Transcript Chapter 9 Review

Geometry
Chapter 9 Review
Secant
•A line that contains a chord of a circle.
.
P
Tangent
•A line in the plane of a circle that
intersects the circle in exactly one point.
.
Point of tangency
.
P
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.
J
M
22
~ LM.
If m 1 = m 2, then JK =
7
7
~ LM, then m 1 = m 2.
If JK =
11
m
K
k
7
7
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.
35o
70o
Corollary
• If two inscribed angles intercept the same
arc, then the angles are congruent.
~ giants
sf =
x~
=y
y
x
giants
sf
Corollary
• If an inscribed angle intercepts a
semicircle, then it is a right angle.
Why?
180o
90o
Corollary
• If a quadrilateral is inscribed in a circle,
then opposite angles are supplementary.
70o
85o
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
1
50˚
D
B
60˚
m<1 = ½( mAD + mBC )
C
If mAD = 50 and mBC = 60
55
m<1 = 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.
x˚
y˚
m<1 = ½(x – y)
1
x˚
x˚
y˚
y˚
2
m<3 = ½(x – y)
m<2 = ½(x – y)
3
Theorem
• When two chords intersect inside a circle,
the product of the segments of one chord
equals the product of the segments of the
Q
S
other chord.
4
8
P
8x3=6x4
3
xX
6
R
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
E
D
F
G
H
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.
A
PB x PC = (PA)2
P
C
External x Whole Thing = (Tangent)2
B
HW: Start After You Finish the Ch.8 Quiz
• Chapter 9 W.S.