Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t...

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