Special Segments in
Circles
Lesson 9.1B
R.4.G.5 Investigate and use the properties of angles
(central and inscribed) arcs, chords, tangents, and
secants to solve problems involving circles
CGT.5.G.4 Write, in standard form, the equation of a
circle given a graph on a coordinate plane or the center
and radius of a circle
Vocabulary
Tangent line:
A line that touches the circle at exactly one
point.
Tangent line is always perpendicular to the
radius
Vocabulary
Secant line:
A line that intersects the circle at exactly two
points.
The IIII Theorem
If two chords or secant segments intersect
inside a circle, then the products of the
intersected pieces are congruent
a·b=c·d
d
b
a
c
Example
Find the value of x.
8
3
x
6
Now You Try…
Find the value of x.
2
6
11
x
Vocabulary
External Secant Segment
The piece of a secant that is between the circle and
a point outside the circle.
Tangent Segment
The piece of a tangent line that is between the
circle a point outside the circle.
IOIO Theorem
If two secants are drawn from an exterior point
to a circle, then the product of the measure
of one secant’s external segment with the
sum of the internal and external segments is
equal to the product of the measure of the
other secant’s external segment with the sum
of the internal and external segments.
b
b(a+b) = d(c+d)
d
a
c
Example
Find the value of x.
Now You Try…
Find the value of x.
IOO Theorem
If a secant and tangent are drawn from an
exterior point to a circle, then the square of
the measure of the tangent segment is equal
to the product of the measure of the secant’s
external segment with the sum of the internal
and external segments
b
a2 = b(b+c)
a
c
Example
Find the value of x.
12
8
x+4
Now You Try…
Find the value of x.
12
9
x