11.4 Chords
Bell Work:
1. Find m<AQB.
2. Find m<APB.
= ½ (100° + 30°) = 65°
= ½ (100° – 30°) = 35°
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100˚
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30˚
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B
With your partner complete problem 3 question 1ab on page 884.
The points are all equally spaced around the circle.
a. Use a ruler to draw the congruent chords.
c. Use a protractor to measure the central angles.
What do you notice about these angles? These central angles are congruent.
What does this tell you about their intercepted arcs?
Since the central angles are congruent and arcs have the same measure as the angle
that intercepts them, then the intercepted arcs will also be congruent
Highlight the Congruent Chord–Congruent Arc Theorem at the bottom of page 885.
Congruent Chord-Congruent Arc Theorem
If two chords of the same circle or congruent circles are congruent,
then their corresponding arcs are congruent.
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Given: CH  DR
Prove: CH  DR
Strategy: Draw radii OC, OH, OR, and OD. Prove ΔCOH  ΔDOR,
<COH & <DOR are corresponding parts. Use the way arcs are measured to get CH  DR.
Proof:
1.
2.
3.
4.
5.
6.
7.
8.
Statements
CH  DR
OC  OH  OR  OD
ΔCOH  ΔDOR
<COH  <DOR
m<COH = m<DOR
m<COH = mCH, m<DOR = mDR
mCH = mDR
CH  DR
1.
2.
3.
4.
5.
6.
7.
8.
Reasons
Given
All radii of a circle are congruent.
SSS
CPCTC
Definition of congruent angles
Definition of an inscribed angle
Substitution Property
Definition of congruent arcs
Highlight the Congruent Chord-Congruent Arc Converse Theorem at the top of page 886.
Congruent Chord–Congruent Arc Converse Theorem
If two arcs of the same circle or congruent circles are congruent, then their corresponding
chords are congruent.
D
Given: CH  DR
Prove: CH  DR
This proof is almost identical to the above proof so we will not do it.
Any time both a theorem and its converse are true, the two of them can be put into one theorem by
using a form called a bi-conditional.
Formula: p if and only if q.
With your partner complete problem 3 question 4 on page 886.
Two chords of the same circle or congruent circles are congruent, if and only it, their
corresponding arcs are congruent.
Read problem 1 on page 878 and complete question 1abcdefg with the class.
1b. Use a ruler to find the midpoint of YR.
Use a protractor to draw the perpendicular
at the midpoint.
1d. Use a ruler to find the midpoint of BR.
Use a protractor to draw the perpendicular
B
at the midpoint.
1f. Use a ruler to find the midpoint of BY.
Use a protractor to draw the perpendicular at the midpoint.
1g. How are these 3 perpendicular bisectors related?
They all pass through the center of the circle.
Thus each of these perpendicular bisectors contain diameters of the circle.
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Diameter–Chord Biconditional Theorem
A circle’s diameter is perpendicular to a chord, if and only if, the diameter
bisects the chord and bisects the arc determined by the chord.
Given: Circle O w/diameter MI, MI | DA
Prove: MI bisects DA, MI bisects DA
Strategy: Draw radii OD and OA. Prove ΔODE  ΔOAE.
Then DE & AE and <DOI &<AOI are corresponding parts.
Given: Circle O w/ MI | bisect DA and/or MI | bisector DA
Prove: MI is a diameter
Complete the proof of the forward part of this theorem with the class using the space
provided at the top of page 879.
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Statements
1. MI | DA
2. <OED and <OEA are rt. <’s
3. ΔOED and ΔOEA are rt. Δ’s
4. OD  OA
5. OE  OE
6. ΔOED  ΔOEA
7. DE  AE
8. MI bisects AD
9. <DOI  <AOI
10. DI  AI
11. MI bisects DA
Reasons
1. Given
2. Definition of perpendicular
3. Definition of right triangle
4. All radii of a circle are congruent.
5. Reflexive Property
6. HL
7. CPCTC
8. Definition of segment bisector
9. CPCTC
10. Congruent Chords-Congruent Arcs Theorem
11. Definition of arc bisector
With your partner complete problem 1 question 3ab on page 880.
Use a ruler to draw two chords that are the same length. (congruent)
For example: HC = 2 cm and RD = 2 cm
a. Use a ruler to locate the midpoint.
C
Use a protractor to draw the perpendicular at this point.
b. Use a ruler instead of a compass to measure the lengths.
For example: O to midpt CH = 0.5 cm and O to midpt RD = 0.5 cm
The distances are the same.
Equidistant Chord Biconditional Theorem
Two chords of the same circle or congruent circles are congruent,
if and only if, they are equidistant from the center of the circle.
Given: CH  DR, OE | CH, OI | RD
Prove: OE  OI
Given: OE  OI, OE | CH, OI | RD
Prove: CH  DR
We will not prove this biconditional theorem.
This biconditional theorem has many applications.
1. Carpenters use it to construct circular openings in buildings and arches.
2. Archeologists use it to restore pottery found at digs.
Homework:
Skills Practice Problem Set 11.4 prob. 2 – 4, 8 – 10, 14 – 16, 20 – 22.
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