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10.5 and 10.6
Tangents and Secants
Glencoe Geometry Interactive Chalkboard
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Objectives
• Use properties of tangents
• Solve problems using circumscribed
polygons
• Find measures of angles formed by lines
intersecting on, inside, or outside a circle.
Tangents and Secants
• A tangent is a line in the
plane of a circle that
intersects the circle in
exactly one point. Line j
is a tangent.
• A secant is a line that
intersects a circle in two
points. Line k is a
secant. A secant contains
a chord.
k
j
Tangents
Theorem 10.9:
If a line is tangent to a , then it is ┴ to the
radius drawn to the point of tangency.
The converse is also true.
r┴j
r
j
ALGEBRA
is tangent to
at point R. Find y.
Because the radius is perpendicular to the tangent at the
point of tangency,
. This makes
a right
angle and 
a right triangle. Use the Pythagorean
Theorem to find QR, which is one-half the length y.
Pythagorean Theorem
Simplify.
Subtract 256 from each side.
Take the square root of each
side.
Because y is the length of the diameter, ignore the
negative result.
Answer: Thus, y is twice
.
is a tangent to
Answer: 15
at point D. Find a.
Determine whether
is tangent to
First determine whether ABC is a right triangle by using
the converse of the Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Because the converse of the Pythagorean Theorem did
not prove true in this case, ABC is not a right triangle.
Answer: So,
is not tangent to
.
Determine whether
is tangent to
First determine whether EWD is a right triangle by using
the converse of the Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Because the converse of the Pythagorean Theorem is
true, EWD is a right triangle and EWD is a right angle.
Answer: Thus,
making
a tangent to
a. Determine whether
Answer: yes
is tangent to
b. Determine whether
Answer: no
is tangent to
Tangents (continued)
Theorem 10.11:
If two segments from the same exterior point
are tangent to a circle, then they are congruent.
W
Z
XW  XY
X
Y
ALGEBRA Find x. Assume that
segments that appear tangent to
circles are tangent.
are drawn from the
same exterior point and are
tangent to
so
are drawn from the same
exterior point and are tangent to
Definition of congruent segments
Substitution.
Use the value of y to find x.
Definition of congruent segments
Substitution
Simplify.
Subtract 14 from each side.
Answer: 1
ALGEBRA Find a. Assume that segments that
appear tangent to circles are tangent.
Answer: –6
Triangle HJK is circumscribed about
perimeter of HJK if
Find the
Use Theorem 10.10 to determine the equal measures.
We are given that
Definition of perimeter
Substitution
Answer: The perimeter of HJK is 158 units.
Triangle NOT is circumscribed about
perimeter of NOT if
Answer: 172 units
Find the
Assignment
• Pre-AP Geometry
Pg. 556 #8 – 20, 23 - 26
• Geometry:
Pg. 556 #8 – 18, 23 - 25
Secants and Interior Angles
Theorem 10.12:
If two secants intersect in the interior of a ,
then the measure of an  formed is ½ the
measure of the sum of the arcs intercepted
by the secants that created the .
A
C
mAOC = ½ (m arc AC + m arc DB)
mAOD = ½ (m arc AD + m arc CB)
D
O
B
Find
Method 1
if
and
Method 2
Answer: 98
Find
if
Answer: 138
and
Secants, Tangents and Angles
Theorem 10.13:
If a secant and a tangent intersect at the
point of tangency, then the measure of each
 formed is ½ the measure of its
D
A
intercepted arc.
O
C
B
mDOB = ½ (m arc AB)
mCOB = ½ (m arc CFB)
F
Find
Answer: 55
if
and
Find
Answer: 58
if
and
Secants – Tangents and Exterior Angles
Theorem 10.14:
If two secants, a secant and a tangent, or two
tangents intersect in the exterior of a circle, then
the measure of the  formed is ½ the measures of
the difference of the intercepted arcs.
A
G
E
mBOC = ½ (m arc BC – m arc HI)
mAOC= ½ (m arc AC – m arc AI)
mAED= ½ (m arc ABD – m arc AD)
O
F
B
H
I
D
C
Find x.
Theorem 10.14
Multiply each side by 2.
Add x to each side.
Subtract 124 from each side.
Answer: 17
Find x.
Answer: 111
JEWELRY A jeweler wants to craft a pendant with
the shape shown. Use the figure to determine the
measure of the arc at the bottom of the pendant.
Let x represent the measure of
the arc at the bottom of the
pendant. Then the arc at the top
of the circle will be 360 – x. The
measure of the angle marked
40° is equal to one-half the
difference of the measure of the
two intercepted arcs.
Multiply each side by 2 and
simplify.
Add 360 to each side.
Divide each side by 2.
Answer: 220
PARKS Two sides of a fence to be built around a
circular garden in a park are shown. Use the figure
to determine the measure of
Answer: 75
Find x.
Multiply each side by 2.
Add 40 to each side.
Divide each side by 6.
Answer: 25
Find x.
Answer: 9
Assignment
• Pre-AP Geometry
Pg. 564 #12 - 32
• Geometry:
Pg. 564 #3 – 7, 12 - 26