Download The degree measure of an arc is

Geometry Notes C – 1: Circles and Arcs
Circles
r
A circle is the set of points in a plane that are
C
A circle is usually named by its center: Circle C
Facts: All radii of a given circle are
Two circles are congruent if
Central Angles
A central angle of a circle is any angle
O
Arcs
An arc of a circle is
B
A
A minor arc is any arc that is
O
A major arc is any arc that is
C
D
A semicircle is an arc that is
Y
A
The degree measure of an arc is
Note: degree measure is NOT the same as arc length.
O
Z
B
Facts: A full circle is
A semicircle is
Congruent arcs are two arcs of the same circle (or congruent circles) that have
B
A
Congruent central angles intercept congruent arcs and vice versa.
C
O
Arcs that share an endpoint but no other points may be added.
D
Ex: In circle O, mAB  70 and OB  OC . Find
A
a. mBC 
b. mAC 
P
B
O
c. mAPC 
C
Ex: In circle O, mAOB = 70, AB  CD and mBC and mAD are in the ratio 3:8. Find mBOC.
B
C
A
O
D
Geometry Notes C – 2: Tangents and Chords
Tangents and Chords
A
A tangent to a circle is a line (in the plane of the circle) that
t
D
intersects the circle in
A secant to a circle is a line that intersects the circle in
O
s
D B
C
l
D
A chord of a circle is a line segment
N
Theorem: A tangent and the radius it intersects are
A
T
Given: TAN is tangent to circle O at A; radius OA is drawn
Prove:
O
A tangent segment is a line segment from the point of tangency to another point on the tangent line.
Theorem: Two tangent segments drawn to a circle from the
same external point
Ex: In the diagram at right PA =
x
x  48
, PB =
and
3
x3
A
the radius of circle O is 2.
P
O
a. Find the value of x.
B
b. Find the length of OP .
Theorem: If a radius (or diameter) is perpendicular to a chord, then it
R
B
M
A
O
B
Theorem: If two chords of a circle are equidistant from the center,
then they are
, and conversely.
C
M
A
O
N
D
Ex: In circle O with radius 12, chord AB is 8 units from O.
a. What is the length of the chord?
b. What is the degree measure of AB ?
Ex: In circle O, PA is a tangent segment and mAB  70 .
a. mAOB =
A
b. mAQ 
c. mAQO =
Q
O
d. mP =
B
P
Geometry Notes C – 3: Inscribed Angles
An inscribed angle of a circle is an angle whose vertex is on the circle
and whose sides are contain chords of the circle.
Theorem: The measure of an inscribed angle is
A
B
P
O
Given: Circle O with inscribed angle APB.
1
Prove: mAPO  m AB
2
A
Case 1: One side of the inscribed angle includes a diameter of the circle.
P
B
O
A
Case 2: The center of the circle is in the interior of the angle.
P
O
B
A
B
Case 2: The center of the circle is in the exterior of the angle.
P
O
Corollary: If two inscribed angles intercept the same arc, they are congruent.
Corollary: Congruent inscribed angles intercept congruent arcs and vice versa.
Ex: Find the measure of x in each diagram.
a.
b.
x
.
c.
70
80
d.
x
.
x
.
x
.
120
Ex: In circle O, AOB is a diameter, mBDC = 30 and
mAD : mBD = 5:4. Find
120
C
a. mCAB
A
b. mCBA
.
E
O
30
c. mACD
d. mCEB
D
B
Geometry Notes C – 5: Angles Formed by Chords, Secants and Tangents
“Interior Angle”
D
An interior angle of a circle is an angle formed by
A
E
.
C
B
Theorem: The measure of an interior angle in a circle is half the sum of the
measures of the arcs intersected by the angle and its vertical angle.
D
A
E
C
B
“Exterior Angle”
An exterior angle of a circle is an angle formed by
Theorem: The measure of an exterior angle in a circle is half the difference
of the measures of the arcs intersected by the angle.
A
.
D
P
C
B
Ex: Find the measure of x in each diagram.
a.
110
b.
.
x
c.
x
80
170
.
40
x
.
130
120
d.
e.
f.
40
40
.70
.
70
.
x
x
x
25
Ex: In the diagram at right, GDF is a tangent and mDA : mBC = 3:4.
Find
a. mP
F
D
A
b. mDEC
G
E
110
P
20
c. mFDA
C
B
Geometry Notes C – 7: Chords, Secant/Tangent Segments
Segments Formed by Intersecting Chords
D
A
Theorem: When two chords intersect inside a circle,
E
.
B
C
8
Ex: Solve for x in the diagram.
x
6
.
Ex: In the diagram (which is not drawn to scale), AE = 12, EB = 6 and
CD = 22. Find the lengths of CE and ED if CE > ED.
12
B
C
.
A
E
D
Segments Formed by Intersecting Secants (or a Secant and a Tangent)
Vocabulary: PA and PC are called secant segments.
PB and PD are called external segments of the secants.
A
Theorem: When two secants intersect outside a circle,
B
P
.
D
C
Proof:
A
B
P
.
D
C
12
6
Ex: Solve for x in the diagram.
.
8
x
A
Ex: In the diagram (which is not drawn to scale), PA is a tangent
segment. If PA = 4 6 and MC is 4 more than PM, find the length
of PC .
.
P
E
M
C
Geometry Notes C – 10: Proofs
The following facts/theorems may be helpful on tonight’s homework and Thursday’s test.
1. All radii of a circle
B
2. A radius (or diameter) and a tangent to a circle
A
O
B
3. The arcs between two parallel chords are
C
A
D
4a. Inscribed angles that intercept congruent arcs
b. Inscribed angles that intercept the same arc
C
C
D
E
B
B
D
F
A
A
5. An inscribed angle in a semicircle is
C
B
O
A