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Geometry

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Geometry Chapter 9 Review Secant Tangent Theorem Corollary Theorem In the same circle or in congruent circles, two minor arcs are congruent if and only if their ... – PowerPoint PPT presentation

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Title: Geometry


1
Geometry
  • Chapter 9 Review

2
Secant
  • A line that contains a chord of a circle.

. P
SECANT
3
Tangent
  • A line in the plane of a circle that
  • intersects the circle in exactly one point.

. P
Tangent
.
Point of tangency
4
Theorem
If a line is tangent to a circle, then the line
is perpendicular to the radius drawn to the point
of tangency.
?
5
Corollary
Tangents to a circle from a point are congruent.
?
?
6
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.
7
Theorem
  • In the same circle or in congruent circles
  • congruent arcs have congruent chords
  • congruent chords have congruent arcs

?
O
8
Theorem
A diameter that is perpendicular to a chord
bisects the chord and its arc.
?
O
9
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
10
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
11
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
12
Corollary
  • If two inscribed angles intercept the same arc,
    then the angles are congruent.


sf giants

x y
x
y
giants
sf
13
Corollary
  • If an inscribed angle intercepts a semicircle,
    then it is a right angle.

Why?
180o
diameter
diameter
90o
14
Corollary
  • If a quadrilateral is inscribed in a circle, then
    opposite angles are supplementary.

70o
85o
supplementary
supplementary
95o
110o
15
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) _____
16
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)
17
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
18
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
19
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
20
HW Start After You Finish the Ch.8 Quiz
  • Chapter 9 W.S.
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