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LESSON
LESSON
6.2
You will do foolish things, but
do them with enthusiasm.
SIDONIE GABRIELLA COLETTE
6.2
Chord Properties
In the last lesson you discovered some properties of a tangent, a line that intersects
the circle only once. In this lesson you will investigate properties of a chord, a line
segment whose endpoints lie on the circle.
PLANNING
In a person with correct vision, light rays from
distant objects are focused to a point on the
retina. If the eye represents a circle, then the path
of the light from the lens to the retina represents
a chord. The angle formed by two of these chords
to the same point on the retina represents an
inscribed angle.
LESSON OUTLINE
One day:
25 min Investigation
10 min Sharing
First you will define two types of angles in a circle.
5 min Closing
5 min Exercises
Investigation 1
MATERIALS
Defining Angles in a Circle
Write a good definition of each boldfaced term. Discuss your definitions with
others in your group. Agree on a common set of definitions as a class and add
them to your definition list. In your notebook, draw and label a figure to illustrate
each term.
Step 1 A central Step 1
angle has its vertex at
the center of the circle.
construction tools
protractors
Gear Fragment (W) for One Step
Sketchpad activity Chord Properties,
optional
Sketchpad demonstration Intersecting
Tangents Conjecture, optional
Central Angle.
R
D
O
P
A
Q
S
TEACHING
T
B
AOB, DOA, and DOB
are central angles of circle O.
Step 2 An
Step 2
inscribed angle has its
vertex on the circle and
its sides are chords.
PQR, PQS, RST, QST, and
QSR are not central angles of circle P
Inscribed Angle.
Q
V
A
P
R
C
In this lesson students discover
some properties relating central
angles, chords, and arcs of
circles. Begin with the one-step
investigation, or ask groups to
work through Investigations
1 through 4.
T
B
W
E
X
D
ABC, BCD, and CDE are
inscribed angles.
NCTM STANDARDS
CONTENT
U
PQR, STU, and VWX are
not inscribed angles.
You might have students use
patty paper for Investigations 2
through 4. You can have students
construct circles on patty paper
by tracing circular objects.
LESSON OBJECTIVES
PROCESS
Number
Problem Solving
Algebra
Reasoning
Geometry
Communication
Measurement
Connections
Data/Probability
S
You can replace or extend
Investigations 1 and 2 with the
Dynamic Geometry Exploration
at www.keymath.com/DG.
Representation
Discover properties of chords to a circle
Practice construction skills
Guiding Investigation 1
One Step
Hand out the Gear Fragment
worksheet and pose this problem: “In repairing a machine,
you find a fragment of a circular
gear. To replace the gear, you
need to know its diameter.
How can you find the gear’s
LESSON 6.2 Chord Properties
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diameter?” You may need to
review the term diameter right
away. As you circulate, wonder
aloud as needed whether the
center is somehow related to the
chords of the circle, pointing out
what you mean by chord.
Step 2 If groups are having difficulty defining inscribed angle,
[Ask] “Where have you heard
the word inscribed before?”
[A triangle inscribed in a circle
(whose center is the circumcenter of the triangle) has all
three vertices on the circle, and
its sides are chords of the circle.
An inscribed triangle has three
inscribed angles.]
Guiding Investigation 2
[Alert] If you find that all or
most students in a group are
using a central angle that
measures 60° (because they
didn’t change the compass
setting after drawing the circle),
point out that they don’t have
enough information for good
inductive reasoning; then have
them all change their settings
from the circle’s radius and
begin again.
Page 318
Investigation 2
Chords and Their Central Angles
Next you will discover some properties of chords and
central angles. You will also see a relationship between
chords and arcs.
You will need
●
●
●
●
a compass
a straightedge
a protractor
patty paper
(optional)
Step 1
Step 2
Students may find this investigation and the next two investigations fairly straightforward
and may begin writing conjec-
318
D
O
Construct a large circle. Label the center O. Construct
two congruent chords in your circle. Label the chords
and CD
, then construct radii OA
, OB
, OC
, and OD
.
AB
A
C
With your protractor, measure BOA and COD. How do they compare? Share
your results with others in your group. Then copy and complete the conjecture.
C-55
Chord Central Angles Conjecture
If two chords in a circle are congruent, then they determine two central
? congruent
angles that are Step 3
Step 3 Fold it so that
coincides with OB
and
OD
coincides with OA
.
OC
Step 4
How can you fold your circle construction to check
the conjecture?
Recall that the measure of an arc is defined as the
measure of its central angle. If two central angles are
congruent, their intercepted arcs must be congruent.
Combine this fact with the Chord Central Angles
Conjecture to complete the next conjecture.
Chord Arcs Conjecture
?
intercepted arcs
? are congruent.
If two chords in a circle are congruent, then their “Pull the cord?! Don’t I need
to construct it first?”
[Alert] Students may have trouble
constructing congruent chords.
Challenge them to find the
endpoints without using a ruler.
They might mark two points on
the circle and then use a
constant compass opening to
locate two other points on the
circle the same distance apart.
Step 3 If their paper is too thick
to see through when folded,
students may want to copy their
figure to patty paper.
B
tures without going through the steps or comparing
results with others in their group. Remind them
that the process is more important than the results.
If students finish early, you might challenge them
with the one-step problem.
CHAPTER 6 Discovering and Proving Circle Properties
?
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Investigation 3
Chords and the Center
of the Circle
You will need
●
●
●
a compass
a straightedge
patty paper
(optional)
Guiding Investigation 3
In this investigation you will discover relationships about a
chord and the center of its circle.
Step 1
Construct a large circle and mark the center. Construct
two nonparallel congruent chords. Then construct the
perpendiculars from the center to each chord.
Step 2
How does the perpendicular from the center of a circle
to a chord divide the chord? Copy and complete the conjecture.
Perpendicular to a Chord Conjecture
C-57
? of
The perpendicular from the center of a circle to a chord is the bisector
the chord.
Let’s continue this investigation to discover a relationship between the length of
congruent chords and their distances from the center of the circle.
Step 3
Guiding Investigation 4
Compare the distances (measured along the perpendicular) from the center to
the chords. Are the results the same if you change the size of the circle and the
length of the chords? State your observations as your next conjecture.
Chord Distance to Center Conjecture
C-58
? from the center of the circle. equidistant
Two congruent chords in a circle are Investigation 4
Perpendicular Bisector
of a Chord
You will need
●
●
●
a compass
a straightedge
patty paper
(optional)
Next, you will discover a property of perpendicular
bisectors of chords.
Step 1
Construct a large circle and mark the center. Construct
two nonparallel chords that are not diameters. Then
construct the perpendicular bisector of each chord and
extend the bisectors until they intersect.
Sharing Ideas (continued)
measure can have very different lengths. Ask
whether having the same measure is enough to
make two arcs congruent, as is the case for line
segments and angles. [Ask] “What’s needed to
ensure that two arcs have the same size and shape?”
[Congruent arcs must be on the same or congruent
circles.] (Some students may think that an arc of a
larger circle will have the same size and shape as an
arc of a smaller circle. To the extent possible, let
other students convince them that the curvature
will be different.)
Step 1 Students can use the same
two congruent chords and the
same drawing for Investigations
2 and 3. [Alert] Students may
need some review in how to
construct perpendiculars. Using
compass constructions to
complete all four investigations
may be too time-consuming, but
this construction and the next
can be completed quickly using
patty-paper constructions. The
perpendicular through the center
of the circle to the chord can be
folded.
[Ask] “Congruent chords are equidistant from the
center. Can we say anything about distance from
the center if one chord is longer than the other?”
[In the same circle, shorter chords are farther from
the center.]
The perpendicular bisectors of
the chords can be constructed by
simply folding the chord in half.
The Perpendicular Bisector of a
Chord Conjecture is the converse
of the Perpendicular to a Chord
Conjecture.
SHARING IDEAS
As usual, for presentations select
groups that have a variety of
statements of the conjectures. As
students present, encourage them
to put the ideas in their own
words, not just those of the
conjectures as presented in the
student book. Help the class
reach consensus on the wording
to record in their notebooks.
[Ask] “How would you define
congruent arcs?” Although the
measure of an arc was first
defined in Chapter 1, some
students may still be wondering
why arcs are measured in degrees
based on their central angle.
Drawing a central angle of 90°
may help some students relate
the central angle to a quarter of
a circle. But one source of difficulty may be a natural tendency
to think that measure means
“size” in some direct way, and the
measure of an arc doesn’t give its
length. In fact, in different
circles, arcs with the same
LESSON 6.2 Chord Properties
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Assessing Progress
You can assess students’ understanding of (and use of the vocabulary for) radius, chord, central
angle, and inscribed angle and
their skill at constructing a circle,
measuring an angle with a
protractor, and comparing
segments with a compass. You
might also see how well they
understand the difference between
drawing and constructing.
What do you notice about the point of intersection? Compare your results with
the results of others near you. Copy and complete the conjecture.
Step 2
? . passes through the center
The perpendicular bisector of a chord of the circle
[ For interactive versions of these investigations, see the Dynamic Geometry Exploration Chord Properties
at www.keymath.com/DG . ]
keymath.com/DG
With the perpendicular bisector of a chord, you can find the center of any circle,
and therefore the vertex of the central angle to any arc. All you have to do is
construct the perpendicular bisectors of nonparallel chords.
Closing the Lesson
Summarize that the major conjectures of this lesson are about
congruent chords of a circle: They
determine congruent central
angles, they intercept congruent
arcs, and they are equidistant
from the center. Another pair of
conjectures about chords forms a
biconditional: A line through the
center of a circle is perpendicular
to a chord if and only if it bisects
the chord. If students are still
shaky about these ideas, you
might want to use one of
Exercises 1–6 as an example.
EXERCISES
You will need
Construction tools
for Exercises 16–19
Solve Exercises 1–10. State which conjectures or definitions you used.
? 165°
1. x ? 84°
2. z ? 70°
3. w 20
x
128
165
70 w
z
definition of measure of an arc
4. AB CD
PO 8 cm
? 8 cm
OQ Chord Arcs Conj.
is a diameter. Find
5. AB
and mB.
mAC
B
C
BUILDING
UNDERSTANDING
A
Chord Central Angles Conj.
6. GIAN is a kite.
Find w, x, and y. w 115°
x 115°
y 65°
G
y
I
C
Q
O
68 O
Chord Distance to Center Conj.
130°
7. AB 6 cm OP 4 cm
8. mAC
CD 8 cm OQ 3 cm
Find w, x, y, and z.
w 110°
BD 6 cm
x 48°
What is the perimeter of
y 82°
OPBDQ? 20 cm
D
w
z 120°
w
? Chord Arcs Conjecture
9. x ?
y
?
z
72
F
C
1–14, 17, 18
O
A
Q
110
y O
x
Portfolio
20
Journal
13, 14, 16
Group
15, 21
Review
22–28
Algebra review 15, 21, 26
MATERIALS
circular objects and patty paper
(Exercise 17)
Exercises 18 and 19 (T), optional
D
Perpendicular to a Chord Conj.
48
x
A
C
definition of arc measure
|
Helping with the Exercises
68°; mB 34° (Because OBC is
5. mAC
isosceles, mB mC, mB mC 68°, and
therefore mB 34°.)
Exercise 7 [Alert] Some students may neglect to add
on the length OQ.
Exercise 9 As needed, [Ask] “What do the measures
of all the arcs add up to?” [360°]
CHAPTER 6 Discovering and Proving Circle Properties
O
y
z
E
B
B
17, 18
320
P
T
z
A
ASSIGNING HOMEWORK
N
A
A
B
The exercises focus on applying
the conjectures about chords.
Encourage students to sketch
pictures on their own papers.
They can then mark and label all
information accordingly.
Performance
assessment
x
O
115
P
D
Essential
C-59
Perpendicular Bisector of a Chord Conjecture
9. x 96°, Chord Arcs Conjecture; y 96°,
Chord Central Angles Conjecture; z 42°,
Isosceles Triangle Conjecture and Triangle Sum
Conjecture.
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66°
CO
, mCI
10. AB
Find x, y, and z.
Page 321
11. Developing Proof What’s wrong
with this picture?
12. Developing Proof What’s wrong
with this picture?
C
z
I
37 cm
A
y O
18 cm
x
B
The length of
the chord is
greater than the
length of the
diameter.
The perpendicular
bisector of the
segment does not
pass through the
center of the circle.
O
13. Draw a circle and two chords of unequal length. Which is closer to the center of the
circle, the longer chord or the shorter chord? Explain.
14. Draw two circles with different radii. In each circle, draw a chord so that the chords
have the same length. Draw the central angle determined by each chord. Which
central angle is larger? Explain.
M (?, 3)
15. Polygon MNOP is a rectangle inscribed in a circle centered at the
origin. Find the coordinates of points M, N, and O.
M(4, 3), N(4, 3), O(4, 3)
16. Construction Construct a triangle. Using the sides of the triangle as
chords, construct a circle passing through all three vertices. Explain.
Why does this seem familiar?
P (4, 3)
O (?, ?)
N (?, ?)
Exercise 10 [Alert] Students may
miss the fact that the two radii in
ABO are congruent, making it
an isosceles triangle.
10. x 66°, y 48°, z 66°;
Corresponding Angles
Conjecture, Isosceles Triangle
Conjecture, Linear Pair
Conjecture
Exercise 12 If students are having
difficulty, [Ask] “What does the
perpendicular bisector have
to go through?” [the center of
the circle]
13. The longer chord is closer
to the center; the longest chord,
which is the diameter, passes
through the center.
17. Construction Trace a circle onto a blank sheet of paper
without using your compass. Locate the center of the circle
using a compass and straightedge. Trace another circle onto
patty paper and find the center by folding.
18. Construction Adventurer Dakota Davis digs up a piece of a
circular ceramic plate. Suppose he believes that some
ancient plates with this particular design have a diameter of
15 cm. He wants to calculate the diameter of the original
plate to see if the piece he found is part of such a plate.
He has only this piece of the circular plate, shown at right,
to make his calculations. Trace the outer edge of the plate
onto a sheet of paper. Help him find the diameter.
19. Construction The satellite photo at right shows only a portion
of a lunar crater. How can cartographers use the photo to
find its center? Trace the crater and locate its center. Using
the scale shown, find its radius. To learn more about satellite
photos, go to www.keymath.com/DG . They can draw two
chords and locate the intersection of their perpendicular
bisectors. The radius is just over 5 km.
14. The central angle of the
smaller circle is larger, because
the chord is closer to the center.
5 cm
0
1
2
3
4
5 km
5 cm
Exercise 16 As needed, remind
students of the meaning of a
triangle inscribed in a circle (or
a circle circumscribed around a
triangle).
16. The center of the circle is
the circumcenter of the triangle.
Possible construction:
Exercise 17 Have available round objects larger than
coins for tracing.
17. possible construction:
Exercises 17–19 If students are having difficulty,
wonder aloud whether there’s a conjecture that ends
with something about the center of a circle.
[Perpendicular Bisector of a Chord]
18. 13.8 cm
O
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20. Developing Proof Complete the flowchart proof shown, which proves that if two
chords of a circle are congruent, then they determine two congruent central
angles.
CD
Given: Circle O with chords AB
A
B
O
Show: AOB COD
C
Flowchart Proof
1
D
AB CD
? Given
2
AOB COD
4
AO CO
All radii of a circle
are congruent
3
COD
5
? ?
? SSS Congruence
Conjecture
?
AOB ? CPCTC
y
BO DO
A (3, 4)
? All radii of a circle are congruent
O
21. Circle O has center (0, 0) and passes through points A(3, 4) and
B(4, 3). Find an equation to show that the perpendicular bisector
passes through the center of the circle. Explain your
of AB
reasoning.
y 17 x; (0, 0) is a point on this line.
x
B (4, –3)
Review
Chapter 5 22. Developing Proof Identify each of these statements as true or false. If the statement is
true, explain why. If it is false, give a counterexample.
a. If the diagonals of a quadrilateral are congruent, but only one is the
perpendicular bisector of the other, then the quadrilateral is a kite. true
b. If the quadrilateral has exactly one line of reflectional symmetry, then the
quadrilateral is a kite. false, isosceles trapezoid
c. If the diagonals of a quadrilateral are congruent and bisect each other, then
it is a square. false, rectangle
Exercise 23 You might use the
Sketchpad demonstration Intersecting Tangents Conjecture to
preview or replace this exercise.
23. Mini-Investigation Use what you learned in the last lesson about the angle formed by a
tangent and a radius to find the missing arc measure or angle measure in each
diagram. Examine these cases to find a relationship between the measure of the
angle formed by two tangents to a circle, P, and the measure of the intercepted arc,
AB. Then copy and complete the conjecture below.
A
140° A
P
40
x
60°
Q
P
y
20
B
160°
Q
B
P
z
A
120 Q
B
Conjecture: The measure of the angle formed by two intersecting tangents
? (Intersecting Tangents Conjecture). 180° minus the measure of the intercepted arc
to a circle is 22a.
C
A
X
B
D
is the perpendicular
Possible explanation: If AB
is
bisector of CD , then every point on AB
equidistant from endpoints C and D. Therefore
322
CHAPTER 6 Discovering and Proving Circle Properties
AD
and BC
BD
. Because CD
is not the
AC
, C is not equidistant
perpendicular bisector of AB
from A and B. Likewise, D is not equidistant from
and BC
are not congruent, and
A and B. So, AC
and BD
are not congruent. Thus ACBD
AD
has exactly two pairs of consecutive congruent
sides, so it is a kite.