Post on 14-Dec-2015
transcript
CIRCLES
Chapter 10
Tangents to Circles lesson 10.1
California State Standards7: Prove and Use theorems involving properties of circles.21: Prove and Solve relationships among chords, secants and tangents.
definitions
CircleThe set of all points in a plane that are equidistant
from a given point.
CenterThe given point.
RadiusA line segment with the center as one endpoint and
a point on the circle as the other endpoint.The distance from the center to a point on the circle.
CircleThe set of all points in a plane that are equidistant
from a given point.
CenterThe given point.
RadiusA line segment with the center as one endpoint and
a point on the circle as the other endpoint.The distance from the center to a point on the circle.
definitions
C
P
the plural of radius is radii
a circle is namedby its center
C
is a radius is the length
CPCP
definitions
Congruent CirclesCircles with the same radius
DiameterA line segment with endpoints on the circle
that contains the center of the circle.The distance across a circle through the center.
2r = d
definitions is a diameterRPCongruent Circles
Circles with the same radius
DiameterA line segment with endpoints on the circle that contains the center of the circle.The distance across a circle through the center. 2r = d
PCR
C R CR RC
2r = d
definitions
ChordA segment whose endpoints are on the circle.A diameter is a “specialized” chord.
SecantA line that intersects a circle in two points.
TangentA line that intersects a circle in exactly one point.The circle and line must lie in the same plane.
definitions
ChordA segment whose endpoints are on the circle.A diameter is a “specialized” chord.
SecantA line that intersects a circle in two points.
TangentA line that intersects a circle in exactly one point.The circle and line must lie in the same plane.
CA
B is a chordAB
F
GFG is a secant,------------- -
T
TU is a tangent,------------- - U
radiusC
P
A
B
Q
X
radiu
s
radiu
sdia
met
er chord
secant
tangent
Identify each line or segment
chord
definition
Tangent CirclesCoplanar circles that intersect in exactly one point.
externally tangent circles
definition
Tangent CirclesCoplanar circles that intersect in exactly one point.
internally tangent circles
definition
Concentric CirclesCoplanar circles with a common center.
definition
Common TangentA line or segment that is tangent to two
coplanar circles• Common Internal Tangent crosses between the circles• Common External Tangent stays along the edges of the
circles
Common TangentA line or segment that is tangent to two
coplanar circles• Common Internal Tangent crosses between the circles• Common External Tangent stays along the edges of the
circles
definitionCommon Internal Tangent
Common External Tangent
example
Is the common tangentinternal or external?
external
C DT
example
C
H
B
A I
D
E
F G
Describe each segment
tangent
diameter
chord
radius
secant
AH
EI
DF
CE
HG
definitions
exterior
interior
Interior of a CircleThe set of points inside the circle
Exterior of a CircleThe set of points outside the circle.
theorem
Circle Tangent-Radius PerpendicularIf a line is tangent to a circle,then it is perpendicular to the radius drawn to the point of tangency.
t
C
P
t CP
C
theorem
Circle Tangent-Radius Perpendicular ConverseIf a line is perpendicular to a radius of a circle at the endpoint on the circle,then the line is tangent to the circle.
tP
is tangent to t C
example
C
A
B
43
45
11
Is tangent to ?AB C2 2 245 ? 43 11
2025 ?1846 1212025 1970
is not to No
AB CB
10r 32 320r
32 256 576r
2 232 256 576r r r
2 2 2( 16) 24r r
example
C
AB
r
r 16
24
is tangent to .
Find the radius of the circle.
AB C
theorem
Congruent TangentsIf two segments from the same exteriorpoint are tangent to a circle,then the segments are congruent.
P
C
Q
S
is tangent to SP C
is tangent to SQ C
SQ SP
2 25x
example
A
B
D
x2 – 4
21
C5x
2 4 21x
and are tangent to .
Find the value of x.
AB AD C
example
B
D
x
36
15
yz
A
C
Find the value of , , and .x y z
AB AD
36x
36
15
CB CD15y
Find the value of , , and .x y z
36
example
B
D
36
15
z
A
C 39z
2 1521z
2 225 1296z
2 2 215 36z
39
15
exampleWhat are the coordinates of each center?
What is the radius of each circle?
x
y
A B
(2,2)A (6,2)B
2A r 2B r
A B
exampleDescribe the intersection of the two circles.
x
y
A B
and are externally tangent
A B
exampleDescribe the common tangents of the circles.
x
y
A B
and have anintenal tangent at 4
A Bx
They have 2 external tangents.
4y 0y
A
y
example
B
What are the coordinates of each center? (0,1)A (2,1)B
What is the radius of each circle? 1A r 3B r
Describe any common tangents.
x _1 is a common external tangent
x
5. SSS 4. reflexive
3. def. radius
2. Tangents Th
5. ABC ADC 4. CA CA
3. CB CD
2. AB AD
1. and are tangent to
AB ADC
Statement
Given: and are tangent to Prove:
AB AD CABC ADC
C
AB
D
Reason
1. given