Download 8.1 Properties of Tangents and Arcs

Name _________________________________ Period _______
8.1
Properties of Tangents and Arcs
Description/Definition
Circle
What does it look like?
- The set of all points in a plane that are
________________ from a given point, called the
_____________.
Radius - The distance from the ____________ of
the circle to a ___________ on the circle.
Diameter - The distance _____________ the circle
through its ____________.
Chord - A segment whose ________________ are
on the circle.
1. List all of the chords and diameters in the figure.
CHORDS
DIAMETERS
2. Determine whether each statement is true or false. If
false identify a counterexample using the diagram to the
left.
1. All chords are diameters.
2. All diameters are chords.
Description/Definition
What does it look like?
Tangent Line - A line that intersects a circle
_____________________
Point of tangency – The point where a tangent line
intersects a circle.
Secant – A line that intersects a circle at ________ points.
Common Tangent Line
A line that is tangent to both
circles.
Common Internal Tangents
Intersects the segment joining the
centers.
1
Common External Tangents
Doesn’t intersect the segment joining
the centers.
Tell whether the line is best described as a radius, chord, diameter, secant, or tangent of
1. ̅̅̅̅
4. ̅̅̅̅
2. ⃡
5. ⃡
3.
6. ̅̅̅̅
MATCHING
_____1.
A. Center
_____2. ⃡
B. Chord
_____3. ̅̅̅̅
C. Diameter
_____4. ̅̅̅̅
D. Radius
_____5.
E. Point of tangency
_____6. ̅̅̅̅
F. Common external tangent
_____7. ⃡
G. Common internal tangent
_____8. ⃡
H. Secant
C.
POINT OF TANGENCY TH
In a plane, a line is tangent to a circle if and only if it is _____________________ to a radius of the circle at
its point of tangency.
Line l is tangent to
→ _________________
●Q
R
_________________ → Line l is tangent to
●Q
l
l
R
(TAN)
1. Determine whether ⃡
●Q
R
( )
is tangent to
●Q
l
( )
.
(TAN)
given ̅̅̅̅ is tangent to
.
2. Find the radius of
, and
A
●
T
R
3. If ̅̅̅̅ tangent to circle R, find…
4. Is ̅̅̅̅ tangent to circle ?
(a) RS
10
(b) area of
2
l
R
,
“CONEHEAD” THEOREM
I
Tangent segments from a common external point are________
K
G
J
14 m
1. ̅̅̅ is tangent to
at K and ̅̅̅̅ is tangent to
Find the value of x.
at M.
6m
2. ̅̅̅̅is tangent to C at S and ̅̅̅̅ is tangent to
17 m
Find the value of x.
THREE TYPES OF ARCS IN A CIRCLE:
1. MINOR ARC: < _____
2. MAJOR ARC:
°
3. SEMICIRCLE:
Names:
Names:
Names:
*Name by 2 letters (endpoints)
*Name by 3 letters (endpoints and
point between)
Name the highlighted arcs.
1.
2.
P
G
3.
O
4.
B
E
C
S
F
A
*Name by 3 letters (endpoints and
point between)
H
R
T
C at T.
A
R
E
D
Every INTERIOR angle has an INTERCEPTED arc
Intercepted arc - The arc that lies in the ________________ of an interior angle.
Example:
𝑅
•𝐹
𝑈
𝐺•
𝑇
•𝑅
has an intercepted arc _____
𝑆
C
D
has an intercepted
arc _____
A
3
C
D Farc ____
has an intercepted
A
B
E
B
Identifying angles and arcs
1) a.
intercepts arc
___________.
b. ̂ is the intercepted
arc of
.
c.
2) a. ̂ is the intercepted arc I
G
of
.
N
H
b.
intercepts arc
D
P
___________.
C
A
c.
intercepts arc
___________.
B
intercepts arc
______ and ______.
E
b. ̂ is the intercepted arc of
H
.
intercepts arc ___________.
d. Do you think
O
is a central angle? Why or why not?
J
M
E
K
L
N
O
J
M
K
4
J
M
K
F
L
D
N
c.
O
I
G
intercepts arc ___________.
F
P•
3)
a.
L
8.2
Constructions
CONSTRUCTION TOOLS
DEMO VERSIO
DEMO VERSION
MO VERSION
• Compass
• Straightedge (a ruler without measuring marks)
Perpendicular Bisector (also segment bisector & midpoint)
1. Place the needle of the compass on
point A. Draw an arc through the
segment.
A
B
2. Without changing the compass
setting, place the needle on point B.
Draw an arc through the segment.
A
3. X marks the spot! Lastly, draw a
line through both intersections of
the two arcs.
A
B
B
Construct a perpendicular bisector for each segment.
1.
2.
EMO VERSION
DEMO VERSI
DEMO VERSION
A Perpendicular from a point not on the line
1. Place the needle of the compass on
point P. Draw an arc that intersects
the line twice.
P
2. Draw points where you arc intersected
the line. From the 1st point (A) draw
an arc. Without changing the
compass, do the same from point (B)
3. X marks the spot! Lastly, draw a
line From point P to the
intersection of the arcs
P
P
B
A
5
A
B
DEMO VERSIO
DEMO VERSION
Construct a line perpendicular to the given line from point P.
1.
2.
P
P
DEMO
VERSIO
EMO VERSION
DEMO VERSION
A Perpendicular through a point on the line
1. Place the needle of the compass on
point P. Draw an arc that intersects
the line twice.
2. Draw points where you arc
intersected the line. From the 1st
point (A) draw an arc. Without
changing the compass, do the same
from point (B)
3. X marks the spot! Lastly, draw a
line From point P to the
intersection of the arcs
A
P
A
B
P
Construct a line perpendicular to the given line through point P.
1.
2.
6
P
B
DEMO VERSION
MO VERSION
DEMO VERSION
Angle Bisector
1. Place the needle of the compass on
the vertex of the angel (A). Draw an
arc that intersects the sides of the
angle.
2. Draw points where you arc
intersected the sides. From the 1st
point (A) draw an arc. Without
changing the compass, do the same
from point (B)
3. X marks the spot! Lastly, draw a
line From point A to the
intersection of the arcs
D
D
A
A
E
A
E
Bisect the angles below.
1.
2.
DEMO VERSION
DEMO VERSIO
EMO VERSION
Equilateral Triangle
1. Place the compass needle on one
endpoint and measure the segment.
Use this measurement to draw a
large arc.
2. Without changing your compass,
draw the same large arc from the
other endoint
3. X marks the spot! Lastly, Draw
lines connecting the endpoints of
the segment to the intersection of
the arcs.
MO VERSION
DEMO VERSION
A
B
B
A
Construct an equilateral triangle using the segment below.
1.
2.
7
A
B
O VERSION
DEMO VERSION
DEMO VERSION
Copying an angle
1. Draw a line that will be one side of
the new angle. Label a point on the
line (B).
2. Using point A as the center draw an
arc intersecting both sides of the
angle. Without changing your
compass draw the same arc form
point B.
3. Draw points where the arcs meets
intersect the lines. Use the compass
to measure the distance from point
C to point D. Mark it!
D
DEMO VERSION
O VERSION
A
A
A
B
B
B
4. Without changing your compass,
place your needle on point D and
draw the same arc
C
E
5. X marks the spot! Lastly, draw a line
from point B to the intersection of
the arcs.
ALL DONE!!!
D
D
DEMO
VERSION
O VERSION
A
A
C
C
B
B
E
Copy the angles below.
1.
E
2.
A
A
8
MO VERSION
DEMO VERSION
DEMO VERSION
Parallel lines
1. Draw a line through P that intersects
the line and mark the intersection.
2. Using point A as the center draw an
arc that intersects both lines.
P
3. Without changing your compass
draw the same arc form point P.
P
P
DEMO VERSION
DEMO VERSION
MO VERSION
A
A
A
4. Mark all intersections with a point.
Using your compass measure the
distance from point C to point D.
Mark it!
5. Without changing your compass
place your compass on point D and
draw the same arc.
6. X marks the spot! Lastly, draw a
line from point P to the intersection
of the arcs
D
D
D
P
P
P
C
C
C
MO VERSION
A
B
A
A
B
B
DEMO VERSION
Draw a line through point P parallel to the given line.
1.
2.
P
P
A
B
B
A
9
8.3
Properties of Arcs, Central and Inscribed Angles
ANGLES IN CIRCLES
Central Angle
•
Inscribed angle
An angle whose vertex is the ________________ of
the circle.
• An angle whose vertex is ______ the circle and
whose sides contain chords.
_______ is a
_______ is a central
angle and it intercepts
inscribed angle and it
arc ______.
intercepts arc ______.
Measuring CENTRAL angles
• The measure of a central angle is the measure of its
intercepted arc.
Central
̂
=
The expression
Inscribed
=
̂
80°
̂ is read as “the measure of arc AB.
Example 1: Find the indicated measures.
̂
1. ̂
2.
5. Find the
Measuring INCRIBED angles
• The measure of an inscribed angle is _______the
measure of its intercepted arc.
and
̂ in
3.
4.
6.
̂
a.
b.
BOWTIE THEOREM
D  ________
If two inscribed angles of a circle intercept the same
A  ________
arc, then the angles are congruent.
1. Name all the pairs of congruent angles
2. Find
10
and
̂
ARC ADDITION POSTULATE
̂
The measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs.
Example 2: Find the measures of the indicated arcs
1.
2.
a.
̂
b.
̂
c.
̂
d.
̂
a.
̂
c.
̂
d.
̂
e.
̂
f.
̂
̂
Checkpoint:
a.
P
̂
b.
T
c.
d.
̂
f.
̂
120°
E
e.
̂
A
U
g.
=
h.
F
11
E
=
8.4 Angles inside and outside a circle
RECAP…
Central angle
VS
Vertex is the CENTER of the circle
Angle is EQUAL to its arc
Inscribed angle
Vertex is ON the circle
Angle is HALF its arc
50º
Central
= ARC
50º
“IN-HALF”
50º
Inscribed
HLF
= ARC
̂
O
25º
̂
2
THREE OTHER ANGLE RELATIONSHIPS (Vertex NOT on Center)
Vertex INSIDE circle (but not on center)
Vertex ON the Circle
Tangent-Chord Angle
Inscribed Angle
W
A
C
B
m1 =
P
D
X
N
1
2 1
O
1
B
A
Formula:
Y
m2 =
Q
Formula:
W
C
Vertex OUTSIDE
circle
W
X
B
B
C
X
1
B
D
C
C
1
1
Y
D
Y
A
A
A
Formula:
P
N
P
N
Basic Examples of the NEW ones…
1. Find the measure Oof ̂ 2 2. Find the value of x.
𝐹
Q
𝑇
𝐺
O
2
3. Find the value
Q of x.
4. Find the value of x.
x°
2
58
2
12
110
°
Identify the type of angle and then apply the proper formula to solve.
1. Line m is tangent to the circle. Find …
2. If ̂ 2 , thn find the measure of
.
T
a.
M
C
b.
̂
H
I
3. Find the value of x.
4. Find the value of .
2
C
66°
E
2
F
𝑥
84°
D
5. If
find the value of x.
and
̂
6. Solve for x and y.
, then
G
D
y˚
E
F
Working BACKWARDS!!!
1. Find the value of x.
2. Find the value of x.
M
L
29
𝑥
9
𝑥
2
J
K
1. ̅̅̅̅ is tangent to circle D.
If
, then find…
TRIANGLES WITHIN THE CIRCLES
2.
and
F
J
H
a.
a. .
H
D
b. .
b.
c.
2 E , then find
E
c.
̂
G
d. .
13
LJK
L
̂
̂
K
I
Inscribed Polygon:
Circumscribed Circle:
A polygon with all vertices that lie _______ the circle.
A circle that contains an inscribed polygon
INSCRIBED TRIANGLE THEOREM
ABC is _____________triangle if and only
Its longest side is the diameter.
ABC is a _____ triangle
____ is the diameter
INSCRIBED QUADRILATERAL THEOREM
_____ + _____ = 180º
A quadrilateral can be inscribed in a circle if and only if
_____ + _____ = 180º
its opposite angles are supplementary.
1. a. Find the values of x and y given ̅̅̅̅ is the diameter. Explain.
Ex
xº
yº
2.
is inscribed inside the circle below. Find
the value of each variable. Explain.
b. Find m ̂
27º
3. The circle is circumscribed about the polygon. Find the
value of each variable. Explain.
SPIRAL REVIEW- parallel lines
If 2 lines are parallel, then…
Corresponding Angles are
Alternate Interior s are
_____________________
___________________
1. If
, then find the value of x
and explain.
4. A right triangle is inscribed in a circle. The radius of the
circle is 5.6 centimeters. What is the length of the
hypotenuse of the right triangle? Explain.
Alternate Exterior s are
_____________________
2. Find the values of r and s. Explain.
Consecutive Interior s are
___________________
3. If
, then find the values of g and f.
Explain.
s°
f
x°
n
r°
73°
46°
j
g°
m
k
2
70°
14
130°
8.5
Segments In & Out of Circles
2 CHORDS on the INSIDE:
2 SECANTS:
1 TANGENT, 1 SECANT:
EA • _____ = DE • _____
EA • _____ = EC • _____
EA2 = _____ • _____
Example 1: Find the value of x.
1.
3.
2.
4.
a.
a. 4
b. 2√
b.
c. 2
c.
d.
d.
5.
a. 36
6.
a. 121°
b. 6√
c.
b. 98°
√2
c. 49°
d. 9
7.
9.
d. 85°
8.
a.
a.
b. 12
b. 78°
c. 7
c. 84°
d. 11
d. 80°
a. 8√2
10.
a. 138°
b. 9
b. 69°
c. 128
x
d. 8√
21°
15
c. 42°
d. 159°
PERPENDICULAR BISECTOR TH
If a diameter of a circle is
perpendicular to a chord, then the
diameter bisects the chord and its arc
If ̅̅̅̅ is the perpendicular bisector
of ̅̅̅̅, then ____ is a __________
**the converse is also true**
If ̅̅̅̅ is a diameter and ̅̅̅̅ ̅̅̅̅ ,
then ____≌____ and ____ ≌ ____.
2. Find the measures of ̂ ̂ and ̂
1. Find BD.
DEMO VERSIO
3. A is the center of the circle below. ̅̅̅̅ and ̅̅̅̅ intersect
at F. If
and
2 , what is the length of
̅̅̅̅ ?
4. O is the center of the circle below. ̅̅̅̅ and ̅̅̅̅ intersect
at D. If
and
, what is the length of
̅̅̅̅?
K
O
D
A
Y
CONSTRUCTION – Center of a Circle
1. Draw a chord on circle
given. Label the endpoints
A and B.
2. Construct the perpendicular
bisector of the chord. (Put
compass needle on point A and
draw arc. Do the same forB).
B
A
3. Draw a second chord and
repeat. (Construct the
perpendicular bisector of the
second chord).
B
B
A
A
Construct the center of the circle.
16
4. Find the point where the
two perpendicular bisectors
intersect. That the center of
the circle.
B
A
C
SPIRAL REVIEW
1. What would be the new coordinates of point S if the
figure were rotated 90° counterclockwise about the
origin?
ROTATIONS
(-b, a)
(a, b)
(-a, -b)
(b, -a)
• For every 90° rotation, _______ the numbers!
• Look at the quadrant the point is in to determine the
________.
2. If
is rotated 180° counterclockwise about the
origin, what are the coordinates of
SHADED AREA
Triangle
Ashaded – Aunshaded
Trapezoid
1. Find the area of the shaded region.
3. If
2
is rotated 90° clockwise about the origin,
what are the coordinates of
Parallelogram
Rhombus
Kite
Regular Polygon
Circle
Equilateral
2. Find the area of the shaded region.
12m
•
8 cm
12m
13m
23 cm
18m
17
16m
8.6
Equation of Circles and Spiral Review
STANDARD EQUATION OF A CIRCLE
*NOTE* in the standard equation there is a
minus sign before the h and k. Therefore:
• If (+h, +k) ⟶
• (h, k) =
• If (–h, –k) ⟶
•r=
Write the standard equation of the circle with given center and radius.
1. Center (-3, 4); radius 5
2. Center (6, 0); radius 2
Write the standard equation of the circle shown.
3.
4.
Find the center and the radius of the circle.
5.
6.
Graph the equation.
7.
2
8.
9. (1, 2) is the center of a circle. If ( 2
is point on the
circle. What is the length of its radius?
10.
18
2 is the center of a circle. If (9
is point on
the circle. What is the length of its radius?
ANGLE MEASURES IN POLYGON
SUM of EXTERIOR s
EACH EXTERIOR s
SUM of INTERIOR s
1. The measures of the interior angles of a hexagon are
,
2
and
°. What is the measure
of the largest angle?
2. If one interior angle of a regular polygon is
168°:
a) What is the measure of one exterior
angle?
b) Classify the polygon.
TRIGONOMETRIC RATIOS
1. The body of a dump truck is rasied to empty a
load of sand. How high is the 14 foot body from
the frame when it is tipped upward at an angle
of 34°.
2. Crazy Jake is standing at the tip of a 284 ft drawbridge. If the
bridge is raise to a 42° angle of elevation. When the draw
bridge is closed it is 40 feet above the water level. If Crazy Jake
spits down from the top of the bridge how far would his spit
travel?
COUNTEREXAMPLES
REMEMBER: a good counterexample is TRUE for the 1st part of the statement and FALSE for the 2nd part.
1. Which of the following is a counterexample to the
following statement?
2. Which of the following is a counterexample to the
following statement?
If a polygon has four sides, then the
diagonals bisect each other.
a.
b.
c.
d.
If a quadrilateral is a parallelogram, then its
diagonals are perpendicular.
square
trapezoid
parallelogram
hexagon
a.
b.
c.
d.
19
square
rectangle
kite
rhombus
DEMO VERSION
REVIEW DAY!!!
1. Find ̂ if
point B
̅̅̅̅ is tangent to
2
2. Find the value of x
at
F
A. 52
B. 128
C. 64
D. 232
DEMO VERS
A. 5√
B. 50
C. 5√2
D. 2√
E
A
C
D
B
3. Find the measure of 1.
4. Solve for x.
A. 55
B. 17.5
C. 57.5
D. 85
A. 55
B. 118
C. 42
D. 186
5. If
̅̅̅̅
, and
, then find the length of
6. If
, then find
.
A. 37
B. 53
C. 104
D. 16
G
I
7. Find the value of x and
̂
8. If
A. 33
B. 66
C. 132
D. 264
̂
76°
34°
J
A. 4.8
B. 12
C. 7
D. 35
9.
̂
x°
is inscribed in
2 . Find
2
A.
B.
C.
D.
where
and
.
P
A. 6
B. 1
C. 2.6
D. 2
K
O
N
11. Find the coordinates of
if
is rotated 9
clockwise around the origin.
A.
2
B. 2
C.
2
D. 2
2
10. Find the value of x.
&
M
9
9
and
and
and
and
, then find measure of arc 2.
A. 312
B. 96
C. 132
D. 264
xº
H
𝐼
𝐹
y
6
12. Which polygon that could serve as a conterexample to the
staterment below.
“If a polygon has four sides, then at least one pair of
opposite angles are congruent”
5
4
3
2
𝐺
1
𝑇
–6 –5 –4 –3 –2 –1–1
–2
–3
1
2
3
4
5
6
A. iscoseles trapezoid
x kite
B.
C. rhombus
D. square
–4
–5
–6
20
13. If lines t and r are parallel, then find the value of y.
𝑦
14. Find the area of the shaded region given the circle is
tangent to the bases of the parallelogram.
14 m
6m
𝑟
17 m
𝑡
Partner Up!
1. Find the value of x.
2. In
of x.
̂
we are given
2
Find the value
DEMO VERSION
xº
3. Annie was working on her cross stitch and wanted to make a design with
4. Name the construction shown below.
two additonal lines at the bottom. Find
the length of the lengthof each additional
string of yarn.
P
A
B
𝟖
𝟏𝟐
𝟑
Answer: Shorter string = ____ & Longer string =____
5. Find the value of x.
6. A light post was blown over by the winds. It remained at a tilt
where its leans at
. The robin sitting on the light post is 13
feet above the ground. What is the length of the light post?
7. Find x and y.
8. If each interior angle of a regular polygon measures
, then what is its interior angle sum?
𝑦
21
Class Practice for CONSTRUCTIONS!
1. Using a compass and a straightedge, draw a
perpendicular to the line through point K on the line.
2. Using a compass and a straightedge, construct a line parallel
to the given line through point P.
P
K•
3. Using a compass and a straightedge, construct a
perpendicular bisector of the given segment.
4. Using a compass and a straightedge, copy the angle given
below.
5. Using a compass and a straightedge, construct the
angle bisector of the angle below.
6. Construct center of the circle below.
MO VERSION
7. Jim was given
. What constuction
is completing below?
8. What is the first step in constructing a perpendicular from a point on the
line?
𝑇
D
𝐽
A
C
B
E
𝑃
𝐾
A. From points J and K, draw arcs above point P creating point T.
B. Draw a line segment through points T and P.
C. From point P, draw an arc that intersects the line at J and K.
D. None of the above.
22
23