Chapter 10 Circles
10.1 Properties of Tangents
Circle:
the set of all points in a plane that are
equidistant from a given point.
(Center)
A circle is named by its center.
(show symbol on board.)
Radius:
A segment from the center of a circle to any
point on the circle.

Chord: A segment whose endpoints are on the circle.
Diameter: a chord containing the center
of the circle.
Diameter = two times radius.
Diameter = 2 times Radius.
Lines in circles:
Secants:
Tangents:
a line that intersects a circle at two points.
a line that intersects a circle in exactly one
point.
Point of tangency
Tell whether the line,
ray, or segment is
best described as a
radius, chord,
diameter, secant, or
tangent of circle C.
B
A
C
E
D
Segment BC
b. Line DA
c. Line DE
a.
a. Radius
b. Tangent
c. Secant
Common Tangents
Common internal tangents
Common external tangents
Tell how many common tangents
the circles have and draw them.
A.
D.
C.
A. 3
B.
B. 2
C. 1
E.
D. 4
E. 0
Geometry CP
Page 655(1-17, 27, 28, 35, 36)
Sophomore Math
Page 655(1-17, 35, 36)
Two Tangent Rules (theorems)
1.
In a plane, a line is tangent to a
circle if and only if the line is
perpendicular to a radius of the
circle at its endpoint on the
circle.
Two Tangent Rules
•
Tangent segments from an
external point are congruent.
In the diagram, RS is a radius of
circle R. Is ST tangent to circle R?
10
24
26
Use the converse of the pythagorean theorem to find out
whether it is a right triangle.
26² = 10² + 24²
That proves that it is a right triangle making the line
perpendicular to the radius. Therefore, it must be a
tangent.
QR is tangent to the circle at R and QS
is tangent at point S. Find the value of
x.
R
32
C
Q
3x+5
S
QR + QS
32= 3x + 5
27 = 3x
9=x
Tangent segments from the same point are congruent.
EXAMPLE 2
Find lengths in circles in a coordinate plane
Use the diagram to find the given lengths.
a.
Radius of
b.
c.
Diameter of
d.
Diameter of
Radius of
A
A
B
B
SOLUTION
a.
The radius of
A is 3 units.
b.
The diameter of
c.
The radius of
d.
The diameter of
A is 6 units.
B is 2 units.
B is 4 units.
EXAMPLE 4
Verify a tangent to a circle
In the diagram, PT is a radius of
tangent to P ?
P. Is ST
SOLUTION
Use the Converse of the Pythagorean Theorem.
Because 122 + 352 = 372,
PST is a right triangle and
ST PT . So, ST is perpendicular to a radius of P at
its endpoint on P. By Theorem 10.1, ST is tangent
to P.
EXAMPLE 5
Find the radius of a circle
In the diagram, B is a point of tangency. Find the
radius r of C.
SOLUTION
You know from Theorem 10.1 that AB BC , so
ABC
is a right triangle. You can use the Pythagorean
Theorem.
AC2 = BC2 + AB2 Pythagorean Theorem
(r + 50)2 = r2 + 802
r2 + 100r + 2500 = r2 + 6400
100r = 3900
r = 39 ft .
Substitute.
Multiply.
Subtract from each side.
Divide each side by 100.
EXAMPLE 6
Find the radius of a circle
RS is tangent to C at S and RT is tangent to
Find the value of x.
C at T.
SOLUTION
RS = RT
28 = 3x + 4
8=x
Tangent segments from the same point are
Substitute.
Solve for x.
Geometry CP

Page 656(18-26,29,33,37,43-47)
Sophomore Math

Page 656(18-20,24,27,28,30,37,43-47)