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 / Reason
Construct radii AP and BP
AC  BC / Given
AP  BP / all radii of a circle are congruent
PC  PC / reflexive property
 APC  BPC / SSS
PCAPCB / CPCTC
PCA + PCB = 180 / Definition of Linear Pair
PCA + PCA = 180 / Substitution
PCA = 90 / Algebra

Proof of: If PC  AB, then AC  BC.

Claim / Reason
Construct radii AP and BP
PC  AB / Given
AP  BP / all radii are congruent
PC  PC / reflexive property
 APC  BPC / HL
AC  BC / CPCTC

Assessments:

Do the “Check for Understanding”

CW WS 11.3

HWpg 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)