GeometryGeometry

Chapter 9 ReviewChapter 9 Review

Secant

•A line that contains a chord of a circle.

SECANT .P

Tangent

•A line in the plane of a circle that intersects the circle in exactly one point.

Tangent .P.Point of tangency

Theorem

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

●

Corollary

Tangents to a circle from a point are congruent.

●●

TheoremTheorem

• In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

If m 1 = m 2, then JK = LM.

M

J

21

K

7 7 ~

If JK = LM, then m 1 = m 2.

7 7~

k

m

12

TheoremTheorem

In the same circle or in congruent circles:• congruent arcs have congruent chords• congruent chords have congruent arcs

●O

TheoremTheorem

A diameter that is perpendicular to a chord bisects the chord and its arc.

●O

TheoremTheorem

In the same circle or in congruent circles:• chords equally distant from the center (or centers) are congruent• congruent chords are equally distant from the center (or centers)

●O

Inscribed Angle TheoremInscribed Angle Theorem

• The measure of the inscribed angle is half the measure of its central angle (and therefore half the intercepted arc).

30o

60o

60o

A Very Similar Theorem A Very Similar Theorem

• The measure of the angle created by a chord and a tangent equals half the intercepted arc.

70o

tangent

chord

35o

CorollaryCorollary

• If two inscribed angles intercept the same arc, then the angles are congruent.

xyx = y~

sf

giants

sf = giants~

CorollaryCorollary

• If an inscribed angle intercepts a semicircle, then it is a right angle.

diameter

Why?

diameter

180o

90o

CorollaryCorollary

• If a quadrilateral is inscribed in a circle, then opposite angles are supplementary.

70o

110o95o

85o

supplementary

supplementary

If mAD = 50 and mBC = 60

Interior AngleInterior Angle TheoremTheorem

The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

A B

C

D

1

m<1 = ½( mAD + mBC )

50˚

60˚

m<1 = ½(50 + 60) = _____55

Exterior Angle TheoremExterior Angle Theorem

The measure of the angles formed by intersecting secants and tangents outside a circle is equal to half the difference of the measures of the intercepted arcs.

1y˚x˚

m<1 = ½(x – y)

2y˚x˚

m<2 = ½(x – y)

3y˚x˚

m<3 = ½(x – y)

TheoremTheorem

• When two chords intersect inside a When two chords intersect inside a circle, circle, the product of the segmentsthe product of the segments of of one chord one chord equalsequals the the product of the product of the segmentssegments of the other chord. of the other chord.

X

P

S Q

R8

34

6x

8 x 3 = 6 x 4

TheoremTheorem

• When two secant segments are When two secant segments are drawn to a circle from an external drawn to a circle from an external point, point, the product of one secant the product of one secant segment and its external segmentsegment and its external segment equalsequals the product of the other the product of the other secant segment and its external secant segment and its external segmentsegment..

G

H

FD E

FD x FE = FH x FG

External x Whole Thing = External x Whole Thing

TheoremTheorem

• When a secant segment and a When a secant segment and a tangent segment are drawn to a tangent segment are drawn to a circle from an external point, the circle from an external point, the product of the secant segment and product of the secant segment and its external segmentits external segment is is equalequal to the to the square of the tangent segmentsquare of the tangent segment..

BP

C

A

PB x PC = (PA)2

External x Whole Thing = (Tangent)2

HW: Start After You Finish the Ch.8 QuizHW: Start After You Finish the Ch.8 Quiz

• Chapter 9 W.S.Chapter 9 W.S.