Transcript 2 - SchoolRack

I. Definitions
10-1 Circles
Circle
The set of all points in a
plane that are at a given distance
from a given point in that plane.
C
Symbol ○R
Radius
The distance between
the center of a circle and any point
on the edge of the circle. AR,BR
Diameter The distance across
B
A
the circle that goes through the
radius AB
Chord A segment that goes from
one side of a circle to another. AC
Circumference the distance
around the circle C= 2  r
Total degrees 360
R
II. EXAMPLES
 1. Find the circumference if
a. r = 8
b. d = 12
2. Find d and r if C = 136.9 cm.
3. Circle A radius=8, Circle B radius =
14, and JE =4. Find EB and DC.
D
A
J
E
B
C
10-2 Angles and Arcs
• I. Central
Angle
an angle whose
vertex
contains the
center of a
circle.
II. Arc
Part of a circle; the curve between
two points on a circle.
• If circle is divided into two unequal
parts or arcs, the shorter arc (in
red) is called the minor arc and the
longer arc (in blue) is called the
major arc.A
Minor arc- 2 letters
Major arc- 3 letters
C
B
•III. Semicircle
– a semicircle is an arc that makes up
half of a circle
180°
Arc Addition Postulate - The measures
of an arc formed by two adjacent arcs
is the sum of the measures of the two
arcs. That is, if B is a point on
,
then
+
=
.
IV. ARC MEASURE
The measure of a
minor arc = central
angle.
The measure of a
major arc = 360 minus
the measure of its
central angle.
V. ARC LENGTH
LENGTH OF THE ARC is a part of the
circumference proportional to the measure of the
central angle when compared to the entire circle
VII. CONCENTRIC CIRCLES
CONCENTRIC
CIRCLES lie in the
same plane and have
the same center, but
have different radii.
ALL CONCENTRIC CIRCLES ARE SIMILAR BC ALL
CIRCLES ARE SIMILAR!
VIII. CONGRUENT
ARCS
TWO ARCS WITH
THE SAME
MEASURE AND
LENGTH
Example 1: Find the length of
arc RT and the measurement
in degrees.
2. a. Find the length of arcs
RT and RST
b. Find the measurement in
degrees of both.
3. Find the arc length of RT
and the degrees measurement
of RT.
4. If <NGE  < EGT, <AGJ =2x, <JGT
= x + 12, and AT and JN are
diameters, find the following:
a. x
b. m NE
c. m JNE
A
J
E
N
T
5.
6. Find x.
N
M
9x
Q
8x
A
19x
O
R
10-3 Arcs and Chords


I. Arc of the chord
When a minor arc and a chord share the same
endpoints, we called the arc the ARC OF THE CHORD.
II. Relationships
–
2 minor arcs are  if their chords
are .
If a diameter is perpendicular to a
chord, it bisects the chord and the arc.
2 chords are  if they are equidistant
from the center
Inscribed polygons must have
vertices on the circle.
1. Circle N has a radius of 36.5 cm.
Radius is perpendicular to chord FG,
which is 53 cm long.

a. If m FG= 85, find m HG.

b. Find NZ.
2. Chords FG and LY are equidistant
from the center. If the radius of M is 32,
find FG and BY.
FG = 46.4
LY = 23.2
3. mWX = 30, mXY = 50, mYZ = 30.
WY= 14, FIND XZ.
4. RT is a diameter. If US = 9, find SV.
5. XZ= 12, UV = 8, WY is a diameter.
Find the length of a radius.
6. IF AB and DC are both parallel and
congruent and MP = 7, find PQ.
10-4 Inscribed
Angles
• I. Definitions
• Inscribed angle — An
angle that has its vertex
on the circle and its
sides are chords of the
circle
• Intercepted arc — An
intercepted arc is the arc
that lies "inside" of an
inscribed angle
• If an angle is inscribed in a circle, then the
measure of the angle equals one-half the
measure of its intercepted arc
If two inscribed angles of a circle or
congruent circles intercept congruent
arcs or the same arc, then the angles
are congruent
If an inscribed angle of a circle
intercepts a semicircle, then the
angle is a right angle
If a quadrilateral is inscribed in a
circle, then its opposite angles are
supplementary
1.
2.
3. If mLM=120, mMN=45, and
mNQ=105, find the numbered angles.
< 1= 22.5
< 2 = 60
< 3 = 45
< 4 = 22.5
< 5 = 112.5
4. If <2= 3a + 2 and < 3= 12 a – 2, find
the measures of the numbered angles
m1 = 45, m4 = 45
<2 = 20, < 3= 70
5. If mW = 74 and mZ = 112, find
mY and mX.
68= mX
106 = mY
10-5 Tangents
• I. A line is TANGENT to a circle if it intersects
the circle in EXACTLY ONE point. This point
is called the POINT OF TANGENCY.
If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the
point of tangency.
II. Common External Tangents & Common
Internal Tangents
A line or line segment that is tangent to two
circles in the same plane is called a common
tangent
COMMON
EXTERIOR
COMMON
INTERIOR
If two segments from the same exterior
point are tangent
to the circle, then they are congruent
Examples
• 1. AZ is tangent to O at point Z. Find x.
x
14
17.5
2. Determine whether BC is
tangent to A.
22
16.5
10
3. Determine whether is tangent
to H.
24
18
12
4. Solve for x.
3x-5
35 – 2 x
5. Triangle SCW is circumscribed
about A. Find the
perimeter of SCW if WT =
0.5(BC).
15
26
6. Find x so that the segment is a
tangent.
8
x
10-6 Secants and
Tangents

A SECANT is a line that
intersects a circle in
exactly two points.
Every secant
forms a chord
A secant that goes through the
center of the circle forms a
diameter.
If a secant and a tangent intersect at
the point of tangency, then the
measure of each angle formed is onehalf the measure of its intercepted arc
If two secants intersect in the
interior of a circle, then the measure
of an angle formed is one-half the
sum of the measures of the arcs
intercepted by the angle and its
vertical angle.
If two secants, a secant and a
tangent, or two tangents intersect in
the exterior of a circle, then the
measure of the angle formed is onehalf the positive difference of the
measures of the intercepted arcs.
(
) / 2 = BVA
(Big Arc - Little Arc)
divided by 2
(
) / 2 = BVA
(Big Arc - Little Arc) divided by 2
5.
6.
7.
8.
9.
10.
11.
11.
12.
13.
14.
10-7 Special
Segments
Iftwo
chords intersect in a circle,
I. Chords
then the products of the measures
of the segments of the chords are
equal
(AO)(DO) = (BO)(CO).
If two secant segments are drawn to a
circle from an exterior point, then the
product of the measures of one secant
segment and its external secant
segment is equal to the product of the
measures of the other secant segment
and its external secant segment.
(EA)(EC) = (EB)(ED)
If a tangent segment and a secant
segment are drawn to a circle from an
exterior point, then the square of the
measures of the tangent segment is
equal to the product of the measures of
secant segment and its external secant
segment
(DC)2 = (DB)(DA)
EXAMPLES: SOLVE.
1.
2.
3.
2x
X+ 3
4
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
10-8 Equations of
Circles
I. Graph of a circle
• You can graph a
circle if you know:
• Its center point
(h,k)
• Its radius or
diameter
II. The equation
• (x - h)2 + (y - k)
2
= r2
– Where (h,k) is your center point
– And r is the radius
– Ex: (x - 3)2 + (y - 2) 2 = 42
or (x - 3)2 + (y - 2) 2 = 16
– So center is (3,2) and radius=4
Your turn
• Ex 1: name the
center and the
radius for:
• (x - 1)2 + (y -3) 2 =
25
Answer:
• Center (1,3)
• Radius 5 ( why? b/c 52 = 25 )
So, how do u write the
equation?
• Center ( -1, 2) radius = 7
• remember : (x - h)2 + (y - k)
– So
–
2
= r2
(x - -1)2 + (y - 2) 2 = 72
(x +1)2 + (y - 2) 2 = 49
Your turn
• Write the equation of a circle with
center (-5,-3) and diameter 16.
Answer
• (x +5)2 + (y +3)
64
2
=
III. What if u are just
given some points?
• Find an equation of the circle that has a
diameter with endpoints at
(6, 10) and (-2, 4).
• Step 1: Use the distance formula to find
how long the diameter is:
• √(-2-6)2 + (4-10) 2
• =√100= 10 so radius is 5
• Or just graph it an count how long it is!
• Step 2:
• Graph it so you can see the center, or
find the half way point like this:
– Take the x’s: 6+ -2 = 4 divide by 2=2
– The y’s : 10+4= 14 then divide by 2 = 7
– Center (2,7)
Finally!
• Write the equation:
(x -2)2 + (y -7) 2 = 25