Download 11.3 Arcs and Chords

Geometry – Chapter 11 Lesson Plans
Section 11.3 – Arcs and Chords
Enduring Understandings: The student shall be able to:
1. Identify and use the relationships among arcs, chords, and diameters
Standards:
30. Circles
Identifies and defines circles and their parts (center, arc, interior, exterior); segments and
lines associated with circles (chord, diameter, radius, tangent, secant); properties of circles
(congruent, concentric, tangent); relationship of polygons and circles (inscribed,
circumscribed); angles (central; inscribed; formed by tangents, chords, and secants).
31. Circles
Apples geometric relationships to solving problems, such as relationships between lines
and segments associated with circles, the angles they form, and the arcs they subtend; and
the measures of these arcs, angles, and segments.
Essential Questions: What is the relationship between an arc, chord and diameter?
Warm up/Opener:
Activities:
Review Thm 11.3 from last section: In a circle or in congruent circles, two minor arcs
(the arc lengths) are congruent iff their corresponding central angles are congruent.
This comes from the relationship:
Arc length = central angle
Circumference
360
If the circles are congruent, the circumferences will be the same. Then we have one equation
and two unknowns. If the central angles are the same then the arc lengths are the same, and
visa versa.
Thm 11.4: In a circle or in congruent circles, two minor arcs are congruent iff their
corresponding chords are congruent. They are again talking about the arc length. If the
chords are congruent, then the triangles are congruent by SSS, and the central angles are
congruent by CPCTC, and by Thm 11.3 the arcs are congruent. Reversing, if the arcs are
congruent (in the same or congruent circles), then the central angles are congruent, and the
triangles are congruent by SAS, and the chords are congruent by CPCTC.
Do the hands-on on page 469. This gives Theorem 11-5
Thm 11.5: In a circle, a diameter bisects a chord and its arc iff it is perpendicular to the
chord. Refer to the drawing below.
Proof of: If, then PC  AB.
Claim
Construct radii AP and BP
AC  BC
AP  BP
PC  PC
 APC   BPC
PCA  PCB
PCA + PCB = 180
PCA + PCA = 180
PCA = 90
Reason
Given
all radii of a circle are congruent
reflexive property
SSS
CPCTC
Definition of Linear Pair
Substitution
Algebra
Proof of: If PC  AB, then AC  BC.
Claim
Construct radii AP and BP
PC  AB
AP  BP
PC  PC
 APC   BPC
AC  BC
Reason
Given
all radii are congruent
reflexive property
HL
CPCTC
A
C
P
B
Assessments:
Do the “Check for Understanding”
CW WS 11.3
HW pg 472-473, # 9-29 by 4’s (6)
HW pg 456-457 of the red book (in his packet), # 29 - 35 odd and 51 (5)
HW Study Guide (in his packet) # 11-3, # 1 – 6 (6)