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10.1 The Circle After studying this section, you will be able to identify the characteristics of...
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![Page 1: 10.1 The Circle After studying this section, you will be able to identify the characteristics of circles, recognize chords, diameters, and special relationships.](https://reader035.fdocuments.us/reader035/viewer/2022080917/56649eb65503460f94bbf5a6/html5/thumbnails/1.jpg)
10.1 The Circle
After studying this section, you will be able to identify the characteristics of circles, recognize chords, diameters,
and special relationships between radii and chords.
![Page 2: 10.1 The Circle After studying this section, you will be able to identify the characteristics of circles, recognize chords, diameters, and special relationships.](https://reader035.fdocuments.us/reader035/viewer/2022080917/56649eb65503460f94bbf5a6/html5/thumbnails/2.jpg)
Basic Properties and Definitions
Definition A circle is the set of all points in a plane that are a given distance from a given point in the plane. The given point is the center of the circle, and the given distance is the radius. A segment that joins the center to a point on the circle is also called a radius.
(The plural of radius is radii).
circle
center radius
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Definition Two or more coplanar circles with the same center are called concentric circles.
Definition Two circle are congruent if they have congruent radii.
5 5
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Definition A point is inside (in the interior of) a circle if its distance from the center is less than the radius.
Definition A point is outside (in the exterior of) a circle if its distance from the center is greater than the radius.
BA
Points A and B are in the interior of Circle B
C
Point C is in the exterior of Circle B
Definition A point is on a circle if its distance from the
center is equal to the radius.
D
Point D is on Circle B
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Chords and Diameters
Definition A chord of a circle is a segment joining any two points on the circle.
diameter
chord
Points on a circle can be connected by segments called chords
Definition A diameter of a circle is a chord that passes through the center of the circle.
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Circumference and Area of a Circle
Area of a Circle
where r is the radius of the circle and
2A rCircumference of a Circle
2 or C r C d
3.14
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Radius-Chord RelationshipsOP is the distance from O to chord AB
O B
AP
Definition The distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the chord
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TheoremsTheorem If a radius is perpendicular to a chord, this
it bisects the chord.
O B
AE
D
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TheoremsTheorem If a radius of a circle bisects a chord that is
not a diameter, then it is perpendicular to that chord.
O F
EG
H
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TheoremsTheorem The perpendicular bisector of a chord
passes through the center of the circle.
O D
CQ
P
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Given: Circle Q
Prove
PR STPS PT
QR
T
S
P
Example 1
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The radius of circle O is 13 mm. The length of chord PQ is 10 mm. Find the distance from chord PQ to the center, O
Example 2
O Q
P
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Example 3
Given: Triangle ABC is isosceles Similar circles P and Q
Prove: Circle Q Circle P
BC PQ
AB AC
P Q
B
A
C
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Summary
Explain how to determine whether a point is on a circle, in the interior of a circle, or in the exterior of a circle.
Homework: worksheet