Chords and Tangents - Full List
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.
7
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.
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
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 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 not on a line segment
construct a perpendicular from a point 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 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
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
26
m∠1
35
m∠1
37
m∠1
42
m∠1
56
m∠1
63
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
Pi
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.
1
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.
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
AF
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
TU
If Chord RS = Chord TU, then Arc RS = Arc ____
6
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:
13
If JH is 13, then FH is __________.
equal
If Line AB is tangent to the circle, then 8²+15² will _____________ 17²
5
If Segment TV is 5, then Segment VU is _____.
8
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.
FC
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.
F
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.
25
Find the measure of angle A.
32
Find the measure of angle A.
40
Find the measure of angle A.
15
Find the measure of angle B.
125
Find the measure of angle W.
29
Find the measure of angle W.
53
Find the measure of angle W.
67
Find the measure of angle W.
71
Find the measure of angle W.
85
Find the measure of angle W.
80
Find the measure of arc AC.
84
Find the measure of arc WV.
230
Find the measure of arc XVY.
31
Find the measure of the arc or the angle indicated.
90
Find the measure of the arc or the angle indicated.
12
Find x.
14
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
CJ
Name a chord that is not a diameter.
FG
Name a chord.
JH
Name a secant that goes outside the circle.
LK
Name a secant that goes outside the circle.
DE
Name the chord that is not a diameter.
DA
Name the chord.
EB
Name the secant that goes outside the circle.
FE
Name the secant that goes outside the circle.
GE
Name the secant that goes outside the circle.
HG
Name the secant that goes outside the circle.
BD
Name the tangent.
KI
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)
34
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°
5
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.
6
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?
16
What is the length of DE?
14
What is the length of GE?
11
What is the length of line BD?
12
What is the length of line EC?
8
What is the length of line ED?
17
What is the measure of Angle ABC?
77
What is the measure of Angle BEC?
37
What is the measure of Angle KLM?
176
What is the measure of Arc JML?
82°
What is the measure of arc AB?
73
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?
4
What is the value of x?
7.7
What is the value of x?
9
What is the value of x?
9.6
What is the value of x?
3
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
2r
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
61
x
67
x
70
x
Angles Made by Intersecting Chords
x = (arc1 + arc2)/2
Angles Made by Intersecting Secants
x = (arc1 - arc2)/2
1
x = ?
16
x = ?
2
x = ?
4
x = ?
5
x = ?
8
x = ?
9
x = ?
Angle between 2 chords made inside circle
x= a°+b° ÷ 2
angle between secant- secant & secant - tangent
x= b°- a°÷ 2