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6.1
Use Properties of Tangents
Example 1 Identify special segments and lines
A
Tell whether the line, ray, or segment
is best described as a radius, chord,
diameter, secant or tangent of C?
a. BC
Solution
b. EA
c. DE
C
F
E
B
D
radius because C is the center and B is a point
a. BC is a _________
on the circle.
secant because it is a line that intersects the
b. EA is a _________
circle in two points.
tangent ray because it is contained in a line that
c. DE is a _________
intersects the circle in exactly one point.
6.1
Use Properties of Tangents
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
Solution
2 units.
a. The radius of A is ___
b. The diameter of A is ___
4 units.
c. The radius of
d. The diameter of
4 units.
B is ___
B is ___
8 units.
A B
6.1
Use Properties of Tangents
Checkpoint. Complete the following exercises.
A
1. In Example 1, tell whether AB is best
described as a radius, chord,
diameter, secant, or tangent. Explain.
C
F
E
B
D
AB is a diameter because it is a chord that contains the
center C.
6.1
Use Properties of Tangents
Checkpoint. Complete the following exercises.
2. Use the diagram to find (a) the
radius of C and (b) the
diameter of D.
a. The radius of
b. The diameter of
C is 3 units.
D is 2 units.
D
C
6.1
Use Properties of Tangents
Example 3 Draw common tangents
Tell how many common tangents the circles have and draw them.
a.
Solution
a. ___
3 common
tangents
b.
c.
2 common
b. ___
tangents
c. ___
1 common
tangents
6.1
Use Properties of Tangents
Checkpoint. Tell how many common tangents
the circles have and draw them.
4.
3.
no common
tangents
4 common
tangents
6.1
Use Properties of Tangents
Theorem 6.1
If a plane, a line is tangent to a circle if and only
if the line is _____________
perpendicular to the radius of
the circle at its endpoint on the circle.
P
O
m
6.1
Use Properties of Tangents
Example 4 Verify a tangent to a circle
In the diagram, RS is a radius of
Is ST tangent to R?
R.
T
26
R
10
Solution
24
Use the Converse of the Pythagorean
S
Theorem. Because 102 + 242 = 262,  RST is
right triangle and RS  ____.
a _____________
ST
So, _____
ST is perpendicular to a radius of R at
its endpoint on R. By ____________,
Theorem 6.1ST is
tangent to
_________
R.
6.1
Use Properties of Tangents
Checkpoint. RS is a radius of
tangent to R?
5  12  13
25  144  169
2
5.
8
13
R
5
12
T
R. Is ST
2
Therefore, RS
2
 ST.
S
By Theorem 6.1, ST is tangent to
R.
6.1
Use Properties of Tangents
Checkpoint. RS is a radius of
tangent to R?
S
6.
12
T
19
R
12  16  19
144  256  361
2
16
7
R. Is ST
2
2
6.1
Use Properties of Tangents
B
Example 5 Find the radius of a circle
In the diagram, B is a point of
tangency. Find the radius r of
C.
A
77
49
r
r
C
Solution
You know from Theorem 6.1 that AB BC, so ABC is
right triangle You can use Pythagorean Theorem.
a _____________.
2
2
2
AC  BC  AB Pythagorean Theorem
r  49
 r  77
2
2
r  ___
____
2401 r  5929
98 r  ____
___
____
98 r  3528
36
r  ____
2
2
The radius of
2
36
C is _____.
Substitute.
Multiply.
Subtract from
each side.
Divide by ____.
98
6.1
Use Properties of Tangents
Checkpoint. Complete the following exercises.
7. In the diagram, K is a point of
tangency. Find the radius r of L.
r
L
r  32  r
 56
2
2
r  64r  1024  r  3136
64r  2112
r  33
2
2
2
K
56
r
32
J
6.1
Use Properties of Tangents
Theorem 6.2
Tangent segments from a common external point
are _____________.
congruent
R
P
T
S
6.1
Use Properties of Tangents
Example 6 Use properties of tangents
QR is tangent to C at R and
QS is tangent to C at S.
Find the value of x.
Solution
R
C
Q
3x  5
Tangent segments from a common
external point are ___________.
congruent
S
QR  QS
___
32  ______
3x  5
___
9 x
32
Substitute.
Solve for x.
6.1
Use Properties of Tangents
Triangle Similarity Postulates and Theorems
Angle-Angle (AA) Similarity Postulate:
If two angles of one triangle are ___________
congruent to two angles
triangle then the two triangles are _________.
of another _________,
similar
Theorem 6.3 Side-Side-Side (SSS) Similarity Theorem:
If the corresponding side lengths of two triangles are
_____________, then the triangles are _________.
similar
proportional
Theorem 6.4 Side-Angle-Side (SAS) Similarity Theorem:
If an angle of one triangle is _____________
congruent to an angle of a
second triangle and the lengths of the sides including these
angles are ______________,
similar
proportional then the triangles are ________.
6.1
Use Properties of Tangents
Example 7 Use tangents with similar triangles
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 AB  AE
AC AD
Solution
AB  BE and AC  CD
__________
__________

ABE and ACD
are right _s.
__________
______
ABE  ACD
CAD  BAE
__________
______
ABE  ACD
AB AE

__________
______
AC AD
A
E
B
C
D
______________
Theorem 6.1
Definition of .
All right s are  .
Reflexive
Prop
______________
AA Similarity Post
Corr. sides lengths
are prop.
6.1
Use Properties of Tangents
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.
RT  RS
x  49
2
x  7
R
S
49
C
x2
T
6.1
Use Properties of Tangents
Pg. 198, 6.1 #1-34