lesson 10.1
DESCRIPTION
Lesson 10.1. Today, we are going to… > identify segments and lines related to circles > use properties of tangents to a circle. Parts of a Circle. C. Circle C. Diameter = _ radius . Y. N. BN. YX. AB. A. C. X. B. A chord is. Y. YX. AB. A. C. X. B. A secant is. AB. - PowerPoint PPT PresentationTRANSCRIPT
Lesson 10.1Parts of a Circle
Today, we are going to…> identify segments and lines related
to circles> use properties of tangents to a circle
C
Circle C
Diameter = _ radius
C
A chord is
YX
AB
A
B
X
YN
BN
C
A secant is
A
B
X
Y
YX
AB
C
A tangent is
ABA
B
Y X
XY
internal tangents
Common Tangent Lines
external tangents
Common Tangent Lines
Two circles can intersect in 2, 1, or 0 points.
Draw 2 circles that have2 points of intersection
internally tangent circles
Draw two circles that have1 point of intersection
externally tangent circles
Draw two circles that have1 point of intersection
concentric circles
Draw two circles that have no point of intersection
9. What are the center and radius of circle A?
Center: Radius =
10. What are the center and radius of circle B?
Center: Radius =
11. Identify the intersection of the two circles.
12. Identify all common tangents of the two circles.
mABC =
A
B
C
Theorem 10.1 & 10.2A line is tangent to a circle if
and only if it is _____________ to the radius from the point of
tangency.
A
B
C
7
13. Find CA.
15D
C
B
AWhat is DA?
7
14. Find x.
15
x
6
C
B
A
xx
168
What is CA?
7
156
C
B
A
2610
24
How do we test if 3 segments create a right triangle?
15. Is AB a tangent?
7
156
C
B
A
178
12
16. Is AB a tangent?
17. Find the slope of line t.
A
C
A (3,0) and C (5, -1)
Slope of AC?
Slope of line t?
t
C
A tangent segment
A B
One endpoint is the point of tangency.
Theorem 10.3If 2 segments from the same
point outside a circle are tangent to the circle, then
they are congruent.
7x - 2
3x + 8
18. Find x.
A
C
B
x2 + 25
50
19. Find x.
A
C
B
Lesson 10.2Arcs and Chords
Today, we are going to…> use properties of arcs and chords
of circles
C
An angle whose vertex is the center of a circle is a
central angle.
A
B
C
Minor Arc - Major Arc
A
B
D
Minor Arc
AB
Major Arc
ADB
C A
B
D
60˚
m AB =
Measures of Arcs
C
Semicircle
m AED = m ABD = m AD
A
B
D
E
Find the measures of the arcs.
1. m BD
2. m DE
3. m FC
4. m BFD
D
E
F
B
C
100˚52˚
68˚
53˚?
AD and EB are diameters.
F
A
B
D
E
C
5. Find x, y, and z.
30˚
z˚
x˚
y˚
x =
y =
z =
Theorem 10.4
Two arcs are congruent if and only if their chords
are congruent.
(2x + 48)°(3x + 11)°
B
ADC
6. Find m AB
Theorem 10.5 & 10.6
A chord is a diameter if and only if it is a
perpendicular bisector of a chord and bisects its arc.
7. Is AB a diameter?A
B
8. Is AB a diameter?A
B
8
8
9. Is AB a diameter?A
B
Theorem 10.7
Two chords are congruent if and only if they are equidistant from the
center.
AB = 12
10. Find CG.
DE = 12
7D
G BA
C
F
E6
x
?
Lesson 10.3Inscribed Angles
Today, we are ALSO going to…> use properties of inscribed angles
to solve problems
An inscribed angle is an angle whose vertex is on the
circle and whose sides contain chords of the circle.
Theorem 10.8If an angle is inscribed,
then its measure is half the measure of its intercepted
arc.
x2x
1. Find x.
x°
120°
x = 60°
2. Find x.
x°
70°
x = 140°
Theorem 10.9If 2 inscribed angles
intercept the same arc, then the angles are
congruent.
3. Find x and y.
y°
45°
x°
InscribedPentagon
x°
A
D
C
B
4. DC is a diameter. Find x.
Theorem 10.10If a right triangle is inscribed in a circle, then the hypotenuse is a
diameter of the circle.
5. Find the values of x and y.
x°
y°A
42 D
C
B
Theorem 10.11If a quadrilateral is inscribed in a
circle, then its opposite angles are
supplementary.
21
4 3
m 1 + m 3 = 180º
m 2 + m 4 = 180º
6. Find the values of x and y.
x°
110°
80° y°
7. Find the values of x and y.
x°
120°
100° y°
Lesson 10.4Angle Relationships
in CirclesToday, we are going to…> use angles formed by tangents and
chords to solve problems > use angles formed by intersecting
lines to solve problems
Theorem 10.12
If a tangent and a chord intersect at a point on a
circle, then...
GSP
Theorem 10.12
… the measure of each angle formed is half the measure of its
intercepted arc.
1A
BC
2
1A
BC
2
1. Find m 1 and m 2.
100°
2. Find and mACB and mAB
95°A
B
C
3. Find x
5x°A
B
C(9x + 20)˚
Theorem 10.13If 2 chords intersect inside a circle, then…
A
B
C
D
1
B
CA
D
1
…the measure of the angle is half the sum of the intercepted arcs.
A
B
C
D
x°
4. Find x.100°
120°
A
B
C
D
x°
5. Find x.130°
160°
A
B
C
D
x°
6. Find x.
80° 90°y°
A
B
C
D
x°7. Find x.
100°
120°
A
B
C
D
x°
8. Find x.
52°74°
Do you notice a pattern?
Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…
A
C
D
1
Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…
A
B
C 1
Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…
A
BC
D
1
A
BC
D1
…the measure of the angle is half the difference of the intercepted arcs.
9. Find x.
20° 80°
A
BC
D
x°
10. Find x.
24°90°
A
BC
Dx°
11. Find x.
200°x°
A
C
D
12. Find x.
135°x°
13. Find x.
100°
3 100°2 1
100°
60°
Lesson 10.5Segment Lengths
in CirclesToday, we are going to…> find the lengths of segments of chords, tangents, and secants
Theorem 10.15
If 2 chords intersect inside a circle, then the product of their “segments” are
equal.
a · b = c · d
a
b
c d
1. Find x.
6
8 4
x
2. Find x.
3x
182x
3
3. Find x.
2x
18x
4
Theorem 10.16 If 2 secant segments share the same endpoint outside
a circle, then…
GSP
…one secant segment times its external part
equals the other secant segment times its external part.
a · c = b · d
b
a
c
d
3. Find x.
5
x
4 6
4. Find x.
9
10x
20
Theorem 10.17 If a secant segment and a tangent segment share an endpoint outside a circle,
then…
…the length of the tangent segment squared equals the
length of the secant segment times its external
part.
a · a = b · d
db
a
a2 = b · d
54
x5. Find x.
15x
106. Find x.
Quadratic Formula?♫♪♫♪♫♪♫♪♫♪♫♪
15x
106. Find x.
x20
317. Find x.
8. Find x.
3
48
x
10x
89. Find x.
Lesson 10.6Equations of
CirclesToday, we are going to…> write the equation of a circle
Standard Equation for a Circle with
Center: (0,0) Radius = r
1. Write an equation of the circle.
2. Write an equation of the circle.
Standard Equation for a Circle with
Center: (h,k) Radius = r
3.Write an equation of the circle.
C =
r =
4.Write an equation of the circle.
C = r =
Graph (x – 3)2 + (y + 2)2 = 9
Center?
Radius =
Identify the center and radius of the circle with the given equation.
5. (x – 1)2 + (y + 3)2 = 100
6. x2 + (y - 7)2 = 8
7. (x + 1)2 + y2 = ¼
Center: (1, -3) radius = 10
Center: (0, 7) radius ≈ 2.83
Center: (-1, 0) radius = ½
Write the standard equation of the circle with a center of (5, -1) if a point on the circle is (1,2).
8. Write the standard equation of the circle with a center of (-3, 4) if a point on the circle is (2,-5).
Is (-2,-10) on the circle (x + 5)2 + (y + 6)2 = 25?
9. Is (0, - 6) on the circle (x + 5)2 + (y – 5)2 = 169?
10. Is (2, 5) on the circle (x – 7)2 + (y + 5)2 = 121?
><
=
Would the point be inside the circle, outside the circle, or on the circle?
(x – 13)2 + (y - 4)2 = 100
11. (11, 13)
12. (6, -5)
13. (19, - 4)
Lessons 11.4 & 11.5Circumference and
Area of CirclesToday, we are going to…> find the length around part of a circle and find the area of part of a circle
Circumference
Arc Length
=
A
B
A
B50°7 cm
1. Find the length of AB
A
B
85°
10 cm
2. Find the radius
3. Find the circumference.
Area
Sector of a circle A region bound by two radii &
their intercepted arc.
A slice of pizza!
Area of a Sector
=
3. Find the area of the sector.A
B50°7 cm
4. Find the radius. A
B
100°
3. Find the area.
Workbook
P. 211 (1 – 10)
P. 215 (1 – 6)