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.

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

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

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

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.

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

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

TheoremsTheorem If a radius is perpendicular to a chord, this

it bisects the chord.

O B

AE

D

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

TheoremsTheorem The perpendicular bisector of a chord

passes through the center of the circle.

O D

CQ

P

Given: Circle Q

Prove

PR STPS PT

QR

T

S

P

Example 1

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

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

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