Chapter 9

Kiara, Chelsea, Angus

9.1 Basic Terms of the Circle

Circle• Set of points in a plane at a given distance (radius)

from a given point (center) • Nothing in interior is considered a circle • 2D

Sphere • Set of points in space w/ distance (radius) from a

given point • 3D

Tangent • A line in a plane of a circle or sphere that intersects the

circle/sphere in exactly one point

Chord• A segment whose endpoints lie on a circle or a sphere• A diameter is a chord that contains the center of the circle

– Longest chord of a circle

Secant • A line in a plane of a circle or sphere that intersects a

circle in 2 points • A line that contains a chord

Congruent Circles • Circles that have the same radii but not

necessarily the same center Concentric Circles• Circles that lie in the same plane and have the

same center but not the same radii • Ex: rock thrown in water

Polygons and Circles

• A polygon is inscribed in a circle or the circle is circumscribed about the polygon when each vertex of the polygon lies on the circle

When 2 circles do not touch • Can construct 4 common tangents

– 2 External Tangents – 2 Internal Tangents

• Internal line of center of circle

When 2 circles touch externally • Can construct 3 tangents

– 2 external tangents– 1 internal tangent

When 2 circles intersect each other at 2 points (overlap) • Can construct 2 external tangents

When 2 circles touch internally • Can construct only one tangent

– Intersects both

2 concurrent circles • Can NOT construct any common tangent

– Zero

9.2 TangentsTangent Perpendicular Theorem• If a line is tangent to a circle, then the line is

perpendicular to the radius drawn to the point of tangency

• Angle <ATO = 90

Corollary• Tangents to a circle from a point outside the circle are

congruent ≅• Prove by H-L and CPCTC

C. Tangent Perpendicular Theorem• If a line is perpendicular to the radius at its outer end

point, then the line is tangent to the circle

A

O

E

B

D

Name each of the following:

Two radii: BO, OD A secant : ABTwo chords: AB, BDA diameter : BDA tangent: EDA point of tangency: D

B CA D

14 7

B and D are the centers of the circles.

Find the length of AC

9-3 : Arcs and Central Angles

Central Angle

● An angle with its vertex at the center of the circle

Arc

● An unbroken part of the circle

Minor Arc

● Points in the interior of central angle

● Named by endpoints

Major Arc

● 2 points & remaining points of center circle

● Named by 3 points

● If 2 points are the endpoints of a diameter then 2 arcs are called semicircles

Measure of a Semicircle

● M = 180

Measure of a minor angle = The measure of its central angle

Measure of a Major Arc

● 360 – measure of minor arc

Semicircles

Adjacent Arcs● Arcs that have exactly one point in common

Arc Addition Postulate● The measure of the arc formed by 2 adjacent arcs is the sum of the

measures of these 2 arcs

Congruent ArcsArcs, in the same circle, or in congruent circles that have equal

measures

Th. 9.3In the same circle or in congruent circles, two minor arcs are congruent if and

only if their central angles are ≅

Problems 9-3

150

Find measure of central angle 1

1

At 11 o'c lock the hands of a clock form anangle of _ __ _ ?

The hands of a c lock form a 120 angle at __ _o'c lock and at __ _ o'c lock.

mCB 60 70 ? ? ?

M<1 ? ? 56 ? ?

M<2 ? ? ? 25 x

9.4 Arcs and Chords:

•Theorem 9.4

–In the Same Circle or in congruent circles congruent arcs have congruent chords–In the Same Circle or in congruent circles, congruent chords have congruent arcs

•Theorem 9.5 –A diameter that is perpendicular to a chord bisects the chord & its arc

•Theorem 9.6 –In the same circle or in congruent circles, chords equally distant from the center are congruent–In the same circle or in congruent circles, congruent chords are equally distant from the center

•

•

Practice Problems 9-4

PQ = 24; OM = ____?

13

N

M

BC = 18; OM = 12ON=10 ; DE = ___?

O

E

D

B

C

9.5 Circles – Inscribed Angles

Inscribed angle :• Angle with vertex on circle and sides which are chords of the circle.

Intercepted Arc • The points on the circle in the interior of an angle of the circle.

Theorem 9-7• The measure of the inscribed angle is half as much as the intercepted arc

Corollary 1: • If two inscribed angles intercept the same arc or congruent arcs, then the angles are

congruent

Example … Find X : • Since X is an inscribed arc, then the measure of the angle is ½ the

measure of the arc• So, x = 54

Try it! Find X:

Corollary 2: • An angle inscribed in a semicircle is a right angle

Corollary 3: • The opposite angles of an inscribed quadrilateral are supplementary

Corollary 4: • The angle formed by a tangent and a chord equal 1/2 the measure of the intercepted arc

9.6 Other Angles Angle Formed by 2 chords: • Is the average of the 2 arcs • M<X = (mBD + mCA) ÷ 2

Angle Formed by 2 secants• Is the difference of the 2 arcs • m<X = (mAB – mCD) ÷ 2

Angle formed by 2 tangents

• Is (major arc – minor arc) ÷ 2 • M<X = (mACB – mAB) ÷ 2

A

Angle formed by a secant & a tangent

• M<X = (mAD – mAC) ÷ 2

Example Problems… Find X:

9.7 Circles & Lengths of Segments Theorem 9.11 • R x Q = T x S

R

S

T

Q

Theorem 9.12• DE x ME = FE x NE

Theorem 9.13 • R x S = T2

• NR x KR = SR2

RS

T

• Try it! Find X: