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Geometry: Chapter 11 Note Packet Name: __________________________
Page 1 of 14
11 .1 Lines that intersect Circles
Definitions & Examples
Interior of a circle- all pts circle. Ex.
Exterior of a circle- all pts. Circle. Ex.
Chord- a whose endpoints are on the circle.
Ex.
Secant- a line that intersects a circle in
Ex.
Tangent of a circle- a line that intersects a circle in
Ex.
Point of tangency- the point where the touches the .
Ex.
Congruent circles- two or more circles with radii.
Concentric circles- circles with the same .
Tangent circles- circles that in point.
Common tangent- one line that is to circles.
Example #1) Use circle A to draw and label the following:
a. a line ℓ that is tangent to circle A at point C
b. a radius AB
c. a chord CB
d. a secant BD
e. a diameter ED
f. a point R that is on the interior of the circle
g. a point S that is on the exterior of the circle
h. a circle Z that is tangent to Circle A at point E
i. a circle W that is tangent to line ℓ
j. a concentric circle X that lies inside Circle W
Geometry: Chapter 11 Note Packet Name: __________________________
Page 2 of 14
Theorem Hypothesis (If) Conclusion (Then)
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Symbols:
A
B
If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.
Symbols:
A
B
If two segments are tangent to a circle from the same external point, then the segments are congruent.
Symbols:
Ex. 2) AB and BC are both tangent to circle O. Find the value of x and then find the length of BC
Ex. 3) Use the picture at the right to find:
a. the radius of both circles: Circle S = Circle R. =
b. the point of tangency =
c. the equation for the line of tangency
*** Note: The general equation for horizontal lines are y =
The general equation for vertical lines are x =
Ex. 4) The surface of the earth is approximately 4000 miles from the center. If a satellite orbits the earth at 120
miles above the surface find the distance from the satellite to the horizon of the earth.
Geometry: Chapter 11 Note Packet Name: __________________________
Page 3 of 14
Chapter 11 .2 Arcs and Chords
Definitions and Examples
Central angle- an angle whose is at the of a circle.
Arc- an part of the circle with two endpoints.
Example: Use Circle X at the r i g h t :
a) Identify a central angle =
b) Identify two arcs =
Complete the definitions of each type of arc and describe their measures:
Def: Minor Arc-
Def: Major Arc –
Major arc = minus the measure of the central angle xo
Def: Semicircle-
Always equals
Def: Adjacent Arcs – The measure of an arc formed by two adjacent arcs = the sum of the measure of the two arcs.
Def: Congruent Arcs -
Two arcs that have
Geometry: Chapter 11 Note Packet Name: __________________________
Page 4 of 14
Use the pie chart at the right to find:
a. measure of arc FG
b. measure of arc KLF
c. the measure of GJK
Ex. 2)Use circle F and it’s central angles at the right to find the measure of:
a. arc AE =
b. arc ABD =
c. arc AE + arc ED =
d. arc ABC =
e. arc CD =
f. the measure of central AFB
Ex. 3 Use circle D at the right to give examples of the following (use proper arc notation)
a. A minor arc:
b. A major arc:
c. A semicircle: B
d. A radius:
Geometry: Chapter 11 Note Packet Name: __________________________
Page 5 of 14
Def: Congruent arcs- are or more arcs with the same .
Ex 1) Use the above theorem to answer the following questions:
If HLG JLK, then what two statements can you make regarding chords HG and JK and arcs HG
and JK? Write the true statements below:
What must the value of y equal?
Ex 2) Given TV SW
find the value of n.
Ex 3) Circle B and E are congruent and so are arcs AC and DF. Find m DEF.
Geometry: Chapter 11 Note Packet Name: __________________________
Page 6 of 14
12y + 20
10y + 36
Geometry: Chapter 11 Note Packet Name: __________________________
Page 7 of 14
Theorem Hypothesis Conclusion
If a radius is perpendicular to
a chord, then it the chord
and its
The perpendicular bisector of
a chord is the
Ex 4) If RS = 8 and SM = 9 find the length of chord NP
Hint: the radius of circle R is the hypotenuse of a right triangle that can be used to find NS.
Ex. 5) Find the length of chord CE
Geometry: Chapter 11 Note Packet Name: __________________________
Page 8 of 14
Chapter 11 .3 Sector Area and Arc Length
Recall how to find the Area of a Circle: A = r 2
Definitions and Examples
Area of a Sector is a fraction of the containing the
sector.
Sector of a circle- is a region bound by radii and their
intercepted _.
Write the formula to find the Area of a Sector with a central angle mo
Read example 1 on page 764. The answer the following: Ex.
1) Find the area of the sectors to the nearest tenth.
a. b.
Segment of a circle- is a region bound by an and its _.
Segment of a circle
Area of a segment of a Circle = Area of sector – Area of triangle
Recall how to find the area of a triangle: A =
b
h
2
Area of a Sector =
Geometry: Chapter 11 Note Packet Name: __________________________
Page 9 of 14
Example 2) Find the area of the segment.
Recall how to find the circumference of a circle: C = d or C= 2 r
The Arc length is a fraction of the of a circle.
Arc length- is the measured along an
in linear units.
Ex 3) Find the length in centimeters of the arc FG.
Ex. 4) Find the length of the arc BD to the nearest tenth.
Segment
Geometry: Chapter 11 Note Packet Name: __________________________
Page 10 of 14
Chapter 11.4 Inscribed Angle
Definitions - complete the definitions below and use the picture to give an example of each definition.
Inscribed angle- an whose
vertex is the circle and whose sides are
of the circle.
Intercepted arc- is the arc whose points lie on the
sides of the angle and all the
points between them.
Subtends- a chord or arc whose endpoints are on the
sides of an angle.
Study and complete the Theorems and their conclusions.
= inscribed angle
= intercepted arc
= subtends
Inscribed Angle Theorem: The
measure of an inscribed angle is
the measure of its
intercepted arc.
A
C
B
Conclusion:
Corrollary: If inscribed angles of a
circle intercept the arc
or are subtended by the same chord
or arc, then the angles are
.
A
D
C
F
B
Conclusion:
Inscribed Semicircle: An inscribed angle subtends a semicircle if and
only if the angle is .
C
A B
conclusion
Inscribed Quadrilateral: If a
quadrilateral is inscribed in a circle
then its
angles are
.
Conclusion:
Geometry: Chapter 11 Note Packet Name: __________________________
Page 11 of 14
Use the Inscribed Angle Thm. to answer the following. Ex1)
Find the measure of arc PS and inscribed angle PRU
Use the Corollary of the Inscribed Angle Thm. to answer the following
Ex. 2) Find the m LJM and arc LM
Use the Inscribed Semicircle Thm. to answer the following
Ex3) Find the value of a.
(5a +20)
Use the Inscribed Quadrilateral Thm. to answer the following Example 4:
Find the measures of the angles in the quadrilateral.
Geometry: Chapter 11 Note Packet Name: __________________________
Page 12 of 14
1020
1
900
Then the measure of the angle is :
Then the measure of the angle is :
Chapter 11.5 Angle Relationships in Circles
**Study and complete the theorems of this section**
IF THE VERTEX OF THE ANGLE IS…
Theorem 11-5-1
ON the circle
This could be two chords, two secants, or
one tangent and one secant
Theorem 11-5-2
INSIDE the circle
This could be two secants or two chords.
Theorem 11 – 5- 3
OUTSIDE the circle
This could be two secants, one secant and one tangent or
two tangents.
Then the measure of the angle is :
1120
mDCB
mC
Geometry: Chapter 11 Note Packet Name: __________________________
Page 13 of 14
Ex 1) Find the measure of FGH
Ex 2) Find the measure of arc LM
Ex 3: Find m ABD.
Ex 4) Find the measure of arc KM
Ex 5) find the measure of ACD
Geometry: Chapter 11 Note Packet Name: __________________________
Page 14 of 14
Chapter 11.7 Circles in the Coordinate Plane
The equation of a circle formula is: where r stands
for , h stands for and k stands for .
Ex 1: State the center and radius of the circle.
a. (x-1)2 + (y+2)2 = 9 b. (x+ 4)2 + (y-2)2 = 34
Ex 2: Graph the circle in Ex. 1 pt. a.
Ex 3: List four other points on the circle in part a.
How could you determine those points without relying on the graph?
Ex 4: What are the equations of the circles graphed below?
–2
–3
–4
–5
–6
–7
–1
–1