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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?
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
6.1 Use Properties of Tangents
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.
6.1 Use Properties of Tangents
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
6.1 Use Properties of 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
6.1 Use Properties of Tangents
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
6.1 Use Properties of Tangents
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
6.1 Use Properties of Tangents
Theorem 6.2Tangent segments from a common external point
are _____________.congruent
P S
R
T
C
Q
R
S
32
53 x
6.1 Use Properties of Tangents
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
6.1 Use Properties of Tangents
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
6.1 Use Properties of Tangents
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
6.1 Use Properties of Tangents
Pg. 198, 6.1 #1-34