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Geometry

- Chapter 9 Review

Secant

- A line that contains a chord of a circle.

. P

SECANT

Tangent

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

. P

Tangent

.

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.

?

?

Theorem

- In the same circle or in congruent circles, two

minor arcs are congruent if and only if their

central angles are congruent.

1

k

2

7

7

If m 1 m 2, then JK LM.

m

7

7

If JK LM, then m 1 m 2.

Theorem

- In the same circle or in congruent circles
- congruent arcs have congruent chords
- congruent chords have congruent arcs

?

O

Theorem

A diameter that is perpendicular to a chord

bisects the chord and its arc.

?

O

Theorem

- 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 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

- The measure of the angle created by a chord and a

tangent equals half the intercepted arc.

tangent

35o

chord

70o

Corollary

- If two inscribed angles intercept the same arc,

then the angles are congruent.

sf giants

x y

x

y

giants

sf

Corollary

- If an inscribed angle intercepts a semicircle,

then it is a right angle.

Why?

180o

diameter

diameter

90o

Corollary

- If a quadrilateral is inscribed in a circle, then

opposite angles are supplementary.

70o

85o

supplementary

supplementary

95o

110o

Interior Angle Theorem

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

mlt1 ½( mAD mBC )

60

1

C

50

If mAD 50 and mBC 60

D

55

mlt1 ½(50 60) _____

Exterior 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.

1

y

x

y

x

3

2

y

x

mlt1 ½(x y)

mlt3 ½(x y)

mlt2 ½(x y)

Theorem

- When two chords intersect inside a circle, the

product of the segments of one chord equals the

product of the segments of the other chord.

3

4

x

8

6

8 x 3 6 x 4

Theorem

- When two secant segments are drawn to a circle

from an external point, the product of one secant

segment and its external segment equals the

product of the other secant segment and its

external segment.

FD x FE FH x FG

External x Whole Thing External x Whole Thing

Theorem

- When a secant segment and a tangent segment are

drawn to a circle from an external point, the

product of the secant segment and its external

segment is equal to the square of the tangent

segment.

PB x PC (PA)2

External x Whole Thing (Tangent)2

HW Start After You Finish the Ch.8 Quiz

- Chapter 9 W.S.