Chapter 6 Test - Review

115°

x 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.

Chapter 6 Test - Review

115°

P Q

R

S

Tangent Conjecture

A tangent to a circle is perpendicular to the

radius drawn to the point of tangency.

Line r is tangent to both circles. v = ____.

v 115°

The angles at the point of tangencies S and R are 90°. Therefore, the sum of angle v + 115° = 180°, making v = 65°

65° r

T

S

R

c

X 112°

Tangent Conjecture

A tangent to a circle is perpendicular to the

radius drawn to the point of tangency.

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

Linear Pair

68°

Solution 1: The angle at the point of tangency S is equal to 90°. Therefore: 112° = X + 90° 22° = X

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

Ray c is tangent to circle T. Find X

Chapter 6 Test - Review

O

Find the value of X

Chapter 6 Test - Review

A

B

C

D X 78°

Chord Central Angles Conjecture If two chords in a circle are congruent, then they determine two central angles that are congruent.

O

290°

35° 6 in. X

Find the value of X

The arcs have to add up to 360°. So, the missing arc is 35° making the two arcs congruent.

Chord Arcs Conjecture If two chords in a circle are congruent, then their intercepted arcs are congruent

Therefore, X = 78°

Another solution is that you have two congruent isosceles triangles. The radii are congruent and the chords are congruent. So by CPCTC, X = 78°

Therefore the chords are congruent. So , X = 6 in.

O

Find the value of x and y

Chapter 6 Test - Review

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°. x

y 68°

Since the radii make the triangle an isosceles triangle 68° = X + X 68° = 2X

34° = X

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

x

If the mRST = 40°, what is the mRQT ?

Chapter 6 Test - Review

80°

T

S

R

Q

Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc.

If the mRST = 40°, then measure of the intercepted arc RT = 80°

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°, mRQT = 80°.

40°

If the mRST = 47°, what is the mRST ?

Chapter 6 Test - Review

94°

T

S

R

Q

Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc.

If the mRST = 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°, mRQT = 94° and the

reflex measure of the central angle = 266° making mRST = 266°

47°

Find x, y and z.

Chapter 6 Test - Review

L K

J

Q

Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc.

95°

M

88°

102°

x y

z

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°

Chapter 6 Test - Review

P

K

R Q

A

84°

KP and RA are parallel. Find mKAR

Parallel Lines Intercepted Arcs Conjecture Parallel lines intercept congruent arcs on a circle.

Therefore, if mAP = 84°, then mKR = 84°

Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc.

Therefore, the mKAR = 42°

Chapter 6 Test - Review If C = 14 cm, find d.

14 cm = d 14 cm = d

C= d

If d = 12 m, find C. Leave the answer in terms of .

C= d

C= (12m) C= 12 m

Q

What is the approximate circumference of circle Q? Round to the nearest whole unit.

C= 2r C= 2(45) in. C= 90 in. C 90(3.14) in.

C 282.7 in. C 283 in.

Chapter 6 Test - Review The circumference of a circle is 35 meters. What is the approximate radius of the circle? Round to nearest 0.01 unit.

35 m = 2r 35 m 2(3.14)r 35 m 6.28r 𝟑𝟓 𝐦

𝟔.𝟐𝟖 r

5.57 m r

C= 2r

Q

What fraction of the circle is JK.

20 cm

J

K

The arc measure of a minor arc is the same as the measure of the central angle.

Therefore mJK = 90°.

and, 𝟗𝟎°

𝟑𝟔𝟎° =

𝟗

𝟑𝟔 =

𝟏

𝟒

Chapter 6 Test - Review

Q

The length of CPU is ____.

45 cm

C

U

= 𝟗𝟎°

𝟑𝟔𝟎° =

𝟗

𝟑𝟔 =

𝟏

𝟒

P

90°

Arc Length = (𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑐)

360 C

C= 2(45 cm) C= 90 cm C 90(3.14) C 282.7 cm C 283 cm 𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝑪𝑷𝑼

𝟑

𝟒 C

𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝑪𝑷𝑼𝟑

𝟒 (283) cm

𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝐂𝐏𝐔 212.25 cm

C= 2r

So the ratio of CPU is: 𝟑

𝟒

The ratio of CU

Chapter 6 Test - Review

6797.45 km d

6797 km d

C= d

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)

21,344 km = d

21,344

km = d

21,3443.14

km d

Chapter 6 Test - Review

Y W

X

Z

Given the mX=110° and mY=110° , find the mZ

Cyclic Quadrilateral Conjecture The opposite angles of a cyclic quadrilateral are supplementary.

Therefore, Z = 180° - 110° Z = 70°

85° 86°

M

O N

P

a b

c d

Find the measure of the unknown angles.

Cyclic Quadrilateral Conjecture The opposite angles of a cyclic quadrilateral are supplementary.

d = 86°, b = 94° c = 85°, a = 95°

Note that since a is supplementary with 85° then c = 85°. And since b is supplementary with 86° then d = 86°.

Chapter 6 Test - Review 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)

C = 2(57 m) C = 114 m C 114(3.14) m C 358.1 m C 358 m

C= 2r

57 m

Find the Circumference: Run 9 times around the track.

Find the Distance:

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.

° Central Angle

Arc Measure

d = (C)(number of times around track)

Chapter 6 Test - Review

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.

Which of the following show 𝟏

𝟑 of a circle.

J

L K

J

L

K

J

L

K

M

120°

J

L

K

M

160°

𝟗𝟎°

𝟑𝟔𝟎° =

𝟏

𝟒

𝟏𝟖𝟎°

𝟑𝟔𝟎° =

𝟏

𝟐

𝟏𝟐𝟎°

𝟑𝟔𝟎° =

𝟏

𝟑

𝟏𝟔𝟎°

𝟑𝟔𝟎° =

𝟒

𝟗

Chapter 6 Test - Review What is the arc length if the radius is 15 meters? Length of QD _____.

40° C = 2r

C=2 (15 m) C=30 m

Arc Length = (𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄)

𝟑𝟔𝟎 C

(𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄)

𝟑𝟔𝟎

Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. Q

D

𝟖𝟎

𝟑𝟔𝟎 =

𝟐

𝟗

Arc Length = 𝟐

𝟗 C

Arc Length = 𝟐

𝟗 (30 m)

Arc Length = 𝟐

𝟑 (10 m)

Arc Length = 𝟐𝟎

𝟑 m

8𝟎°

Chapter 6 Test - Review 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?

C = 2r

C=2 (8 in) C=16 in C 16(3.14) in C 50.27 in

Arc Length = (𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄)

𝟑𝟔𝟎 C

(𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄)

𝟑𝟔𝟎

𝟏𝟖

𝟑𝟔𝟎 =

𝟗

𝟏𝟖𝟎 =

𝟏

𝟐𝟎

Arc Length 𝟏

𝟐𝟎 C

Arc Length 𝟏

𝟐𝟎 (50.24 in)

Arc Length 2.512 in Arc Length 2.512 in

8 in.

18°

Chapter 6 Test - Review In circle Q, PA and PB are tangents. Find mP

B

Q

P

A

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.

Therefore m Q = 100°

Tangent Conjecture

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

100°

Since A and B are right angles, then Q and P are supplementary.

mQ + mP = 180°

100°+ mP = 180°

mP = 180° - 100°

mP = 80°

Chapter 6 Test - Review

CD = EF QM = 7 cm QN = _____

M

Q

N C

Chord Distance to Center Conjecture Two congruent chords in a circle are equidistant from the center of the circle

Therefore, QM = QN.

So, QN = 7 cm

D

E

F

Chapter 6 Test - Review

96°

T

S

R

Q

Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc.

Therefore mRST = 48°,

The arc measure of a minor arc is the same as the measure of the central angle. Therefore, if the arc measure is 96°, mRQT = 96°

48°

If the mRST = 264°, what is the m RST ?

264°

If the mRST = 264°, then m RT = 96°

Chapter 6 Test - Review