Download Circles

Bell work
What is a circle?
Bell work Answer

A circle is a set of all points in a plane that
are equidistant from a given point, called
the center of the circle
Unit 3 : Circles:
10.1 Line & Segment Relationships
to Circles (Tangents to Circles)
Objectives: Students will:
1. Identify Segments and lines related
to circles.
2. Use Properties of a tangent to a
circle
Lines and Segments related to
circles
Exterior Point
CHORD
CENTER
•
•
TANGENT LINE
•
DIAMETER
ALSO A CHORD
Interior Point
SECANT
RADIUS
Point of tangency
Lines and Segments related to
circles
Center of the circle
CENTER
•P
Circle P
Lines and Segments related to
circles
Diameter – from one point on the
circle passing through the center
(2 times the radius)
CENTER
•
DIAMETER
ALSO A CHORD
Lines and Segments related to
circles
Radius – Segment from the center of
the circle to a point on the circle
(1/2 the diameter)
CENTER
•
RADIUS (I)
= 1/2 the Diameter
Lines and Segments related to
circles
Chord – a segment from one point on
the circle to another point on the circle
CHORD
•
DIAMETER
ALSO A CHORD
Lines and Segments related to
circles
Secant – a line passing through two
points on the circle
•
SECANT
Lines and Segments related to
circles
Tangent – is a line that intersects the
circle at exactly one point
TANGENT LINE
•
•
Point of Tangency
Label Circle Parts
1.
2.
3.
4.
Semicircles
Center
Diameter
Radius
5. Exterior
6. Interior
7. Diameter
8. Chord
9. Tangent
10. Secant
11. Minor Arc
12. Major Arc
Point of Tangency Theorem
If a line is tangent to a circle, then it is
perpendicular ( _|_ ) to the radius
drawn to the point of tangency.
Q
Tangent line
P
k
If line k is tangent
to circle Q at point P,
Then line k is _|_ to
Segment QP.
Example 1
Find the distance from Q to R, given
that line m is tangent to the circle Q at
Point P, PR = 4 cm and radius is 3 cm.
Q
•
3 cm
P
4 cm
•R
m
Example 1 answer
Use the Pythagorean Theorem
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
√25 = √c²
c =5
Example 2
Given that the radius (r) = 9 in,
PR = 12, and QR = 16 in. Is the line m
tangent to the circle?
Q
9 in
•
P
12 in
16 in
•R
m
Example 2 answer
No, it is not tangent.
Use the Pythagorean Theorem
a² + b² = c²
9² + 12² = 16²
81 + 144 = 256
225 = 256
Since they are not = then the triangle is
not a right triangle and thus the radius is
not perpendicular to the line m, therefore the
line is not tangent to the circle.
Intersections of Circles
No Points of Intersection
CONCENTRIC CIRCLES –
Coplanar circles that share
a common center point
•
Intersections of Circles
One Point of Intersection
The Circles are
tangent
to each other
at the point
•
Internal
Tangent
External
Tangent
Common Tangents
•
Intersections of Circles
Two Points of Intersection
•
•
Tangents drawn from a point not on
the circle theorem
If two segments from the same
exterior point are tangent to a circle,
then they are congruent.
R
•
•
•P
•
T
S
__
__
If SR and ST are
tangent to circle P,
__ __
SR  ST
Example 3
Segment SR and Segment ST are
tangent to circle P at Points R and T.
Find the value of x.
2x + 4
R
•
•
•P
•
T
3x – 9
S
Example 3 Answer
__
__
Since SR and ST are tangent to the
circle, then the segments are , so
2x + 4 = 3x – 9
-2x
-2x
4=x–9
+9
+9
13 = x
Unit 3 : Circles:
10.2 Arcs and Chords
Objectives: Students will:
1. Use properties of arcs and chords to
solve problems related to circles.
Bell work
Find the value of radius, x, if the diameter
of a circle is 25 ft.
25 ft
x
Arcs of Circles
CENTRAL ANGLE – An angle with its
vertex at the center of the circle
Central Angle
•
CENTER P
P
•
A
60º
•
B
Arcs of Circles
Minor Arc AB and Major Arc ACB
Central Angle
•
CENTER P
P
•
MAJOR ARC
ACB
C
•
A
MINOR ARC
AB
60º
•
B
Arcs of Circles
The measure of the Minor Arc AB = the measure of the Central
Angle
The measure of the Major Arc ACB = 360º - the measure of the
Central Angle
Central Angle
•
CENTER P
P
The measure of the
MAJOR ARC =
360 – the measure of C
the MINOR ARC
ACB = 360º - 60º
= 300º
•
•
300º
A
60º
•
B
Measure of the
MINOR ARC =
the measure of the
Central Angle
AB = 60º
Label Circle Parts
1.
2.
3.
4.
Semicircles
Center
Diameter
Radius
5. Exterior
6. Interior
7. Diameter
8. Chord
9. Tangent
10. Secant
11. Minor Arc
12. Major Arc
Arcs of Circles
Semicircle – an arc whose endpoints
are also the endpoints of the diameter
of the circle; Semicircle = 180º
180º
•
Semicircle
Arc Addition Postulate
The measure of an arc formed by two
adjacent arcs is the sum of the
measures of the two arcs
A
•
170º + 8 0º = 2 5 0º
ARC ABC = 250º
•
170º
•
AB + BC = ABC
80º
•B
C
Example 1
Find m XYZ and XZ
X
•
•
75º
P• 110°
•
Z
Y
Congruent Chords Theorem
In the same circle or in congruent circles
two minor arcs are congruent iff their
corresponding chords are congruent
Congruent Arcs and Chords
Theorem
Example 1: Given that Chords DE is
congruent to Chord FG. Find the value
Arc DE = 100º
of x.
D
E
F
G
Arc FG = (3x +4)º
Congruent Arcs and Chords
Theorem
Example 2: Given that Arc DE is
congruent to Arc FG. Find the value
of x.
E
D
Chord DE = 25 in
Chord FG = (3x + 4) in
F
G
Diameter through a chord
If a diameter of a circle is perpendicular to a chord,
then the diameter bisects the chords and its arcs.
Chord
Diameter
Congruent Arcs
P
•
Congruent
Segments
Diameter through a chord
If one chord is the perpendicular bisector of another
chord then the first chord is the diameter
Chord 2
Chord 1: _|_ bisector
of Chord 2, Chord 1 =
the diameter
P
•
Diameter
Distance Congruent Chords are
from the Center
In the same circle or in congruent circles, two
chords are congruent iff they are equidistant from
the center. (Equidistant means same perpendicular
distance)
Q
T
Chord TS  Chord QR
__
__
iff PU  VU
V
•
U
P
S
R
Center P
Example
Find the value of Chord QR, if TS = 20
inches and PV = PU = 8 inches
Q
T
V
8 in
•
U
P
8 in
S
R
Center P
Unit 3 : Circles:
10.3 Arcs and Chords
Objectives: Students will:
1. Use inscribed angles and properties
of inscribed angles to solve
problems related to circles
Bell work 1
Find the measure of Arc ABC, if Arc AB = 3x,
Arc BC = (x + 80º), and
__
__
AB  BC
A
C
AB = 3xº
BC = ( x + 80º )
B
Bell work 2
You are standing at point X. Point X is 10
feet from the center of the circular water
tank and 8 feet from point Y. Segment XY is
tangent to the circle P at point Y. What is the
radius, r, of the circular water tank?
Y
r
P
8 ft
10 ft
•
•
X
Words for Circles
1.
2.
3.
4.
Inscribed Angle
Intercepted Arc
Inscribed Polygons
Circumscribed
Circles
Are there any words/terms that you are
unsure of?
Inscribed Angles
Inscribed angle – is an angle whose
vertex is on the circle and whose sides
contain chords of the circle.
A
INSCRIBED ANGLE
INTERCEPTED ARC,
AB
B
Vertex on the circle
Intercepted Arc
Intercepted Arc – is the arc that lies in
the interior of the inscribed angle and
has endpoints on the angle.
A
INSCRIBED ANGLE
INTERCEPTED ARC,
AB
B
Vertex on the circle
Inscribed Angle Theorem.
The measure of an inscribed angle is equal
half of the measure of its intercept arc.
Central Angle
•
CENTER P
B
= ½ m AC
P
•
•
Inscribed angle
A
m ∕_ ABC
C
Example 1
The measure of the inscribed angle ABC = ½ the
measure of the intercepted AC.
Central Angle
•
B
m ∕_ ABC
= ½ mAC = 30º
•
30º
•
A
60º
•C
Measure of the
INTERCEPTED ARC =
the measure of the
Central Angle
AC = 60º
Example 2
Find the measure of the intercepted TU, if the
inscribed angle R is a right angle.
T
•
R
•
•
U
Example 3
Find the measure of the inscribed angles Q , R ,and
S, given that their common intercepted TU = 86º
Q
•
R
T
•
S
•U
TU = 86º
Inscribed Angles Intercepting
the Same Arc Theorem
If two inscribed angles of a circle intercepted
the same arc, then the angles are congruent
Q
•
IF ∕_ Q and ∕_ S both
intercepted TU, then
T
•
∕_ Q  ∕_ S
S
•U
Inscribed vs. Circumscribed
Inscribed polygon – is when all of its
vertices lie on the circle and the
polygon is inside the circle. The Circle
then is circumscribed about the
polygon
Circumscribed circle – lies on the
outside of the inscribed polygon intersecting
all the vertices of the polygon.
Inscribed vs. Circumscribed
The Circle is circumscribed about the
polygon.
Circumscribed Circle
Inscribed Polygon
Inscribed Right Triangle
Theorem
If a right triangle is inscribed in a
circle, then the hypotenuse is the
diameter of the circle.
Hypotenuse
= Diameter
•
Converse
If one side of an inscribed triangle is a
diameter of the circle, then the triangle is a
right triangle and the angle opposite the
diameter is a right angle.
The triangle is
inscribed in the circle
and one of its sides
is the diameter
Angle B is a right angle
and measures 90º
•
Diameter
= Hypotenuse
B
Example
Triangle ABC is inscribed in the circle
Segment AC = the diameter of the
circle. Angle B = 3x. Find the value of x.
A
•
C
3xº
B
Inscribed Quadrilateral Theorem
A quadrilateral can be inscribed in a circle iff its
opposite angles are supplementary.
X
•
•
Y
/ X + / Z = 180º, and
•
P
W
•
The Quadrilateral
WXYZ is inscribed in
the circle iff
•
/ W + / Y = 180º
Z
Example
A quadrilateral WXYZ is inscribed in circle P, if
∕_ X = 103º and ∕_ Y = 115º , Find the measures of
∕_ W = ? and ∕_ Z = ?
X
•
•
103º
Y
115º
/ X + / Z = 180º, and
•
P
W
•
The Quadrilateral
WXYZ is inscribed in
the circle iff
•
/ W + / Y = 180º
Z
Example
From Theorem 10.11
∕_ W = 180º – 115º = 65º and
∕_ Z = 180º – 103º = 77º
X
•
•
103º
Y
115º
/ X + / Z = 180º, and
•
P
W
•
The Quadrilteral WXYZ
is inscribed in the circle
iff
•
/ W + / Y = 180º
Z
Unit 3 : Circles:
10.4 Other Angle Relationships in
Circles
Objectives: Students will:
1. Use angles formed by tangents and
chords to solve problems related to
circles
2. Use angles formed by lines
intersecting on the interior or
exterior of a circle to solve
problems related to circles
Bell work 1
Find the measure of the inscribed angles , R,
given that their common intercepted TU = 92º
T
•
TU = 92º
R
•
•
U
Bell work 2
A quadrilateral WXYZ is inscribed in circle P, if
∕_ X = 130º and ∕_ Y = 106º , Find the measures of
∕_ W = ? and ∕_ Z = ?
X
•
•
130º
Y
106º
/ X + / Z = 180º, and
•
P
W
•
The Quadrilateral
WXYZ is inscribed in
the circle iff
•
/ W + / Y = 180º
Z
Angle Formed by a Tangent and
a Chord Theorem
If a tangent and a chord intersect at a point on a circle, then
the measure of each angle formed is ½ the measure of
its intercepted arc
A
•
m ∕_ 2
B
•
= ½ m Major ABC
Angle 2
P
Angle 1
•
•
C
m
m ∕_ 1
= ½ m minor AC
Example 1
Find the measure of Angle 1 and Angle 2, if
the measure of the minor Arc AC is 130º
A
•
B
•
P
•
Angle 1
Angle 2
•
C
m
m minor AC =
130º
Example 2
Find the measure of Angle 1, if Angle 1 = 6xº, and
the measure of the minor Arc AC is (10x + 16)º
m minor AC =
(10x + 16)º
A
•
P
•
B
Angle 1= 6xº
•
•
m
C
Intersections of lines with respect
to a circle
There are three places two lines can
intersect with respect to a circle.
•
•
•
On the circle
In the circle
Outside the cirlce
Angles Formed by the Intersection
of Two Chords Theorem
If two chords intersect in the interior of a circle , then the
measure of each angle is ½ the sum of the measures of
the arcs intercepted by the angle and its vertical angle.
D
m ∕_ 1
Angle 1
•
= ½ (m AB + m CD)
•
P
C
•
Angle 2
A
•
•
B
m ∕_ 2
= ½ (mBC + mAD)
Example
Find the value of x.
m CD = 16º
D
•
•
P
A
m AB = 40º
•
C
•
Angle 1
xº
•
B
Angles formed by secants and tangents:
If a tangent and a secant, two tangents, or two secants
intersect in the exterior of a circle, then the measure of the
angle formed is ½ the difference of the intercepted arcs.
1 Tangent
and 1 Secant
2 Tangents
P
B
•
A
•
1
•
C
2 Secants
X
•
•
2
•
•
R
•
•
W
•Q
•
m ∕_ 1
m ∕_ 2
= ½ (m BC – m AC)
B
= ½ (m PQR – m PR)
3
•
•
Z
•
Y
m ∕_ 3
= ½ (m XY – m WZ)
Example 1
Find the value of x
P
•
•
Major Arc PQR
= 266º
xº
•Q
•
R
m ∕_ x
= ½ (m PQR - mPR)
Example 2
Find the value of x, GF. The m EDG = 210º
The m angle EHG = 68º
E
•
Major Arc EDG
= 210º
F
•
D
•
xº
68º
•
H
•
m ∕_ EHG = 68º
= ½ (m EDG – m GF)
G
Unit 3 : Circles:
10.5 Segment Lengths in Circles
Objectives: Students will:
1. Find lengths of segments of chords,
secants and tangents
Bell work 1
Find the value of x.
m CD = 20º
D
•
•
P
A
m AB = 65º
•
C
•
Angle 1
xº
•
B
Bell work 2
Find the value of x, GF. The m EDG = 300º
The m angle EHG = 54º
E
•
Major Arc EDG
= 200º
F
•
D
•
xº
54º
•
H
•
m ∕_ EHG = 54º
= ½ (m EDG – m GF)
G
Answer
m ∕_ EHG = 68º = ½ (m EDG – m GF)
54º = ½ ( 200º - xº )
108º = 200º - xº
E
xº = 200º - 108º
•
xº = 92º
Major Arc EDG
= 200º
F
•
D
•
xº
54º
•
H
•
m ∕_ EHG = 54º
= ½ (m EDG – m GF)
G
Bell work 3
A quadrilateral WXYZ is inscribed in circle P, if
∕_ X = 130º and ∕_ Y = 106º , Find the measures of
∕_ W = ? and ∕_ Z = ?
X
•
•
130º
Y
106º
/ X + / Z = 180º, and
•
P
W
•
The Quadrilateral
WXYZ is inscribed in
the circle iff
•
/ W + / Y = 180º
Z
Bellwork 3 Answer
From Theorem 10.11
∕_ W = 180º – 106º = 74º and
∕_ Z = 180º – 130º = 50º
X
•
•
130º
Y
106º
/ X + / Z = 180º, and
•
P
W
•
The Quadrilteral WXYZ
is inscribed in the circle
iff
•
/ W + / Y = 180º
Z
Segments of Chords, Secants and
Tangents
External
Segment
Segments of
Chords
Segments of
Secants
Segments of
Tangents
Lengths of Intersecting Chords
Theorem
If two chords intersect in the interior of a circle, then the product of
the lengths of the segments of one chord is equal to the product of
the lengths of the segments of the other chord
A
•
EA • EB = EC • ED
C
•
E
D
•
B
Example
Chord AB intersects Chord CD in the interior
of the circle at Point E, Find the measure of segment EA.
Given ED= 6cm, EC = 16 cm, and EB = 8 cm.
A
•
EA • EB = EC • ED
16 cm
C
•
?
6 cm
E
D
8 cm
•
B
Example Answer
EA = 12 cm
A
•
EA • EB = EC • ED
EA • 8 = 16 • 6
8EA = 96
8
8
EA = 12 cm
12 cm
16 cm
C
•
6 cm
E
D
8 cm
•
B
Lengths of Intersecting Secants
If two secant segments Theorem
share the same endpoint outside the
circle, then the product of the length of one secant segment
and the length of the of its external segment equals the
product of the length of the other secant segment and the
length of its external segment.
B
A
EA • EB = EC • ED
E
C
D
Example
EA = 9 ft, AB = 11 ft, and EC = 10 ft.
Find CD, the value of x and ED.
B
9 ft
EA • EB = EC • ED
A
11 ft
E
10 ft
C
x
D
Example Answer
CD = x = 8 ft
ED = 18 ft
EA • EB = EC • ED
9 (9+11) = 10(10 + x)
9 (20) = 100 + 10x
180 = 100 + 10x
-100 -100
E
80 = 10x
10
10
x = 8 ft and ED = (10+8)
ED = 18 ft
B
9 ft
10 ft
A
C
11 ft
x
D
Lengths of Intersecting Tangent
and Secants Theorem
If a secant segment and a tangent segment share an endpoint
outside a circle, then the product of the length of one secant
segment and the length of its external segment equals
the square of the length of the tangent segment
A
(EA)² = EC • ED
E
C
D
Example
EC = 9 in, CD = 15 in
Find EA, the value of x.
A
x
(EA)² = EC • ED
E
9
C
15 in
D
Example Answer
Find EC, the value of x = 5, and ED= 20
(EA)² = EC • ED
A
10² = x • (x + 15)
10 in
100 = x² + 15x
0 = x² + 15x - 100
x = -15 ± √15²- 4(1)(-100)
2
x = -15 + √625 = -15 +25
2
2
x = (10/2) = 5
E
x
C
15 in
D
10.6 Equations of Circles


TSW write the standard equation of a
circle given the radius and center
TSW determine the radius and center
of a circle given the equation in
standard form
Standard form of a Circle

(x - h)2 + (y – k)2 = r2

Center of Circle: ( h, k)

Radius: r
Example 1

Find the radius and center of the
given circle:
(x - 3)2 + (y + 2)2 = 36

x2 + (y- 5)2 = 9

Example 2


Write the equation of a circle given a
radius of 5 and center (3, -1)
Write the equation of a circle with a radius
of
and7a center (-1, 0)