Transcript Lesson 6.2 Lecture

6.2- Chord Properties
In lesson 6.1 we discovered some properties of a tangent.
 a line that intersects the circle only once
In this lesson we will investigate properties of a chord.
 a line segment whose endpoints lie on the circle.
.
. .
First we will define two types of angles in a circle.
JRLeon
Geometry Chapter 6.1 – 6.2
HGHS
6.2- Chord Properties
Two types of angles in a circle.
Defining Angles in a Circle
CENTRAL ANGLE A central angle has its vertex at the center of the circle.
AOB, DOA, and
DOB are central
angles of circle O.
 PQR, PQS, RST, QST,
and  QSR are not central
angles of circle P.
INSCRIBED ANGLE An inscribed angle has its vertex on the circle and its
sides are chords.
 ABC,  BCD,
and  CDE are
inscribed angles.
JRLeon
Geometry Chapter 6.1 – 6.2
 PQR,  STU, and
 VWX are not
inscribed angles.
HGHS
6.2- Chord Properties
Chords and Their Central Angles
 We will discover some properties of chords and central angles.
 We will also see a relationship between chords and arcs.
1. Construct a large circle. Label the center O.
2. Construct two congruent chords in your circle.
3. Label the chords AB and CD, then construct radii OA, OB, OC,
and OD.
4. With a protractor, measure  BOA and  COD. How do they
compare?
How can you fold the circle construction to check the conjecture?
JRLeon
Geometry Chapter 6.1 – 6.2
HGHS
6.2- Chord Properties
Chords and Their Central Angles
 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.
JRLeon
Geometry Chapter 6.1 – 6.2
HGHS
6.2- Chord Properties
Chords and the Center of the Circle
 Discover relationships about a chord and the center of its circle.
1. Construct a large circle and mark the center.
2. Construct two nonparallel congruent chords.
3. Construct the perpendiculars from the center to each chord.
How does the perpendicular from the center of a circle
to a chord divide the chord?
 Discover a relationship between the length of congruent chords and their distances
from the center of the circle.
 Compare the distances (along the perpendicular) from the center to the chords.
JRLeon
Geometry Chapter 6.1 – 6.2
HGHS
6.2- Chord Properties
Perpendicular Bisector of a Chord
 Discover a property of perpendicular bisectors of chords.
1. Construct a large circle and mark the center.
2. Construct two nonparallel chords that are not diameters.
3. Construct the perpendicular bisector of each chord and
extend the bisectors until they intersect.
What do you notice about the point of intersection?
 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.
JRLeon
Geometry Chapter 6.1 – 6.2
HGHS
6.2- Chord Properties
Lesson 6.2: Pages 320 - 322, Problems 1 thru 12,
Homework: 18, 23
JRLeon
Geometry Chapter 6.1 – 6.2
HGHS