Chapter 10Chapter 10CIRCLESCIRCLES

Ms. Watson

Geometry

Banneker Academic High School

GOAL #1GOAL #1Identify segments and lines Identify segments and lines

related to circles.related to circles.

Who’s N’ The Circle Fam?Who’s N’ The Circle Fam?VOCABULARY DEFINITION

CIRCLE -the set of all points in a plane that are equidistant from a given point called the center of a circle

RADIUS -the distance from the center to a point on the circle

DIAMETER -the distance across the circle, through its center (the diameter is 2x the radius)

radius

diameter

Who’s N’ The Circle Fam?Who’s N’ The Circle Fam?VOCABULARY DEFINITION

CHORD -a segment whose endpoints are points on the circle and are chords

DIAMETER -a chord that passes through the center of the circle

SECANT -a line that intersects a circle in two points

TANGENT -a line in the plane of a circle that intersects the circle in exactly one point

PS PR

secant

tangentP

Q

R

S

Common Tangents

A line or line segment that is tangent to

two circles in the same plane is called

a common tangent.

Who’s N’ The Circle Fam?Who’s N’ The Circle Fam?Common External Tangent

Common Internal Tangent

2 Types

GOAL #2GOAL #2Use properties of tangent Use properties of tangent

to a circle.to a circle.

Theorem 10.1

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

What do we know about right triangles???

How can we use what we know to solve the length of sides of a triangle???

Theorem 10.3

If two segments from the same exterior point are tangent to a circle, then they are congruent (equal).

10.2

Arcs & Chords

GOAL #1GOAL #1Use properties of arcs of Use properties of arcs of

circles.circles.

Who’s N’ The Circle Fam?Who’s N’ The Circle Fam?VOCABULARY DEFINITION

CENTRAL ANGLE -an angle whose vertex is the center of a circle

MINOR ARC -part of a circle that measures less than 180°

MAJOR ARC -part of a circle that measures between 180° and 360°

SEMICIRCLE -an arc whose endpoints are the end points of a diameter of the circle

Minor Arc

Major Arc

A

B

C

K

Name That Arc!Name That Arc!Arcs are named by their endpoints.Arcs are named by their endpoints.

Minor Arc

Major Arc

A

B

C

K

MINOR ARCS MAJOR ARCS & SEMICIRCLES

-named by their endpoints-the minor arc associated with AKC is AC

-named by their endpoints and by a point on the arc-the major arc associated with AKC is ABC

Measure That Arc!Measure That Arc!MINOR ARCS MAJOR ARCS & SEMICIRCLES

-the same as the measure of its central angle

-

-the difference between 360° and the measure of its associated minor arc-

Minor Arc

Major Arc

A

B

C

K

60 °

60 °

300 °

60 ° 360° - 60 ° = 300 °

Arc Addition PostulateArc Addition PostulateDiscoveryDiscovery

Step 1: Draw a circle.

Step 2: Place three points on the circle named A, B, and C.

Discover that…

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

mABC = mAB + mBC

Arc for ThoughtArc for Thought

• Two arcs of the same circle or of congruent circles are congruent arcs if they have the same measure. So, two minor arcs of the same circle or of congruent circles are congruent if their central angles are congruent.

Which minor arcs are congruent?

Why are they congruent?

GOAL #2GOAL #2Use properties of chords of Use properties of chords of

circles.circles.

Theorem 10.4Theorem 10.4

B

A

C

Theorem 10.5Theorem 10.5

Theorem 10.6Theorem 10.6

If one chord is a perpendicular bisector of another chord, the first chord is a diameter.

M

K

L

J

Theorem 10.7Theorem 10.7

Hands on ActivityHands on Activity

10.3

Inscribed Angles

GOAL #1GOAL #1Use inscribed angles to Use inscribed angles to

solve problems.solve problems.

Inscribed Angle

-an angle whose vertex is on a circle and whose sides contain chords of the circle

Intercepted Arc

-an arc that lies in the interior or an inscribed angle and has endpoints on the angle

Who’s N’ The Circle Fam?Who’s N’ The Circle Fam?

Theorem 10.8

Measure of an Inscribed Angle

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

m ABC = mAC2

1

B

A

C

Theorem 10.9

If two inscribed angles of a circle intercept the same are, then the angles are congruent. A

BC

D

C D

GOAL #2GOAL #2Use properties of inscribed Use properties of inscribed

polugons.polugons.

If all of the vertices of a polygon lie on a circle, the polygon is INSCRIBED in the circle and the circle is CIRCUMSCRIBED about the polygon.

Who’s N’ The Circle Fam?Who’s N’ The Circle Fam?

Theorem 10.10

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Also the angle opposite the diameter is a right angle.

B is a right angle if and only if is a diameter of the circle.

AC

A

B C

Theorem 10.11

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

E F

G

D

D, E, F, and G lie on the circle if and only if

m D + m F = 180Andm E + m G = 180