Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Section 4.5:
Circles
Derek W. Hein
Math 3130–1: Modern Geometries
Southern Utah University
Spring 2014
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definition
Propositions
Comments
Definition of circle
A circle is the set of points that are equidistant from a given
point. The given point is called the center, and the given
distance is the radius.
We recall that two distinct points are sufficient to determine a
unique line (in neutral geometry). However, it should be clear
that two distinct points determine many circles.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definition
Propositions
Comments
Proposition A
However, the following can be proven:
Theorem
In the Euclidean plane, three distinct (non–collinear) points
determine a unique circle.
Proof.
Existence: Use the intersection of perpendicular bisectors of two
lines to find the center; this also determines the radius.
Uniqueness: Proceed by way of contradiction ...
(The proof is in the text, pp. 157–159.)
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definition
Propositions
Comments
Proposition B
We also note that three distinct non–collinear points determine
a triangle!
Corollary
In Euclidean geometry, every triangle can be circumscribed.
Note that the center of the circumscribing circle (called the
circumcenter of the triangle) is where the perpendicular
bisectors of its sides meet.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definition
Propositions
Comments
Comments
Note that this statement was listed earlier (on p. 127) as being
equivalent to the Euclidean parallel postulate.
Hence, there are triangles (in hyperbolic geometry) that cannot
be circumscribed. (More details in Chapter 6...)
Also, we denote the circle C with center O and radius r = OP (determined by a point P on C) by C(O, r).
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Propositions
Definitions of chord and diameter
Definition
A chord of a circle is a line segment that joins any two points
on a circle.
Definition
A diameter of a circle is a chord that contains the center of the
circle.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Propositions
Proposition C
Theorem
If AB is a diameter of a circle and CD is any
chord
of
other
the same circle that is not a diameter, then AB > CD.
Proof.
Text, p. 160.
(on board)
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Propositions
Proposition D
Theorem
If a diameter of a circle is perpendicular to a chord of the
circle, then the diameter bisects the chord.
Proof.
Text, p. 161.
(on board)
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Propositions
Proposition E
Theorem
If a diameter of a circle bisects a chord of the circle (that is not
a diameter), then the diameter is perpendicular to the chord.
Proof.
Assigned homework.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Propositions
Proposition F
Theorem
The perpendicular bisector of a chord of a circle contains a
diameter of the circle.
Proof.
Unassigned homework.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Proposition
Definitions of secant and tangent
Definition
A secant of a circle is a line that contains exactly two points of
a circle.
Definition
A tangent of a circle is a line that contains exactly one point of
a circle.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Proposition
Proposition G
Theorem
If a line is tangent to a circle, then it is perpendicular to the
radius drawn to the point of tangency.
Proof.
Text, p. 162.
(on board)
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Definition of central angle
Definition
Any angle θ whose vertex is the center of a circle is called a
central angle for the circle.
θ
Note that the Protractor Postulate gives us the restriction
0◦ ≤ θ ≤ 180◦ .
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Definition of minor arc
Definition
If A and B are points on C(O, OA) that are not endpoints of
the same diameter, then the union of A, B and all points of
C(O, OA) that are in the interior of central angle ∠AOB is
called a minor arc of C(O, OA).
A
←
O
B
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Definition of major arc
Definition
If A and B are points on C(O, OA) that are not endpoints of
the same diameter, then the union of A, B and all points of
C(O, OA) that are in the exterior of central angle ∠AOB is
called a major arc of C(O, OA).
A
→
O
B
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Definition of semicircle
Definition
If A and B are points on C(O, OA) that are endpoints of a
diameter, then the union of A, B and all points of C(O, OA)
←→
that lie on either half–plane defined by AB is called a
semicircle of C(O, OA).
A
O
B
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Definition of arc measure
_
We denote the circular arc between points A and B by AB.
Definition
Let A and B be points on C(O, OA). Then the degree measure
_
of AB is
_
a) the degree measure of the angle ∠AOB if AB is a minor
arc;
_
b) 180◦ if AB is a semicircle;
_
c) 360◦ minus the degree measure of the angle ∠AOB if AB is
a major arc.
_
_
Also, mAB means the measure of arc AB.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Proposition H
Theorem
If two chords of a circle are congruent, then their corresponding
minor arcs have the same measure.
Proof.
Immediate, from the above definitions.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Proposition I
(The converse of the last one:)
Theorem
If two minor arcs of a circle are congruent, then so are the
corresponding chords.
Proof.
Homework!
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Proposition J
Many properties of angles carry over to arcs (by SMSG
Postulate 13). For example:
Theorem (Arc Addition Theorem)
_
_
_ _ _
B, then mAPB + mBQC = mABC.
If APB and BQC are arcs of a circle sharing only the endpoint
Proof.
Text, pp. 166–167.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Definition of inscribed angle
Definition
_
An angle ∠SP T is said to be inscribed in an arc AQB iff
_
a) the vertex P of the angle is a point of AQB,
b) one ray of the angle contains point A, and
c) the other ray contains point B.
A
S
P
Q
B
Math 3130
T
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Definition of intercepted arc
Definition
_
An angle ∠SP T is said to intercept an arc ARB iff
a) both A and B are points of the angle,
b) each ray of the angle contains at least one endpoint of
_
ARB, and
_
c) (excepting A and B) each point on ARB lies in the interior
of ∠SP T .
A
S
P
R
B
Math 3130
T
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Proposition K
Theorem (Inscribed Angle Theorem)
The measure of an angle inscribed in an arc is one–half the
measure of its intercepted arc.
Proof.
Discussion, pp. 168–169.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Proposition L
Corollary
An angle inscribed in a semicircle is a right angle.
Proof.
Assigned homework.
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Definitions
Arcs
Arc Measure
Propositions
Other Ingredients
Proposition M
Corollary
Angles inscribed in congruent (or same) arcs are congruent.
Proof.
Unassigned homework (?)
Math 3130
Section 4.5
Circle
Chords and Diameters
Secants and Tangents
Angles
Conclusion
Homework
Your homework assignment is:
pp. 176–180 #3, 5, 7, 8, 11; 17, 22
Math 3130
Section 4.5
Bibliography
c
Roads to Geometry, 3rd edition, Wallace & West, 2004,
Pearson Education Inc., ISBN 978–0–13–041396–3
Math 3130
Section 4.5