# Ch 7

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```Mth 97
Fall 2013
Chapter 7
Section 7.1 – Central Angles and Inscribed Angles
An ___________ of a circle consists of the points A and B of the circle together with the portion of the
circle contained between the two points. Points A and B are called the __________________ of the arc.
A __________________________ is an arc whose endpoints are the endpoints of a diameter and is
named by an arc symbol above three points: endpoint, another point on the arc between the
endpoints, endpoint.
An arc that is shorter than a semicircle is called a __________________ arc and is named by an arc
symbol above just the endpoints.
An arc that is longer than a semicircle is called a __________________ arc and is named by an arc
symbol above three points: endpoint, another point on the arc between the endpoints, endpoint.
E
A
D
G
B
Minor arc AB
Semicircle DEF
Major arc GHI
H
F
I
Symbols
A Central Angle is an angle whose vertex is the ______________ of the circle
and whose sides intersect the circle. The measure of an arc of a central angle
is equal to the measure of the angle.
mAOB  x if and only if AB =
A
O
B
Without using a protractor find the measures of the following arcs and central angles in circle O.
40°
C
AB =
B
110°
55°
O
D
ACD =
BOC =
DOE =
CAD =
AOE =
AE =
A
E
Postulate 7.1 – Length of an Arc – The ratio of the length of an arc l , of an arc of a circle to the
circumference, C, of the circle equals the ratio of the measure of the central angle of the arc, x°, to 360°.
r x°
O
l
l
x

C 360
or
l
x
 2 r 
360
A grandfather clock has a pendulum 32 inches long. If the pendulum swings through and arc of 12°, how
far does the pendulum travel?
1
Mth 97
Fall 2013
Chapter 7
Postulate 7.2 – Area of a Sector – The ratio of the area of a sector of a circle to the area of the circle is
equal to the ratio of the measure of the central angle of the sector, x°, to 360°.
B
A
A sector
x
x
x

or A sector 
 Acircle    r 2 
360
360
A circle 360
r
x°
O
Find the area of the minor sector if mAOB  75 and the radius is 10 cm.
An ____________________ angle is an angle whose vertex lies on the circle and its sides each intersect
the circle in another point.
Inscribed Angle Theorem – The measure of an inscribed angle in a circle is equal to half the measure of
its intercepted arc.
A
mABC 
B
1
AC
2
Find mABC if AC = 66°.
C
If AB = 140°, find mACB and mAOB .
C
O
B
mACB =
mAOB =
A
Find y, if mTPR  55
If AB = 150° and BC = 110°, find x.
A
P
x°
B
A
y°
T
R
C
2
Mth 97
Fall 2013
Chapter 7
Corollary 7.2 – Inscribed angles that intercept the same arc (or congruent arcs) are congruent.
A
mABD  mACD 
B
C
1
AD
2
If AD = 48°, find the measure of each angle.
D
mABD 
mACD 
Corollary 7.3 – An angle is inscribed in a semicircle if and only if it is a right angle.
C
A
B
O
C is inscribed in semicircle ACB
mC=
Find the measure of the following angles where O is the center of the circle and AC and DF are
diameters. The figure is not drawn to scale.
120°
D
C
O
A
F
mCOD 
mC 
mDBC 
mF 
mADF 
mDAF 
mDOA 
mDAB 
60°
B
Do ICA 12, problems 1 and 2
3
Mth 97
Fall 2013
Chapter 7
Section 7.2 – Chords of a Circle
A __________________ is a segment of a circle whose endpoints are on the circle. A
B
O
In circle O, __________, __________, and _________ are chords.
C
D
A _____________________ is a chord of the circle that goes through the center of the circle. ________
A ____________________ is half of a diameter of a diameter (from the center to a point on the circle).
The plural of radius is ______________. In circle O _______, ________, and ________ radii.
Theorems important for constructions:
l
A
Theorem 7.4 – The perpendicular bisector of a chord contains the center of the circle.
C
Theorem 7.5 - The intersection of the perpendicular bisectors of any two
nonparallel chords is the center of the circle.
m
B
O
D
Using a compass and straight edge draw a circle with two nonparallel chords. Label the center of the
circle P and your chords AB and CD . Construct the perpendicular bisector of each chord. (See the
bottom of page 375 for different methods you can use to bisect a segment.)
Corollary 7.6 – If two circles, O and P, intersect in two points A and B, then the line containing O and P is
the perpendicular bisector of AB .
AC =
A
P
O
C
B
4
Mth 97
Fall 2013
Chapter 7
Measures of Angles formed by Chords
Theorem 7.7 – If two chords intersect, then the measure of any one of the vertical angles formed is
equal to half the sum of the measures of the two arcs intercepted by the two vertical angles.
B
A



1
AB  CD  mCED
2
mAED 
1
AD  BC  mBEC
2
C
E

mAEB 
D
If AD  50 and BC  30 , then mAED 
mBEC 
mAEB 
mCED 
Measures of Segments of Chords
Theorem 7.8 – If two chords of a circle intersect the product of the lengths of the two segments formed
on one chord is equal to the product of the lengths of the two segments formed on the other chord.
A
(AE)(EC) = (BE)(ED)
E
B
If AE = 8, EC = 3, and BE = 4, find ED.
C
D
If AE = 9.8 cm, CE = 5.7 cm, and DE = 3.8 cm,
find BE.
If AC = 10 cm, BE = 8 cm, DE = 3 cm,
find AE.
Use the circle below to find the following measures, if possible.
A
40°
B
6
8
E
16
mAEB 
mCEB 
mCED 
mAED 
EB 
AD 
C
60°
D
5
Mth 97
Fall 2013
Chapter 7
More practice. Figures are not drawn to scale.
If AB  24 and mAEB  35 , find CD .
D
E
C
A
C
If mCED = 41 and mDAC = 25 , find AB .
D
E
B
A
B
Section 7.3 – Secants and Tangents
A _________________ line intersects a circle in two points.
A ___________________ line intersects a circle in exactly one point, called the point of tangency.
A
B
C
Chord
Secant
m
Tangent
Point of tangency
Angles formed by Secants and Tangents
The measure of the angles formed outside of circles by
secant lines, a secant and a tangent line, or two tangents is half the difference of the intercepted arcs.
Theorem 7.9 – If two secant lines intersect outside a circle, the measure of the acute angle formed is
half the difference of the measures of the intercepted arcs.
B
A
E
1
D
mE =
O
O
2
C
Theorem 7.13 – If a secant and a tangent line intersect outside a circle, the measure of the angle
formed is half the difference of the measures of the intercepted arcs.
A
B
mB =
1
2
C
D
6
Mth 97
Fall 2013
Chapter 7
Theorem 7.14 – If two tangent lines intersect outside a circle, the measure of the angle formed is half
the difference of the measures of the intercepted arcs.
A
B
mB =
1
2
C
Theorem 7.11 – The radius or diameter of a circle is perpendicular to a tangent line at its point of
tangency.
B
O
A
OA 
which means ____________ is a right angle.
Theorem 7.12 – The measure of the angle formed when a chord intersects a tangent line at the point
of tangency is half the measure of the arc intercepted by the chord and tangent line.
A
C
mABC 
1
2
B
Measure of Segments formed by Secant and Tangent Lines
A
B
E
Theorem 7.10 – If two secants intersect outside a circle, then
the product of the lengths of the two segments formed on one
secant (vertex to point on the circle) is equal to the product of
the lengths of the corresponding segments on the other secant.
D
C
(AE)(BE) =
A
B
C
AB =
A
B
Theorem 7.16 – If we draw a tangent and a secant line from the
same point in the exterior of the circle, the length of the
tangent segment is the mean proportional between the length
of the external secant segment and the length of the secant
segment inside the circle.
D
C
BD
Theorem 7.15 – If two tangent lines are drawn to a circle from
the same point in the exterior of the circle, the distances from
the common point to the points of tangency are equal.

BC
or
(BA)(BA) = ( ) ( )
7
Mth 97
Fall 2013
Chapter 7
Secant and Tangent Practice
Find BC, mBD , and mCAE .
Find the value of x, mBC , and mM .
120°
A
6
C
A
154
B
8
o
10
78o
90°
D
18
E
C
x
B
12
M
110°
Find the value of x, mA , and AD.
A
Find the values of AC, x and y in circle below.
D
4 E
5
A
x°
B
52°
P
y°
7
C
x°
D
84°
50°
C
12
B
P is the center of the circle.
8
Mth 97
Fall 2013
Chapter 7
Section 7.4 - Constructions Involving Circles
Given an arc, construct the circle it is part of.
First draw two nonparallel chords. Construct the
perpendicular bisector of each chord to find
the circle’s center. Set your compass for the radius and
draw the rest of the circle.
The circumscribed circle of a triangle is the unique circle that contains the triangle’s _______________.
Constructing the circumscribed circle of a triangle
First construct the perpendicular bisector of two
sides of the triangle. Use the distance from the
intersection point of the bisectors to any vertex as
the radius to circumscribe the triangle in a circle.
Theorem 7.17 – Circumcenter of a Triangle
The perpendicular bisectors of the sides of a triangle
________________in a single point, the circumcenter.
A
P is the circumcenter of ∆ABC.
P
B
C
9
Mth 97
Fall 2013
Chapter 7
Construct an altitude from each vertex to the opposite side. You may need to extend a side to help you
construct the perpendicular.
B
Theorem 7.18 – Orthocenter of a Triangle
The ______________ of a triangle intersect in a single point,
the orthocenter.
P is the orthocenter of ∆ABC.
P
A
C
The points A, B, C, and P form an orthocentric set which has the
property that the triangle formed by any three of the four points in the set has the fourth point
as its orthocenter.
Constructing the inscribed circle of a triangle
First construct any two angle bisectors of the triangle to locate
the incenter, the point that is equidistant from all sides of the triangle.
Next construct a perpendicular from the incenter to a side.
Use the distance from the the incenter to the side as a radius to
inscribe a circle within the triangle.
10
Mth 97
Fall 2013
Chapter 7
A
Theorem 7.19 – Incenter of a Triangle
The angle bisectors of a triangle meet
in a single point, the incenter.
P is the incenter of ∆ABC.
P
C
B
Constructing the centroid of a triangle
First construct the midpoints of any two sides and
draw the medians. Their intersection point is the
centroid, or center of gravity, or balance point
of the triangle.
Theorem 7.20 – Centroid of a Triangle
The medians of a triangle intersect in a single point,
the centroid, which is two-thirds of the way from
the vertex to the other endpoint of the median
from that vertex.
A
P is the centroid of ∆ABC
P
C
BP 
B
2
2
 BF  , AP   AE 
3
3
and CP 
2
 CD 
3
Constructing a tangent to a circle
First draw a ray from the center of the circle through
the point on the circle you want the tangent to intersect.
Next construct a line perpendicular to the radius you
drew at the point
11
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