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 DefinitionA 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

DefinitionTwo or more coplanar circles with the same center are called concentric circles. DefinitionTwo circle are congruent if they have congruent radii. 5 5

DefinitionA point is inside (in the interior of) a circle if its distance from the center is less than the radius. DefinitionA point is outside (in the exterior of) a circle if its distance from the center is greater than the radius. B A Points A and B are in the interior of Circle B C Point C is in the exterior of Circle B DefinitionA 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 DefinitionA 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 DefinitionA 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 Circumference of a Circle

Radius-Chord Relationships OP is the distance from O to chord AB O B A P DefinitionThe distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the chord

Theorems TheoremIf a radius is perpendicular to a chord, this it bisects the chord. O B A E D

Theorems TheoremIf a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord. O F E G H

Theorems TheoremThe perpendicular bisector of a chord passes through the center of the circle. O D C Q P

Given: Circle Q Prove Q R 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 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