6.1 use properties of tangents

6.1 Use Properties of Tangents

Example 1 Identify special segments and lines

Solution

Tell whether the line, ray, or segment

is best described as a radius, chord,

diameter, secant or tangent of C?F

BE

A

C

DBC a. EA b. DE c.

a. BC is a _________ because C is the center and B is a point

on the circle.

radius

b. EA is a _________ because it is a line that intersects the

circle in two points.

secant

c. DE is a _________ ray because it is contained in a line that

intersects the circle in exactly one point.

tangent


Example 2 Find lengths in circles in a coordinate plane

Use the diagram to find the given

lengths.

a. Radius of A

b. Diameter of A

c. Radius of B

d. Diameter of B

BA

Solution

a. The radius of A is ___ units.2

b. The diameter of A is ___ units.4

c. The radius of B is ___ units.4

d. The diameter of B is ___ units.8


Checkpoint. Complete the following exercises. 1. In Example 1, tell whether AB is best

described as a radius, chord,

diameter, secant, or tangent. Explain.F

BE

A

C

D

AB is a diameter because it is a chord that contains the

center C.


Checkpoint. Complete the following exercises.

C

D

2. Use the diagram to find (a) the

radius of C and (b) the

diameter of D.

a. The radius of C is 3 units.

b. The diameter of D is 2 units.


Example 3 Draw common tangents

Tell how many common tangents the circles have and draw them.

a. b. c.

Solution

a. ___ common

tangents

3 b. ___ common

tangents

2 c. ___ common

tangents

1


Checkpoint. Tell how many common tangents

the circles have and draw them.

3. 4.

no common

tangents

4 common

tangents


Theorem 6.1

If a plane, a line is tangent to a circle if and only

if the line is _____________ to the radius of

the circle at its endpoint on the circle.

perpendicular

O

P

m


Example 4 Verify a tangent to a circle

In the diagram, RS is a radius of R.

Is ST tangent to R?T

R

S

1024

26

Solution

Use the Converse of the Pythagorean

Theorem. Because 102 + 242 = 262, RST is

a _____________ and RS ____.right triangle ST

So, _____ is perpendicular to a radius of R at

its endpoint on R. By ____________, ST is

_________ to R.

STTheorem 6.1

tangent


Checkpoint. RS is a radius of R. Is ST

tangent to R?

5.

R

S

5

T

12

813

222 1312525 144 169Therefore, RS ST.

By Theorem 6.1, ST is tangent to R.


Checkpoint. RS is a radius of R. Is ST

tangent to R?

T

S

R

12

16

7

6.

19

222 191612144 256 361


Example 5 Find the radius of a circle

In the diagram, B is a point of

tangency. Find the radius r of C.

Solution

right triangle

B

CA r49

77 r

You know from Theorem 6.1 that AB BC, so ABC is

a _____________. You can use Pythagorean Theorem.222 ABBCAC Pythagorean Theorem

Substitute.222

7749 rrMultiply.___________ 22 rrr 98 2401 5929Subtract from each side._______ r98 3528Divide by ____.98____r 36

The radius of C is _____.36


Checkpoint. Complete the following exercises. 7. In the diagram, K is a point of

tangency. Find the radius r of L.

J

L

Kr

32

56

r

2r256

232r

2r r64 1024 31362r

r64 2112

33r


Theorem 6.2

Tangent segments from a common external point

are _____________.congruent

P S

R

T

C

Q

R

S

32

53x


Example 6 Use properties of tangents

Solution

QSQR Tangent segments from a common

external point are ___________.

_________32congruent

QR is tangent to C at R and

QS is tangent to C at S.

Find the value of x.

Substitute.53x

Solve for x.x___9


Triangle Similarity Postulates and Theorems

congruent

Angle-Angle (AA) Similarity Postulate:

If two angles of one triangle are ___________ to two angles

of another _________, then the two triangles are _________.triangle similar

Theorem 6.3 Side-Side-Side (SSS) Similarity Theorem:

If the corresponding side lengths of two triangles are

_____________, then the triangles are _________.proportional similar

Theorem 6.4 Side-Angle-Side (SAS) Similarity Theorem:

If an angle of one triangle is _____________ to an angle of a

second triangle and the lengths of the sides including these

angles are ______________, then the triangles are ________.proportional similar

congruent


Example 7 Use tangents with similar triangles

Solution

CDAC and BEAB ______________

_____________________ s.right are ACD and ABE

Theorem 6.1

In the diagram, both circles are

centered at A. BE is tangent to the

inner circle at B and CD is tangent

to the outer circle at C. Use similar

triangles to show that

________________

A

B

C

E

D

Definition of .

All right are .s ACD ABE

BAECAD ______________Reflexive Prop________________ AA Similarity Post ACD ABE

________________ Corr. sides lengths are prop. AD

AE

AC

AB

AD

AE

AC

AB


Checkpoint. Complete the following exercises. 8. RS is tangent to C at S and RT

is tangent to C at T. Find the

value(s) of x.C

TR

S

49

2xRSRT49

2x

7x

6.1 use properties of tangents

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