Chapter 6 Test Review

Transcription

Chapter 6 Test - Review
x
115°
X = _____
The arc measure of a minor arc is the same as
the measure of the central angle, the angle
with its vertex at the center of the circle, and
sides passing through the endpoints of the arc.
115°
S
R
P
v
115°
Q
r
Line r is tangent to both circles.
65°
v = ____.
Tangent Conjecture
A tangent to a circle is perpendicular to the
radius drawn to the point of tangency.
The angles at the point of tangencies S and R are 90°. Therefore, the sum of
angle v + 115° = 180°, making v = 65°
Ray c is tangent to circle T. Find X
c
S
112° 68°
T
Linear Pair
Tangent Conjecture
A tangent to a circle is perpendicular to the
radius drawn to the point of tangency.
X
R
Solution 1:
Triangle Exterior Angle Conjecture
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the remote interior angles
The angle at the point of tangency S is equal to 90°.
Therefore: 112° = X + 90°
22° = X
Solution 2:
The angle at the point of tangency S is equal to 90°.
Linear Pair: 112°+ 68° = 180°
Therefore: 180° = X + 90° + 68°
22° = X
A
B
Find the value of X
C
Chord Central Angles Conjecture
If two chords in a circle are congruent, then they determine
two central angles that are congruent.
78° X
O
D
Another solution is that you have two congruent isosceles
triangles. The radii are congruent and the chords are congruent.
So by CPCTC, X = 78°
Find the value of X
290°
Chord Arcs Conjecture
If two chords in a circle are congruent, then their
intercepted arcs are congruent
The arcs have to add up to 360°.
So, the missing arc is 35° making the two arcs congruent.
O
X
Therefore, X = 78°
6 in.
35°
Therefore the chords are congruent. So , X = 6 in.
Find the value of x and y
O
x
y
68°
x
The arc measure of a minor arc is the same as the
measure of the central angle, the angle with its vertex
at the center of the circle, and sides passing through
the endpoints of the arc.
Therefore, if the arc measure is 68°, y = 68°.
Triangle Exterior Angle Conjecture
The measure of an exterior angle of a triangle is equal
to the sum of the measures of the remote interior
angles
Since the radii make the triangle an isosceles triangle
68° = X + X
68° = 2X
34° = X
If the mRST = 40°, what is the mRQT ?
R
80°
80°
S
40°
Q
T
Inscribed Angle Conjecture
The measure of an angle inscribed in a circle is one-half
the measure of the intercepted arc.
If the mRST = 40°,
then measure of the intercepted arc RT = 80°
The arc measure of a minor arc is the same as the
measure of the central angle.
Therefore, if the arc measure is 80°, mRQT = 80°.
If the mRST = 47°, what is the mRST ?
R
94°
S
47°
Q
T
If the mRST = 47°,
then measure of the intercepted arc RT = 94°
The arc measure of a minor arc is the same as the measure
of the central angle.
Therefore, if the arc measure is 94°, mRQT = 94° and the
reflex measure of the central angle = 266° making mRST = 266°
Find x, y and z.
z
K
L
88°
95°
102°
Q
x
J
y
M
Cyclic Quadrilateral Conjecture
The opposite angles of a cyclic quadrilateral
are supplementary.
Therefore, x = 180° - 88°
x = 92°
And y = 180° - 95°
y = 85°
Therefore, if x = 92°, the mKLM = 184°
And z = 184° - 102°
z = 82°
KP and RA are parallel. Find mKAR
K
Parallel Lines Intercepted Arcs Conjecture
Parallel lines intercept congruent arcs on a circle.
Q
P
84°
A
R
Therefore, if mAP = 84°, then mKR = 84°
Therefore, the mKAR = 42°
If C = 14 cm, find d.
C= d
14 cm = d
14 cm = d
If d = 12 m, find C. Leave the answer in terms of .
C= d
C= (12m)
C= 12 m
What is the approximate circumference of circle Q?
Round to the nearest whole unit.
Q
C= 2r
C= 2(45) in.
C= 90 in.
C  90(3.14) in.
C  282.7 in.
C  283 in.
The circumference of a circle is 35 meters.
What is the approximate radius of the circle? Round to nearest 0.01 unit.
C= 2r 35 m = 2r
35 m  2(3.14)r
35 m  6.28r
𝟑𝟓 𝐦
r
𝟔.𝟐𝟖
5.57 m  r
What fraction of the circle is JK.
J
The arc measure of a minor arc is the same as
the measure of the central angle.
Q
20 cm K
Therefore mJK = 90°.
and,
𝟗𝟎°
𝟑𝟔𝟎°
=
𝟗
𝟑𝟔
=
𝟏
𝟒
(𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑐)
The length of CPU is ____.
Arc Length =
360
C
𝟗𝟎°
𝟗
𝟏
The ratio of CU =
= =
90°
𝟑𝟔𝟎°
Q
45 cm U
𝟑𝟔
So the ratio of CPU is:
𝟒
𝟑
𝟒
𝟑
𝟒
𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝑪𝑷𝑼 C
P
𝟑
𝟒
𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝑪𝑷𝑼 (283) cm
𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝐂𝐏𝐔 212.25 cm
C
C= 2r
C= 2(45 cm)
C= 90 cm
C  90(3.14)
C  282.7 cm
C  283 cm
The distance around the equator of Mars is about 21,344 km.
What is the equatorial diameter of Mars? (Round to the nearest whole unit)
C= d
21,344 km = d
21,344
km = d

21,344
km  d
3.14
6797.45 km  d
6797 km  d
Given the mX=110° and mY=110° , find the mZ
X
W
Y
are supplementary.
Therefore, Z = 180° - 110°
Z = 70°
Z
Find the measure of the unknown angles.
N
O
86° b
a 85°
are supplementary.
c
M
d = 86°, b = 94°
c = 85°, a = 95°
d
P
Note that since a is supplementary with 85° then c = 85°.
And since b is supplementary with 86° then d = 86°.
Amanda ran 9 times around the circular track that has a radius of 57 meters.
What is the approximate distance that she ran? (Round to the nearest meter)
Run 9 times around the track.
57 m
Find the Circumference:
C= 2r
C = 2(57 m)
C = 114 m
C  114(3.14) m
C  358.1 m
C  358 m
Find the Distance:
d = (C)(number of times around track)
d  (358)(9) m
d  3,222 m
Does the following description represent the arc length or arc measure?
The angle formed by two different radii in a circle.
Arc Measure
Central °
Angle
Does 95° represent arc measure? Explain why or why not.
The measure of an arc is measured in degrees, while the measure
of the arc length is measured in distance.
So, yes, 95° represents an arc measure.
𝟏
𝟑
Which of the following show of a circle.
J
𝟗𝟎°
𝟑𝟔𝟎°
L
J
=
120°
𝟏𝟐𝟎°
𝟑𝟔𝟎°
𝟏
𝟒
K
=
𝟏
𝟑
K
L
𝟏𝟖𝟎°
𝟑𝟔𝟎°
=
J
𝟏
𝟐
K
J
160°
M
𝟏𝟔𝟎°
𝟑𝟔𝟎°
L
L
M
=
𝟒
𝟗
K
What is the arc length if the radius is 15 meters?
Length of QD _____.
Q
8𝟎°
Arc Length =
D
40°
(𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄)
𝟑𝟔𝟎
𝟑𝟔𝟎
C = 2r
C=2 (15 m)
C=30 m
𝟖𝟎
𝟐
=
𝟑𝟔𝟎
𝟗
𝟐
Arc Length =
𝟗
C
C
𝟐
(30 m)
𝟗
𝟐
Arc Length = (10 m)
𝟑
𝟐𝟎
Arc Length =
m
𝟑
Arc Length =
Elizabeth watched a bug crawl through an arc of 18° along the rim of half a melon.
If the radius of the melon was 8 inches, how far did the bug crawl?
Arc Length =
𝟑𝟔𝟎
𝟑𝟔𝟎
8 in.
18°
𝟏𝟖
𝟑𝟔𝟎
𝟗
=
𝟏𝟖𝟎
=
C = 2r
C=2 (8 in)
C=16 in
C  16(3.14) in
C  50.27 in
𝟏
𝟐𝟎
Arc Length 
C
𝟏
𝟐𝟎
C
𝟏
Arc Length 
(50.24 in)
𝟐𝟎
Arc Length  2.512 in
Arc Length  2.512 in
In circle Q, PA and PB are tangents. Find mP
A
P
Q 100°
260°
The measure of an angle is the smallest amount of rotation
about the vertex from one ray to the other, measured in
degrees. According to this definition, the measure of an angle
can be any value between 0° and 180°. The largest amount of
rotation less than 360° between the two rays is called the
reflex measure of an angle.
260° is the reflex measure of Q.
B
Therefore m Q = 100°
Tangent Conjecture
A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
Since  A and  B are right angles, then  Q and  P are supplementary.
mQ + mP = 180°
100°+ mP = 180°
mP = 180° - 100°
mP = 80°
CD = EF
QM = 7 cm
QN = _____
D
Chord Distance to Center Conjecture
Two congruent chords in a circle are equidistant from the
center of the circle
M
F
Q
Therefore, QM = QN.
N
C
E
So, QN = 7 cm
If the mRST = 264°, what is the m RST ?
R
If the mRST = 264°, then m RT = 96°
96°
S
264°
48°
Q
T
The arc measure of a minor arc is the same as the measure
of the central angle.
Therefore, if the arc measure is 96°, mRQT = 96°
Therefore mRST = 48°,

Chapter 6 Test Review

Transcription

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