius and a tangent line intersect on the same circle, then they are _______

Perpendicular

tangent is always

Perpindicular to radius

The measure of an inscribed angle formed by a chord and a tangent is equal to...

half the measure of the intercepted arc.

(Theorem 11.3) the area of a triangle equals

half the product of a base and the height of that base (A=bh)

(Theorem 11.4) the area of a rhombus equals

half the product of its diagonals (A= 1/2(d1d2)

(Theorem 11.6) the area of a regular polygon equals

half the product of the apothem and the perimeter (A=1/2ap)//(A= 1/2asn)

The measure of an angle formed by two chords that intersect inside a circle is equal to...

half the sum of the measures of the incepted arcs.

the measure of an inscribed angle is ______________ the measure of its intercepted ____________

half, arc

Total Area Pyramid

lateral area + area of base

Chord-Chord angle

IN the circle; add arcs and multiply by 1/2

If AD is 9, then ______ is 9.

140

If Angle G = 40, then Angle E = ______.

Circumference Conjecture

If C is the circumference and d is the diameter, C = Pi times d

If Chord RS = Chord TU, then Arc RS = Arc ____

If DB is 6, then EB is ______.

multiply by 2

If I am given an inscribed angle, to find its missing intercepted arc I should:

divide by 2

If I am given an intercepted arc, to find its missing inscribed angle I should:

If JH is 13, then FH is __________.

equal

If Line AB is tangent to the circle, then 8²+15² will _____________ 17²

If Segment TV is 5, then Segment VU is _____.

If ZV=ZW and TU is 8, what is the measure of RS?

Perpendicular Chord Bisector Theorem

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

bisects (cuts in half)

If a diameter or radius is perpendicular to a chord, then it _________________ the chord and its arc.

Theorem

If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

tangent

If a polygon is circumscribed around a circle, then all sides are ____________________.

Theorem (Circumscribed Polygon)

If a polygon is circumscribed around a circle, then all sides are tangent.

Corollary #3

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

supplementary (adds up to 180)

If a quadrilateral is inscribed inside a circle, then it's opposite angles are ___________________________.

A radius perpendicular to a chord

If a radius is perpendicular to a chord, then it bisects it

Radius Perpendicular to Chord

If a radius is perpendicular to a chord, then it bisects it

cosine of an angle

leg adjacent to </ hypotenuse

sine of an angle

leg opposite </ hypotenuse

Tangent of an angle

leg opposite </ leg adjacent to <

Tangent Circles

two coplanar planes that intersect at exactly one point.

tangent-tangent, secant-tangent, secant-secant angle

two lines intersect on the EXTERIOR of a circle, measure of angle formed is 1/2 the difference of its intercepted arc

vertical angles

two nonadjacent angles formed by two intersecting lines

Triangle Angle Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides

To solve for the radius in this problem that involves a tangent line, use _____________________

Pythagorean Theorem

Internally Tangent Circles

two or more circles that intersect at exactly one point; one circle contains the interior of the other.

Concentric Circles

two or more coplanar circles having the same center.

A segment that intersects a circle at two points

Secant

half

The degree of the inscribed angle is equal to ___________________ the measure of the intercepted arc.

perimeter

The distance around a figure.

How do you find the distance from a point to a line or segment?

The distance is the length of a perpendicular line from the point to the segment

extremes

The first and last numbers in a proportion

Arc Length Conjecture

The length of an arc equals the circumference times the measure of the central angle divided by three hundred sixty degrees

Chord

The line segment joining two points on a curve (circle; or arc)

tangent

a line that intersects a circle exactly once

secant lines

a line that intersects a circle twice

Common Tangent

a line that is tangent to two or more circles

tangent to a circle

a line, ray, or segment that intersects a circle exactly one time

Rhombus (quadrilaterals)

a parallelogram with opposite equal acute angles, opposite equal obtuse angles, and four equal sides.

segment of a circle

a part of a circle bounded by an arc and the segment joining its endpoints

arc

a part of the circumference of a circle

altitude of a triangle

a perpendicular segment from a vertex to the line containing the opposite side

quadrilateral

a polygon with four sides

Parallelogram (quadrilaterals)

a quadrilateral whose opposite sides are both parallel and equal in length

two tangent segments

two tangent segments from the same external point are congruent

rectangle (quadrilaterals)

a quadrilateral with four right angles, a plane figure with four straight sides and four right angles, especially one with unequal adjacent sides, in contrast to a square.

Trapezoid (quadrilaterals)

a quadrilateral with only one pair of parallel sides.

Angle Bisector

a ray that divides an angle into two congruent angles

Secant-Tangent Product Theorem

If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared.

Secant-tangent

If a secant line and a tangent line intersect outside a circle, then the product of the outside piece of the secant line and the entire line, is equal to the square of the tangent segment out(whole)=(out)^2

external secant segment

a secant seg that lies in the exterior of the circle with one endpoint on the circle

Tangent and Chord Rule

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

External Secant Segment

a secant segment that lies in the exterior of the circle with one endpoint on the circle

Theorem #2

If a tangent and a chord intersect at a point on the circle, then the measure of the angle they form is one-half the measure of the intercepted arc. angle formed by a tangent and a chord = intercepted arc / 2

Secant Segment

a segment of a secant with at least one endpoint on the circle

secant segment

a segment of a secant with at least one endpoint on the circle

Tangent Segment

a segment of a tangent with one endpoint on the circle

chord

a segment whose endpoints are on a circle

chord

a segment whose endpoints lie on a circle

chord

a segment whose endpoints lie on the circle

perpendicular bisector

a segment, ray, line, or plane that is perpendicular to a segment at its midpoint

Circle

a set of all point in a plane

inscribed

a shape is located inside of a circle

Isosceles angle (triangles)

a triangle with at least two congruent sides

Equidistant Chords Theorem

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

Chord Segments

a x b = c x d

Congruent Corresponding Chords Theorem

In the same circle, two minor arcs are congruent if their corresponding chords are congruent.

In this picture, EC is equal to __________.

Corner A

In this picture, where should I draw in my right angle (boxy box)? Corner A Corner B Corner P

Inscribed Angles Intercepting Arcs Conjecture

Inscribed angles that intercept the same arc are congruent

Inscribed Angles Intercepting Arcs Conjecture

Inscribed angles that intercept the same arc are congruent.

Inscribed Angles

Inscribed angles that intercept the same/ congruent arcs are congruent

Intersecting chords - length

a x b = c x d

Inscribed Angle

Inscribed angles that intercept the same/congruent arcs are congruent

geometric mean

a/x=x/b (x is the geometric mean between a and b)

measure of major arc

360 - x (minor arc)

polygon inscribed in a circle

(polygon is in the circle) each vertex of the polygon lies on the circle

circle circumscribed about a polygon

(polygon is still in circle) each vertex of polygon lies on the circle

Find the equation of the circle with centre (-2,-3) and radius 6.

(x+2)²+(y+3)²=36

Find the equation of the circle with centre (3,0) and radius 6.

(x-3)²+y²=36

Find the equation of the circle with centre (4,2) and radius 5.

(x-4)²+(y-2)²=25

Find the equation of the circle with centre (6, -4) and radius 7.

(x-6)²+(y+4)²=49

Equation of a circle with the center radius r is

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

equation of a circle

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

What form do you usually write the equation of a circle?

(x-h)²+(y-k)²=r²

a radical in simplest form does not have:

1. a perfect square factor under the radical sign 2. a fraction under the radical sign 3. a fraction has a radical in the denominator

two polygons are similar if their vertices can be paired so that:

1. corresponding angles are congruent 2. corresponding sides are in proportion (their lengths have the same ratio)

c:d:e (c to d to e) means:

1. the ratio of the first two numbers is c:d 2. the ratio of the last two numbers is d:e 3. the ratio of the first and last numbers is c:e

(Theorem 11.7) If the scale factor of two similar figures is a:b, then

1. the ratio of their perimeter is a:b 2. the ratio of their areas is a^2:b^2

Secant and Tangent - angles

1/2 (AD - AC)

two secants - angles

1/2 (BD - AC)

Two tangents - angles

1/2 (JAF - JF)

(Theorem 11.5) the area of a trapezoid equals

1/2 the product of the height and the sum of the bases (A= 1/2h(b1 + b2)

Lateral Area Pyramid

1/2 x perimeter of base x slant height

the median of a trapezoid equals

1/2(b1 + b2)

Volume Pyramid

1/3 x base area x height

Volume Cone

1/3πr² x slant height

angles in similar shapes have a scale factor of:

1:1

inscribed angle

2(inscribed angle) = measure of intercepted arc

(Theorem 10.4) the medians of a triangle intersect in a point that is...

2/3 of the distance from each vertex to the midpoint of the opposite side

circumference of a circle (C) with a radius (r) equals

2πr

Lateral Area Cylinder

2πr x height

Isosceles right triangle:

45: 45: 90 degrees

Area of Sphere

4πr²

What happens when a tangent line intersects with a radius?

90 degree angle is created at the point of tangency

Area of a Circle

A = π r²

Tangent

A line in the plane of a circle that intersects the circle in exactly one point.

perpendicular

A line is tangent to a circle if and only if it is ____________________ to a radius drawn to the point of tangency.

Theorem (Prove Tangent)

A line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.

Radius

A line segment that extends from the center to the outside of a circle.

Chord

A line segment whose endpoints are on the circle.

Chord

A line segment with endpoints on the circle.

Secant

A line that goes through two points on a circle

Secant

A line that intersects a circle in two points

Tangent

A line that intersects the circle at exactly one point.

Tangent

A line that intersects the circle in exactly 1 point

Tangent

A line that intersects the circle only once.

diameter of a circle

A line that passes through the center of the circle, connecting any two points

Tangent

A line that touches the circle at one point.

Tangent

A line which touches a circle or ellipse at just one point.

Secant

A line, ray, or a line segment that intersects the circle at exactly two points.

Theorem

A mathematical statement which we can prove to be true

Midpoint

A point that divides a segment into two congruent segments

Vertex

A point where two or more straight lines meet

vertex

A point where two or more straight lines meet.

Decagon

A polygon with 10 sides

Dodecagon

A polygon with 12 sides

Pentagon

A polygon with 5 sides

Hexagon

A polygon with 6 sides

Heptagon

A polygon with 7 sides

Octogan

A polygon with 8 sides

nonagon

A polygon with 9 sides

Quadrilateral

A polygon with four sides

Triangle

A polygon with three sides.

triangle

A polygon with three sides.

Arc

A portion of the circumference of a circle defined by two endpoints.

kite (quadrilaterals)

A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

square (quadrilaterals)

A quadrilateral with 4 sides that are equal and has all 90 degree angles

The measure of an angle formed by a tangent and a radius/diameter is _____

A right angle/90°

How do you know when a point lies on the circumference of a circle?

Insert coordinates into the equation of a circle both sides of eqaution should be equal

Translations (tranformation)

A type of transformation that slides a figure in a given direction for a certain distance.

Intersecting secants - length

A(A + B) = C(C + D)

What is the center of a circle whose equation is x2 + y2 + 4x - 8y + 11 = 0? (-2, 4) (-4, 8) (2, -4) (4, -8)

A) (-2, 4)

Secant Segments

AB x BD = AC x CE

When angles are INSIDE the circle (not center)

ADD

Intersecting Secant and Tangent - length

A^2 = C(C + D)

inscribed angle

An _______________________ is an angle with its vertex ON the circle with two sides that are chords.

intercepted arc

An ___________________________ is an arc that lies between the points of an inscribed angle. (the pink arc in the photo)

obtuse angle (triangles)

An angle that measures more than 90 degrees but less than 180 degrees

Central Angle

An angle whose vertex is at the center of the circle and whose sides are 2 radii intersecting the circle.

inscribed angle

An angle whose vertex is on a circle and whose sides are chords of the circle

Central Angle

An angle whose vertex is the center of the circle.

Major Arc

An arc of a circle that is greater than 180°.

Minor Arc

An arc of a circle that is less than 180°.

Semicircle

An arc of a circle whose endpoints are on the diameter.

major arc

An arc of a circle whose measure is greater than 180 degrees.

minor arc

An arc of a circle whose measure is less than 180 degrees.

Semicircle

An arc whose measure is exactly 180 degrees because its endpoints are on the diameter of the circle. *Use 3 letters to define.

Minor Arc

An arc whose measure is less than 180 degrees. *Use 2 letters to define.

Major Arc

An arc whose measure is more than 180 degrees. *Use 3 letters to define.

intercepted arc

An arc with endpoints on the sides of an inscribed angle, and its other points in the interior of the angle

proportion

An equation stating that two ratios are equal

How do you know a point lies inside the circle?

Insert the coordinates into the equation of a circle. It's when: left hand side < right hand side

Inscribed Angle Conjecture

An inscribed angle measures half of the arc it intercepts

Corollary #1

An inscribed angle that intercepts a semi-circle is a right angle.

Angle D + Angle _____ = 180.

Inscribed Angle

Angle measure is 1/2 the measure of the intercepting arc

Central angle

Angle whose vertex is at the center of the circle

Inscribed angle

Angle whose vertex lies on a circle

Angles Inscribed in a Semicircle Conjecture

Angles inscribed in a semicircle are 90 degrees.

Angles Inscribed In A Semicircle Conjecture

Angles inscribed in a semicircle are right angles

Angles Inscribed In a Semicircle Conjecture

Angles inscribed in a semicircle measure 90 degrees

Intercepted arc

Arc created by an angle

Semi circle

Arc whose endpoint lie on a diameter

If two chords congruent, then the ______ they form are congruent

Arcs

similar

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

similar shapes

Figures with the same shape but not necessarily the same size are similar.

Find the measure of angle A.

Find the measure of angle B.

125

Find the measure of angle W.

Find the measure of arc AC.

Find the measure of arc WV.

230

Find the measure of arc XVY.

Find the measure of the arc or the angle indicated.

Find x.

105 degrees

Find;

234 degrees

Find;

306 degrees

Find;

Bisection of a Chord

Form a Right Angle with the Radius

The measure of an angle formed by a tangent and a chord is ____ the measure of its intercepted arc

Half

The measure of an inscribed angle is _______ the measure of its intercepted arc

Half

Inscribed Angle

Has vertex on the circle; sides of the angle are chords

Multiply by 2

If you know the angle, how do you find the arc?

Divide by 2

If you know the arc, how do you find the angle?

Theorem (Minor Arcs)

In a circle (or congruent circles), 2 minor arcs are congruent if and only if their corresponding chords are congruent.

Theorem (Equidistant Chords)

In a circle (or congruent circles), two chords if and only if they are equidistant from the center.

Radius

In a circle, all radii are congruent

Theorem (Perpendicular Diameter/Radius)

In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.

Pythagorean Theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs a^2+ b^2= c^2

hypotenuse

In a²+b²=c², I should label the __________________ my "c".

Theorem

In the same circle or in congruent circles 1) chords equally distant from the center (or centers) are congruent 2) congruent chords are equally distant from the center or centers

Theorem

In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent

Secant line

Line that passes through the circle (all the way through)

Congruent Chords

Mean congruent central angles and arcs

If vertex is on circle

Measure is half its intercepted arc

Name a chord that is not a diameter.

Name a chord.

Name a secant that goes outside the circle.

Name the chord that is not a diameter.

Name the chord.

Name the secant that goes outside the circle.

Name the tangent.

Circle

Named by the center point

Formula for secants from an external point

O1*W1 = O2*W2

Inscribed angle

ON the circle; 1/2 int. arc

Tangent-Chord angle

ON the circle; 1/2 int. arc

Secant-Secant angle

OUTSIDE the circle; 1/2 (diff. int. arc)

Tangent-Secant angle

OUTSIDE the circle; 1/2 (diff. int. arc)

Tangent-Tangent angle

OUTSIDE the circle; 1/2 (major - minor)

(x+1)²+(y+2)²=100

R: 10, C: (-1, -2)

(x-1)²+(y-2)²=100

R: 10, C: (1, 2)

(x+2)²+(y+1)²=9

R: 3, C: (-2, -1)

(x+2)²+(y-1)²=9

R: 3, C: (-2, 1)

(x-2)²+(y+1)²=9

R: 3, C: (2, -1)

(x-2)²+(y-1)²=9

R: 3, C: (2, 1)

(x-3)²+(y-5)²=9

R: 3, C: (3, 5)

(x-5)²+(y-3)²=9

R: 3, C: (5,3)

(x+3)²+(y-5)²=16

R: 4, C: (-3, 5)

(x-3)²+(y+5)²=16

R: 4, C: (3, -5)

(x+3)²+(y+5)²=25

R: 5, C: (-3, -5)

(x+5)²+(y+3)²=25

R: 5, C: (-5, -3)

RQ = (x+3) QP = 5 TS = (x-5) SP = 6 Solve for x

Fixed Distance

Radius

sketch

Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features.

Angles inscribed in a semicircle are always _______

Right Angle/90°

SK = ?

When angles are OUTSIDE the circle

SUBTRACT

Central Angle

Same measure as minor arc

A segment that intersects a circle at exactly one point

Tangent

External Tangent Congruence Theorem

Tangent segments from a common external point are congruent.

Tangent Segments Conjecture

Tangent segments to a circle from a point outside the circle are congruent

Tangent Segments Conjecture

Tangent segments to a circle from a point outside the circle are congruent.

What do we know about two tangent lines that come from the same point?

Tangents from the same point to the circle are equal. e.g. NA = NG.

Tangent-Chord

The angle formed by a tangent line and chord is 1/2 the measure of the chords intercepted arc

(Postulate 19) Area Addition Postulate

The area of a region is the sum of the areas of its non-overlapping parts

The measure of an angle formed by the intersection of (2 secants/secant+tangent/two tangents) is equal to:

The average of the measure of the farther arc minus the measure of the arc first intercepted.

Central Angle

Vertex is the center of the circle; end points are on the circle.

WS = ?

chords

What circle parts are shown in this picture?

Chords

What circle parts are shown in this picture? Chords Secants

Pythagorean Theorem (a²+b²=c²)

What formula should I use to figure out if Line AB is tangent to the circle?

What is the length of DE?

What is the length of GE?

What is the length of line BD?

What is the length of line EC?

What is the length of line ED?

What is the measure of Angle ABC?

What is the measure of Angle BEC?

What is the measure of Angle KLM?

176

What is the measure of Arc JML?

82°

What is the measure of arc AB?

What is the perimeter of this figure? (fill in the missing parts & add up all the sides)

278°

What is the reflex measure?

11.2

What is the value of x?

25.6

What is the value of x?

7.7

What is the value of x?

9.6

What is the value of x?

What is y?

PA = PB

What segments are congruent.

Inscribed Angle

What type of angle is shown?

2 Secants

What type of lines do you see in this picture? 2 Secants 2 Tangents 1 Secant/1 Tangent

Theorem

When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segments and its external segment is equal to the square of the tangent segment

The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to...

half the difference of the measures of the intercepted arcs.

square root

When solving, to get rid of a "squared", I should use the _______________________ button.

Theorem

When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord (x+y) y= (m+n)n

Theorem

When two secant segment are drawn to a circle from an external point of one secant segment and its external segment equals the product of the other secant and its external segment

(x+4)²+(y-1)²=25

Write the equation of this circle.

(x+5)²+(y+3)²=9

Write the equation of this circle.

(x-2)²+(y-5)²=4

Write the equation of this circle.

(x-5)²+(y-3)²=16

Write the equation of this circle.

diameter

a chord that contains the center of a circle

Diameter

a chord that contains the center of a circle (can refer to THE length or A segment)

Plane

a flat surface that has no thickness and extends forever

semi-circle

a half of a circle or of its circumference.

tangent

a line in the plane of a circle that intersects in exactly one point

Tangent

a line in the plane of a circle that intersects the circle in exactly one point

tangent

a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency

A secant

a line that contains a chord

secant

a line that contains a chord

lines of symmetry

a line that divides a figure into two parts that are mirror images of each other

Secant

a line that intersects a circle at 2 points

secant

a line that intersects a circle at two points

if two tangent segments to a circle share a common _____________________ then the 2 segments are __________________

endpoint, congruent

The measure of an arc is _______________ to its central angle

equal to

(Theorem 10.1) The bisectors of the angles of a triangle intersect in a point that is ______________ from the 3 sides of the triangle

equidistant

(Theorem 10.2) The perpendicular bisectors of the sides of a triangle intersect in a point that is ______________ from the 3 vertices of the triangle

equidistant

In the same circle, or in congruent circles, two chords are congruent if and only if they are _______________________ from the center.

equidistant

within a circle or in congruent circles, chords ______________________ from the center are ______________

equidistant, congruent

congurent

exactly the same size and shape

𝝅d

formula to find the circumference of a circle

formula to find the diameter of a circle

1/2 circumference + diameter

formula to find the perimeter of a semicircle

d/2

formula to find the radius of a circle

the radius

given distance

radius

given distance in a circle

center of circle

given point

center

given point in a circle

semicircle

half of a circle

semi-circle

half of a circle (180°)

semi circles

half the circle through center

Theorem

in the same circle or in congruent circles, 1) congruent circles have congruent chords 2) congruent chords have congruent arcs

Median (of triangle)

is a line segment joining a vertex to the midpoint of the opposing side, bisecting it.

angles of rotation

is a measurement of the amount, the angle, by which a figure is rotated counterclockwise about a fixed point, often the center of a circle.

Line

is straight (no curves) has no thickness, and extends in both directions without end (infinitely).

Total Area Prism

lateral area + 2(area of base)

Total Area Cylinder

lateral area + 2(πr²)

arc length

the unit length of an arc

in the same circle or in congruent circles, 2 minor arcs are congruent if and only if...

their central angles are congruent

In the same circle or in congruent circles, two minor arcs are congruent if and only if...

their central angles are congruent.

Externally Tangent Circles

their interiors do not intersect.

If a line is tangent to a circle...

then the line is perpendicular to the radius drawn to the point of tangency.

If a line in the plane of a circle is perpendicular to a radius at its outer endpoint...

then the line is tangent to the circle.

If the square of one side of a triangle is less than the sum of the squares of the other two sides..

then the triangle is acute

If the square of one side of a triangle is greater than the sum of the squares of the other two sides...

then the triangle is obtuse

(Theorem 8.3) If the square of one side of a triangle is equal to the sum of the squares of the other two sides...

then the triangle is right

If 3 parallel lines intersect 2 transversals,

then they divide the transversals proportionally

Concentric cricles

they share the same center

In a 30: 60: 90 triangle, the hypotenuse is

twice as long as the shorter leg

supplementary angles

two angles that add up to 180 degrees

adjacent angles

two angles that have a common vertex and a common side(meaning that the vertex point and the side are shared by the two angles.)

chord-chord angle

two chords intersect on the INTERIOR of a circle, measure of angle formed is 1/2 the sum of its intercepted arc

A circumcenter is equidistant to the _________ of a triangle.

vertices

What do we know about a line from the center that bisects a chord?

when a line from the center bisects a chord, it also creates a right angle

circumscribed

when each vertex of the polygon lies on the circle

extended proportion

when three or more ratios are equal a:b = c:d = e:f

secant-secant product thm

whole x outside = whole x outside

secant-tangent product thm

whole x outside seg = tangent squared

Angles Made by Intersecting Chords

x = (arc1 + arc2)/2

Angles Made by Intersecting Secants

x = (arc1 - arc2)/2

x = ?

Angle between 2 chords made inside circle

x= a°+b° ÷ 2

angle between secant- secant & secant - tangent

x= b°- a°÷ 2

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