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Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
t
A
Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
t
C
A
D
B
Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB ) t
C
A
D
B
Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB )
Common Tangent Line
- tangent to two coplanar circles
t
C
A
D
B
P Q
a
Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB )
Common Tangent Line
- tangent to two coplanar circles
- common external tangent lines do
not intersect ( lines “a” and “e” ) the
segment joining the circles
t
C
A
D
B
P Q
a
e
Circles – Tangent Lines
A tangent line touches a circle at exactly one point.
In this case, line t is tangent to circle A.
The point at which the line is tangent is called the point of tangency ( point C )
Rays can also be tangent to circles.
Ray CD ( segment CD )
Ray CB ( segment CB )
Common Tangent Line
- tangent to two coplanar circles
- common external tangent lines do
not intersect ( lines “a” and “e” ) the
segment joining the circles
- common internal tangent lines
intersect the segment joining the circles
t
C
A
D
B
P Q
a
e
g h
Circles – Tangent Lines
Tangent circles
- two coplanar circles that are tangent to the same line at the same point
t
A
QS
c
B
C
Circles – Tangent Lines
Tangent circles
- two coplanar circles that are tangent to the same line at the same point
circle A tangent to circle Q
circle A tangent to circle S
t
A
QS
c
B
C
Circle A and Q are internally tangent, one circle is inside the other.
Circles – Tangent Lines
Tangent circles
- two coplanar circles that are tangent to the same line at the same point
circle A tangent to circle Q
circle A tangent to circle S
t
A
QS
c
B
C
Circle A and Q are internally tangent, one circle is inside the other.
Circle A and S are externally tangent, not one point of one circle is in the interior of the other.
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
t
A
B
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
t
A
B
Theorem - tangents to a circle from an exterior point are congruent
A
D
E
P
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
t
A
B
Theorem - tangents to a circle from an exterior point are congruent
A
D
E
P
AEAD
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
A
C
B
S
AB and AC are tangent to circle S.
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
A
C
B
S
AB and AC are tangent to circle S.
ACSC
ABSB
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
A
C
B
S
AB and AC are tangent to circle S.
ACSC
ABSB
25CBS
65 then ,25 If ABCCBS ( 90° - 25° = 65° )
: if Find Am
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
A
C
B
S
AB and AC are tangent to circle S.
ACSC
ABSB
25CBS
65 then ,25 If ABCCBS ( 90° - 25° = 65° )
∆ABC is isosceles from the theorem above about tangents from an exterior point…
: if Find Am
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
A
C
B
S
AB and AC are tangent to circle S.
ACSC
ABSB
25CBS
65 then ,25 If ABCCBS ( 90° - 25° = 65° )
∆ABC is isosceles from the theorem above about tangents from an exterior point…
65ACBABC
: if Find Am
65°
65°
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
Let’s use these two theorems to solve some problems.
A
C
B
S
AB and AC are tangent to circle S.
ACSC
ABSB
25CBS
65 then ,25 If ABCCBS ( 90° - 25° = 65° )
∆ABC is isosceles from the theorem above about tangents from an exterior point…
65ACBABC
: if Find Am
65°
65°
506565180Am
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
A
C
B
S
AB and AC are tangent to circle S.
SC
BA
AC
FIND :
10
26
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
A
C
B
S
AB and AC are tangent to circle S.
SC 10
BA
AC
FIND :
10
26
10 therefore,10
SCBS
SCBS
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
A
C
B
S
AB and AC are tangent to circle S.
SC 10
BA 24
AC
FIND :
10
26
10 therefore,10
SCBS
SCBS
24576
100676
1026 22
BA
BA
BA
∆BAS is a right triangle :
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
Theorem - tangents to a circle from an exterior point are congruent
EXAMPLE # 2 :
A
C
B
S
AB and AC are tangent to circle S.
SC 10
BA 24
AC 24
FIND :
10
26
10 therefore,10
SCBS
SCBS
24576
100676
1026 22
BA
BA
BA
∆BAS is a right triangle :
24 therefore,24
ACAB
ABAC
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 3 :
A
C
D
SAD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25
10.5
25
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 3 :
A
C
D
SAD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25
10.5
25
5.10
5.10
SA
CS
SACS
10.5
( both are radii )
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 3 :
A
C
D
SAD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25
10.5
25
5.10
5.10
SA
CS
SACS
10.5
( both are radii )
21
5.105.10
CA
CA
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 3 :
A
C
D
SAD is tangent to circle S.
Find CD if CS = 10.5 and AD = 25
10.5
25
5.10
5.10
SA
CS
SACS
10.5
( both are radii )
21
5.105.10
CA
CA 65.321066
625441
2521 22
CD
CD
CD
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 4 :
R
C D
SCD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26
14
26
24
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 4 :
R
C D
SCD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26
14
26
24
We can sketch in a parallel line to CD that creates a right triangle SRE.
Since CD was perpendicular to CS and DR, the new line will be perpendicular as well.
E
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 4 :
R
C D
SCD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26
14
26
24
We can sketch in a parallel line to CD that creates a right triangle SRE.
Since CD was perpendicular to CS and DR, the new line will be perpendicular as well.
E
101424
SE
DRCSSE
10
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 4 :
R
C D
SCD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26
14
26
24 E
101424
SE
DRCSSE
10
24576
100676
1026 22
ER
ER
ER
Circles – Tangent Lines
Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency.
EXAMPLE # 4 :
R
C D
SCD is a common tangent to circle S and circle R.
Find CD if CS = 24, DR = 14
and SR = 26
14
26
24 E
101424
SE
DRCSSE
10
24576
100676
1026 22
ER
ER
ER
24
24
CD
ER
CDER