Transcript Circles - MrsMcFadin

Circles
Chapter 9
Tangent Lines (9-1)
• A tangent to a circle is a line in the
plane of the circle that intersects the
circle in exactly one point.
• The point where a circle and a
tangent intersect is the point of
tangency.
B
A
P
Tangent Lines (9-2)
• Theorem: If a line is tangent to a
circle, then the line is perpendicular
to the radius drawn to the point of
tangency.
B
A
P
Tangent Lines (9-2)
• Converse: If a line in the plane of a
circle is perpendicular to a radius at
its endpoint on the circle, then the
line is tangent to the circle.
B
A
A
P
8
4
B
7
D
Tangent Lines (9-2)
• Corollary: The two segments tangent
to a circle from a point outside the
circle are congruent.
• AB = BC
A
Q
B
C
Tangent Lines (9-1)
B
• “Inscribed in
the circle ”
C
A
• “Circumscribed about the circle”
S
T
U
R
V
Tangent Lines (9-1)
• Circle G is inscribed in quadrilateral
CDEF. Find the perimeter of CDEF.
D
8 ft
E
6 ft
11 ft
G
7 ft
C
F
Arcs and Central
Angles 9-3
• Central Angle (of a circle)- angle with
its vertex at the center of the circle
• Arc- unbroken part of a circle
• Minor Arc (less than 180 degrees)
• Name them using the endpoints
• Major Arc (more than 180 degrees)
• Name them using three points
• Semicircles- two arcs formed by the
endpoints of a diameter
Arcs and Central
Angles 9-3
• Measure of a minor arc= measure of
its central angle
• Measure of a major arc= 360 degrees –
measure of its minor arc
• Adjacent arcs- arcs with exactly one
point in common (crust of adjacent
pizza slices)
Arc Addition Postulate
• The measure of the arc formed by two
adjacent arcs is the sum of the
measures of these two arcs
• Similar to the Angle Addition
Postulate
Congruent Arcs
• Arcs in the same circle or congruent
circles
• Have equal measures
• Arcs in two circles of different sizes
cannot be congruent, even if they
have the same measure (to be
congruent, they must be the same
shape and size)
Theorem 9-3
• In the same circle or in congruent
circles, two minor arcs are congruent
if and only if their central angles are
congruent
• STOP
Chords and Arcs (9-4)
• A chord is a segment whose endpoints
are on a circle.
• Each chord cuts off a minor
arc and a major arc
B
C
A
Chords and Arcs (9-4)
• Theorem: Within a circle or congruent
circles
1. Congruent arcs have congruent
chords.
2. Congruent chords have congruent
arcs.
Chords and Arcs (9-4)
• Within a circle or in congruent
circles…
B
C
A
D
Theorem 9-5
• A diameter that is perpendicular to a
chord bisects the chord and its arc.
Converse…
• In a circle, a diameter that bisects a
chord (that is not the diameter) is
perpendicular to the chord.
• Example
86 degrees
Chords and Arcs (9-3)
• Theorem: Within a circle or congruent
circles
1. Chords equidistant from the center
are congruent.
2. Congruent chords are equidistant
from the center.
Chords and Arcs (9-4)
• Find x.
Chords and Arcs (9-4)
• Find HL and QJ.
H
11
26
Q
J
L
•
HL= 22, QJ = 4 √3
Chords and Arcs (9-4)
• In a circle, the perpendicular bisector
of a chord contains the center of the
circle.
• STOP
Inscribed Angles (9-5)
• Inscribed angle – vertex on the circle,
sides of angle are chords of circle
• Intercepted arc – arc formed when the
sides of the inscribed angle cross the
circle
A
C
B
Inscribed Angles (9-5)
• Theorem: The measure of an inscribed
angle is half the measure of its
intercepted arc.
A
C
B
Inscribed Angles (9-5)
D
80
• Find x and y.
70
A
y
• x= ½ *(80+70)
• x= 75°
• m arc BC= 360- (80+70+90)
= 120°
• y= ½ * (70+120)= 95°
C
90
x
B
Inscribed Angles (9-5)
• Corollary- Two inscribed angles that
intercept the same arc are congruent.
Inscribed Angles (9-5)
Corollary- An angle inscribed in a
semicircle is a right angle.
• GeoGebra example
Inscribed Angles (9-5)
Corollary- The opposite angles of a
quadrilateral inscribed in a circle are
supplementary.
Inscribed Angles (9-5)
• Find the value of a and b.
• a= 90°
• 2 *32° = 64°
• b= 180- 64= 116°
32
E
b
a
• 9-5 handout
• Problems 1-9 all
Inscribed Angles (9-5)
• The measure of an angle formed by a
tangent and a chord is half the
measure of the intercepted arc.
I
F
G
H
• 9-5 handout
• Problems 10-21 all
Angle Measure and
Segment Lengths (9-5)
• A secant is a line that intersects a
circle at two points.
A
B
Angle Measure and
Segment Lengths (9-6)
• The measure of an angle formed by
two lines that intersect
1. inside a circle is half the sum of the
measures of the intercepted arcs.
2. outside a circle is half the difference
of the measure of the intercepted
arcs.
The measure of an angle formed by two
lines that intersect
inside a circle is half the sum of the
measure of the intercepted arcs
• Find the measure of <1
• m<1= ½ (45 + 75)
• = 60
The measure of an angle formed by
two lines that intersect
outside a circle is half the
difference of the measure of the
intercepted arcs.
• m <B = ½ (m AFD - m AC)
• 65 = ½ (m AFD – 70)
• 200 = m AFD
Angle Measure and
Segment Lengths (9-6)
• Find the value of x.
• x = ½ (268 – 92)
• x = 88
268
92
x
Angle Measure and
Segment Lengths (9-6)
•
•
•
•
Find the value of x.
94 = ½ (x + 122)
188 = x + 122
x = 66
x
94
112
Angle Measure and
Segment Lengths (9-6)
Where the
angle vertex is
Center of circle
Angle measure
m(arc)
On circle
½ m(arc)
Inside circle
½ sum of m(arcs)
Outside circle
½ difference of m(arcs)
Angle Measure and
Segment Lengths (9-7)
ab=cd
a
c
x
b
w
d
y
z
(w + x)w = (y + z)y
t
y
z
(y + z)y = t2
Angle Measure and
Segment Lengths (9-7)
• Find the value of x.
7
5
3
x
Angle Measure and
Segment Lengths (9-7)
• Find the value of y.
15
8
y
Angle Measure and
Segment Lengths (9-7)
90
O
x
a
8
30
b
34
8
12
Angle Measure and
Segment Lengths (9-7)
98
8
7
x
M
a
21