How do I use the properties of tangents to identify lengths in a circle? How do you use information of a circle to find arc measures?

• How do I use the properties of tangents to identify lengths in a circle?

•How do you use information of a circle to find arc measures?

6.1 Use Properties of Tangents

Example 1 Identify special segments and linesIdentify 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 planeFind lengths in circles in a coordinate planeUse the diagram to find the given lengths.a. Radius of A

b. Diameter of A

c. Radius of B

d. Diameter of B

BA

Solutiona. 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

6.1 Use Properties of TangentsCheckpoint. Complete the following exercises. 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.

6.1 Use Properties of TangentsCheckpoint. Complete the following exercises. 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 tangentsDraw common tangentsTell how many common tangents the circles have and draw them.

a. b. c.

Solutiona. ___ common

tangents3 b. ___ common

tangents2 c. ___ common

tangents1

6.1 Use Properties of TangentsCheckpoint. Tell how many common tangents Checkpoint. Tell how many common tangents the circles have and draw them.the circles have and draw them.

3. 4.

no common tangents

4 common tangents


Theorem 6.1If 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 circleVerify a tangent to a circleIn the diagram, RS is a radius of R. Is ST tangent to R?

TR

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

6.1 Use Properties of TangentsCheckpoint. RS is a radius of R. Is ST Checkpoint. RS is a radius of R. Is ST tangent to R?tangent to R?

5.R

S

5

T

12

813

222 13125 25 144 169Therefore, RS ST.

By Theorem 6.1, ST is tangent to R.

6.1 Use Properties of TangentsCheckpoint. RS is a radius of R. Is ST Checkpoint. RS is a radius of R. Is ST tangent to R?tangent to R?

T

S

R

12

16

7

6.

19

222 191612 144 256 361


Example 5 Find the radius of a circleFind the radius of a circleIn 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

6.1 Use Properties of TangentsCheckpoint. Complete the following exercises. 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

56r

2r 256 232r2r r64 1024 31362 r

r64 211233r


Theorem 6.2Tangent segments from a common external point

are _____________.congruent

P S

R

T

C

Q

R

S

32

53 x


Example 6 Use properties of tangentsUse 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.53 xSolve 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 similarTheorem 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 trianglesUse 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

6.1 Use Properties of TangentsCheckpoint. Complete the following exercises. 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

2xRSRT 492x

7x


Pg. 198, 6.1 #1-34

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How do I use the properties of tangents to identify lengths in a circle? How do you use information of a circle to find arc measures?