Download Topic 12 Theorems About Circles

Topic 12
TOPIC OVERVIEW
VOCABULARY
12-1Tangent Lines
12-2Chords and Arcs
12-3Inscribed Angles
12-4Angle Measures and Segment
Lengths
DIGITAL
Theorems About Circles
APPS
English/Spanish Vocabulary Audio Online:
EnglishSpanish
chord, p. 492cuerda
inscribed angle, p. 499
ángulo inscrito
intercepted arc, p. 499
arco interceptor
point of tangency, p. 486
punto de tangencia
secant, p. 505secante
tangent to a circle, p. 486
tangente de un círculo
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484
Topic 12 Theorems About Circles
3--Act Math
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485
12-1 Tangent Lines
TEKS FOCUS
VOCABULARY
•Point of tangency – the point where a circle and a tangent intersect
•Tangent to a circle – a line in the plane of the circle that intersects
TEKS (12)(A) Apply theorems about circles,
including relationships among angles, radii,
chords, tangents, and secants, to solve noncontextual problems.
the circle in exactly one point
•Analyze – closely examine objects, ideas, or relationships to learn
TEKS (1)(F) Analyze mathematical
relationships to connect and communicate
mathematical ideas.
more about their nature
Additional TEKS (1)(G), (6)(A), (9)(B)
ESSENTIAL UNDERSTANDING
A radius of a circle and the tangent that intersects the endpoint of the radius on the
circle have a special relationship.
Key Concept Tangent Lines
A
B
A tangent to a circle is a line in the plane of the
circle that intersects the circle in exactly one point.
The point where a circle and a tangent intersect
is the point of tangency.
3
BA is a tangent ray, and BA is a tangent segment.
Theorem 12-1
Theorem
If a line is tangent to a
circle, then the line is
perpendicular to the radius
at the point of tangency.
If . . .
< >
AB is tangent to } O at P
Then . . .
< >
AB # OP
A
A
P
P
O
B
O
B
For a proof of Theorem 12-1, see the Reference section on page 683.
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486
Lesson 12-1 Tangent Lines
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Theorem 12-2
Theorem
If a line in the plane of a
circle is perpendicular to
a radius at its endpoint on
the circle, then the line is
tangent to the circle.
If . . .
< >
AB # OP at P
Then . . .
< >
AB is tangent to } O
A
P
O
B
You will prove Theorem 12-2 in Exercise 19.
Theorem 12-3 hsm11gmse_1201_t05174
Theorem
If two tangent segments to
a circle share a common
endpoint outside the circle,
then the two segments are
congruent.
If . . .
Then . . .
BA and BC are tangent to } O
BA ≅ BC
A
B
O
C
You will prove Theorem 12-3 in Exercise 12.
hsm11gmse_1201_t05177
Problem 1
TEKS Process Standard (1)(F)
Finding Angle Measures
Multiple Choice ML and MN are tangent to } O. What is the value of x?
L
O 117
x
M
N
58
What kind of angle
is formed by a radius
and a tangent?
The angle formed is
a right angle, so the
measure is 90.
63
90
117
Since ML and MN are tangent to } O, ∠L and ∠N are right angles (Theorem 12-1).
LMNO is a quadrilateral. Sohsm11gmse_1201_t05167
the sum of the angle measures is 360.
m∠L + m∠M + m∠N + m∠O = 360
90 + m∠M + 90 + 117 = 360
297 + m∠M = 360
m∠M = 63 Substitute.
Simplify.
Solve.
The correct answer is B.
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Problem 2
Finding Distance
How does knowing
Earth’s radius help?
The radius forms a right
angle with a tangent line
from the observation
deck to the horizon. So
you can use two radii, the
tower’s height, and the
tangent to form a right
triangle.
STEM
Earth Science The CN Tower in Toronto, Canada, has
an observation deck 447 m above ground level. About
how far is it from the observation deck to the horizon?
Earth’s radius is about 6400 km.
Step 1 Make a sketch. The length 447 m is about 0.45 km.
T
447 m
0.45 km
E
C
6400 km
Not to scale
Step 2 Use the Pythagorean Theorem.
CT 2 = TE 2 + CE 2
hsm11gmse_1201_t07766
(6400 + 0.45)2 = TE 2 + 64002
(6400.45)2
=
TE 2
+
64002
Substitute.
Simplify.
40,965,760.2025 = TE 2 + 40,960,000
5760.2025 = TE 2
76 ≈ TE
Use a calculator.
Subtract 40,960,000 from each side.
Take the positive square root of each side.
The distance from the CN Tower to the horizon is about 76 km.
Problem 3
Finding a Radius
B
What is the radius of }C?
Why does the value x
appear on each side
of the equation?
The length of AC, the
hypotenuse, is the radius
plus 8, which is on the
left side of the equation.
On the right side of the
equation, the radius is
one side of the triangle.
488
AC 2 = AB2 + BC 2
(x +
8)2
=
122
+
x2
Pythagorean Theorem
Substitute.
x2 + 16x + 64 = 144 + x2 16x = 80
Subtract x 2 and 64 from each side.
x=5
Divide each side by 16.
The radius is 5.
Lesson 12-1 Tangent Lines
Simplify.
12
x
C
A
8
x
Problem 4
TEKS Process Standard (1)(G)
Identifying a Tangent
Is ML tangent to } N at L? Explain.
What information
does the diagram
give you?
•LMN is a triangle.
•NM = 25, LM = 24,
NL = 7
•NL is a radius.
25
N
7
The lengths
of the sides of
△LMN
M
24
L
To determine
ML is a tangent if ML # NL. Use the
whether ML is
Converse of the Pythagorean Theorem to
hsm11gmse_1201_t05175
tangent to } N
determine whether △LMN is a right triangle.
NL2 + ML2 ≟ NM 2
72 + 242 ≟ 252 625 = 625 Substitute.
Simplify.
By the Converse of the Pythagorean Theorem, △LMN is a right triangle with ML # NL.
So ML is tangent to }N at L because it is perpendicular to the radius at the point of
tangency (Theorem 12-2).
Problem 5
Circles Inscribed in Polygons
} O is inscribed in △ABC. What is the perimeter of △ABC?
How can you find the
length of BC?
Find the segments
congruent to BE and
EC. Then use segment
addition.
A 10 cm D 15 cm B
O
F
E
8 cm
C
AD = AF = 10 cm
BD = BE = 15 cm
CF = CE = 8 cm
Thm 12-3: Two segments tangent to a
circle from a point outside the circle are
congruent, so they have the same length.
hsm11gmse_1201_t05180
P = AB + BC + CA
Definition of perimeter
= AD + DB + BE + EC + CF + FA
Segment Addition Postulate
= 10 + 15 + 15 + 8 + 8 + 10
Substitute.
= 66
The perimeter is 66 cm.
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Lines that appear to be tangent are tangent. O is the center of each circle. What is
the value of x?
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
2.
3.
O x
x
O
43
60
STEM
60
O x
h
Apply Mathematics (1)(A) The circle at the right represents Earth. The radius of
d
Earth is about 6400 km. Find the distance d to the horizon that a person can see
r
on a clear
day from each of the following
heights h above Earth. Round your answer
hsm11gmse_1201_t05183
hsm11gmse_1201_t05182
hsm11gmse_1201_t05184 E r
to the nearest tenth of a kilometer.
4.
5 km
5.1 km
6.2500 m
Analyze Mathematical Relationships (1)(F) In each circle, what is the
value of x, to the nearest tenth?
7.
8.
x
14
x
10 cm
7 cm
x
x
O
x
P 15 in.
9.
O
x
hsm11gmse_1201_t05195
Q
9 in.
10
Each polygon circumscribes a circle. What is the perimeter of each polygon?
10.
16 cm
8 cm
1.9 in.
11.
hsm11gmse_1201_t05192
hsm11gmse_1201_t05193
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3.7 in.
3.4 in.
6 cm
3.6 in.
9 cm
A
B
12.Write a paragraph proof to prove Theorem 12-3.
Proof
hsm11gmse_1201_t05190
Given: BA and BC are tangent to } O at A and C, respectively.
O
hsm11gmse_1201_t05189
Prove: BA ≅ BC
C
13.a. A belt fits snugly around the two circular pulleys. CE is an
auxiliary line from E to BD. CE } BA. What type of
quadrilateral is ABCE ? Explain.
b.What is the length of CE?
B
14 in. C
35 in.
A
c.What is the distance between the centers of the pulleys to
the nearest tenth?
490
Lesson 12-1 Tangent Lines
8 in.
E
hsm11gmse_1201_t05202
D
hsm11gmse_1201_t07773
A
14.Explain Mathematical Ideas (1)(G) A nickel, a dime,
and a quarter are touching as shown. Tangents are
drawn from point A to both sides of each coin. What
can you conclude about the four tangent segments?
Explain.
15.Analyze Mathematical Relationships (1)(F) Leonardo da Vinci wrote, “When each of two
squares touches the same circle at four points, one is
double the other.” Explain why the statement is true.
16.Two circles that have one point in common are
tangent circles. Given any triangle, explain how to draw
three circles that are centered at each vertex of the triangle and
tangent to each other.
17.Given: BC is tangent to }A at D.
Proof DB ≅ DC
18.Given: }A and }B with common tangents
DF and CE
Proof
Prove: AB ≅ AC Prove: △GDC ∼ △GFE
D
A
A
B
D
E
G
C
B
F
C
19.Write an indirect proof of Theorem 12-2.
A
Proof
Given: AB # OP at P.
P
hsm11gmse_1201_t05203
Prove: AB is tangent to } O.
hsm11gmse_1201_t05204
O
B
TEXAS Test Practice
hsm11gmse_1201_t05174
Lines in }O that appear to be tangent are tangent. What is the value of x?
20.
21.
O 114 x
O
56
x
22.The perimeter of an equilateral triangle is 90 in. What is its area to the nearest
square inch?
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491
12-2 Chords and Arcs
TEKS FOCUS
VOCABULARY
•Chord – a segment whose endpoints are
TEKS (12)(A) Apply theorems about circles, including relationships
among angles, radii, chords, tangents, and secants, to solve noncontextual problems.
on a circle
TEKS (1)(C) Select tools, including real objects, manipulatives, paper
and pencil, and technology as appropriate, and techniques, including
mental math, estimation, and number sense as appropriate, to solve
problems.
•Number sense – the understanding of
what numbers mean and how they
are related
Additional TEKS (1)(A), (1)(G), (5)(A), (5)(C), (6)(A), (9)(B)
ESSENTIAL UNDERSTANDING
You can use information about congruent parts of a circle (or congruent circles) to
find information about other parts of the circle (or circles).
Theorem 12-4 and Its Converse
Theorem
Within a circle or in congruent circles, congruent
central angles have congruent arcs.
Converse
Within a circle or in congruent circles, congruent
arcs have congruent central angles.
B
A
C
O
D
¬ ¬
If ∠AOB ≅ ∠COD, then AB ≅ CD.
¬ ¬
If AB ≅ CD, then ∠AOB ≅ ∠COD.
You will prove Theorem hsm11gmse_1202_t08234
12-4 and its converse in Exercises 7 and 24.
Theorem 12-5 and Its Converse
Theorem
Within a circle or in congruent circles, congruent
central angles have congruent chords.
Converse
Within a circle or in congruent circles, congruent
chords have congruent central angles.
B
A
C
O
D
If ∠AOB ≅ ∠COD, then AB ≅ CD.
If AB ≅ CD, then ∠AOB ≅ ∠COD.
You will prove Theorem 12-5 and its converse in Exercises 8 and 25.
hsm11gmse_1202_t08235
492
Lesson 12-2 Chords and Arcs
Theorem 12-6 and Its Converse
Theorem
Within a circle or in congruent circles, congruent
chords have congruent arcs.
¬ ¬
If AB ≅ CD, then AB ≅ CD .
¬ ¬
If AB ≅ CD, then AB ≅ CD.
C
B
O
Converse
Within a circle or in congruent circles, congruent
arcs have congruent chords.
A
D
You will prove Theorem 12-6 and its converse in Exercises 9 and 26.
hsm11gmse_1202_t08236
Theorem 12-7 and Its Converse
Theorem
Within a circle or in congruent circles, chords
equidistant from the center or centers are congruent.
BC
E
A
Converse
Within a circle or in congruent circles, congruent
chords are equidistant from the center (or centers).
O
F
If OE = OF, then AB ≅ CD.
If AB ≅ CD, then OE = OF.
D
For a proof of Theorem 12-7, see the Reference section on page 683.
You will prove the converse of Theorem 12-7 in Exercise 27.
hsm11gmse_1202_t08242.ai
Theorem 12-8
Theorem
In a circle, if a diameter is
perpendicular to a chord,
then it bisects the chord and
its arc.
If . . .
AB is a diameter and AB # CD
Then . . .
¬ ¬
CE ≅ ED and CA ≅ AD
C
C
E
A
O
B
E
A
B
O
D
D
You will prove Theorem 12-8 in Exercise 10.
Theorem 12-9
Theorem
In a circle, if a diameter
bisects a chord (that is
not a diameter), then it is
perpendicular to the chord.
hsm11gmse_1202_t08238.ai hsm11gmse_1202_t08239.ai
If . . .
AB is a diameter and CE ≅ ED
Then . . .
AB # CD
C
C
A
E
O
D
B
A
E
O
B
D
For a proof of Theorem 12-9, see the Reference section on page 683.
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Theorem 12-10
Theorem
In a circle, the perpendicular
bisector of a chord contains
the center of the circle.
If . . .
AB is the perpendicular bisector
of chord CD
C
A
Then . . .
AB contains the center of }O
C
A
B
B
O
D
D
You will prove Theorem 12-10 in Exercise 11.
hsm11gmse_1202_t08244.ai
Problem 1
hsm11gmse_1202_t08245.ai
Using Congruent Chords
Why is it important
that the circles are
congruent?
The central angles will
not be congruent unless
the circles are congruent.
B
In the diagram, }O @ }P. Given that BC @ DF, what can
you conclude?
∠O ≅ ∠P because, within congruent circles, congruent chords have
¬
¬
congruent central angles (conv. of Thm. 12-5). BC ≅ DF because,
within congruent circles, congruent chords have congruent
arcs (Thm. 12-6).
O
D
P
C
F
hsm11gmse_1202_t06826.ai
Problem 2
Finding the Length of a Chord
What is the length of RS in }O?
S
O 9
The diagram indicates that
PQ = QR = 12.5 and PR and RS
are both 9 units from the center.
P
9
Q
R
PR ≅ RS, since they are the same distance from the
center of the circle. So finding PR gives the length of RS.
The length of chord RS
494
12.5
hsm11gmse_1202_t06828
PQ = QR = 12.5
Given in the diagram
PQ + QR = PR Segment Addition Postulate
12.5 + 12.5 = PR 25 = PR RS = PR Chords equidistant from the center of a circle are
congruent (Theorem 12-7).
RS = 25 Lesson 12-2 Chords and Arcs
Substitute.
Add.
Substitute.
2 5
. . . . . . .
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9
Problem 3
TEKS Process Standard (1)(C)
Investigating Special Segments of Circles
Choose from a variety of tools (such as a compass, straightedge, geometry
software, and pencil and paper) to investigate the perpendicular bisectors of
chords of a circle. Draw a circle with two chords that are not diameters. Construct
the perpendicular bisectors of the chords. Then make a conjecture about the
perpendicular bisector of a chord.
You can construct perpendicular bisectors using paper, a pencil, a compass, and a
straightedge. You can draw the circle and chords using a compass and straightedge,
and construct the perpendicular bisectors using paper folding.
Step 1 Use a compass to draw a circle on a piece of paper.
Step 2Use a straightedge to draw two chords that are not
diameters.
Why should you
draw more than two
chords?
The more examples you
can find to support your
conjecture, the stronger
your conjecture becomes.
Step 3Fold the perpendicular bisector for each chord. The
perpendicular bisectors appear to intersect at the center
of the circle.
Step 4Draw a third chord and construct its perpendicular bisector.
The third perpendicular bisector also appears to intersect the
other two.
Conjecture: The perpendicular bisector of any chord of a circle goes through the
center of the circle.
Problem 4
TEKS Process Standard (1)(A)
Using Diameters and Chords
How does the
construction help find
the center?
The perpendicular
bisectors contain
diameters of the circle.
Two diameters intersect
at the circle’s center.
Archaeology An archaeologist found pieces of a jar. She wants
to find the radius of the rim of the jar to help guide her as she
reassembles the pieces. What is the radius of the rim?
Step 1 Trace a piece of
the rim. Draw two chords
and construct perpendicular
bisectors.
Step 2 The center is
the intersection of the
perpendicular bisectors.
Use the center to find
the radius.
0 1 2 3 4 5 6
inch
The radius is 4 in.
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Problem 5
Finding Measures in a Circle
Find two sides of a right
triangle. The third side
either is the answer or
leads to an answer.
Algebra What is the value of each variable to the nearest tenth?
LN = 12(14) = 7
A diameter # to a chord bisects the
chord (Theorem 12-8).
A r
L
K
3 cm
N
r 2 = 32 + 72 M
r ≈ 7.6
14 cm
E
B A
Use the Pythagorean Theorem.
Find the positive square root of each side.
BC # AF A diameter that bisects a chord that
is not a diameter is # to the
chord (Theorem 12-9).
15
11
hsm11gmse_1202_t06831
B
C y
BA = BE = 15 Draw an auxiliary BA. The auxiliary BA ≅ BE
because they are radii of the same circle.
F
y 2 + 112 = 152 y 2 = 104 11
Use the Pythagorean Theorem.
Solve for y 2.
y ≈ 10.2
Find the positive square root of each side.
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
In Exercises 1 and 2, the circles are congruent. What can you conclude?
1.
B
2.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
C
A
X
Y
E
Z
M
T
H
F
G
L
J
K
N
Find the value of x.
3.
4.
5.
hsm11gmse_1202_t06834
O
7
5
x
5
15
3.5
x
x
hsm11gmse_1202_t06835.ai
5
8
K
G
6.
Justify Mathematical Arguments (1)(G) In the diagram
at the right, GH and KM are perpendicular bisectors of
the chords they intersect. What can you conclude
hsm11gmse_1202_t06841.ai
hsm11gmse_1202_t06836.ai
about the center of the circle? Justify
your answer.
hsm11gmse_1202_t06840.ai
M
H
496
Lesson 12-2 Chords and Arcs
hsm11gmse_1202_t08251
7.Prove Theorem 12-4.
Proof
Given: } O with ∠AOB ≅ ∠COD
¬ ¬
Prove: AB ≅ CD B
A
Given: }O with ∠AOB ≅ ∠COD
Prove: AB ≅ CD
A
C
O
8.Prove Theorem 12-5.
Proof
O
D
9.Prove Theorem 12-6.
Proof
B
Proof
hsm11gmse_1202_t08252
Given: / is the # bisector of WY.
Prove: / contains the center of }X. D
10.Prove Theorem 12-8.
Proof
11.Prove Theorem 12-10.
C
hsm11gmse_1202_t06853.ai
12.Given: }A with CE # BD
Proof
¬ ¬
Prove: BC ≅ DC
C
X
W
Z
A
Y
B
E
F
D
13.a.Select Tools to Solve Problems (1)(C) Select a tool, such as compass and
straightedge or geometry software, to construct two chords in a circle such that
the chords are perpendicular and have a common endpoint. Draw the line
hsm11gmse_1202_t06861.ai
hsm11gmse_1202_t06863.ai
segment that joins the other two endpoints.
b.In the same circle, repeat part (a) several times, changing the lengths of the
perpendicular chords each time. Then make a conjecture about the line segment
that joins the endpoints of two perpendicular chords with a common endpoint.
14.Apply Mathematics (1)(A) You are building a circular patio table. You have to drill
a hole through the center of the tabletop for an umbrella. How can you find the center?
}A and }B are congruent. CD is a chord of both circles.
15.If AB = 8 in. and CD = 6 in., how long is a radius?
16.If AB = 24 cm and a radius = 13 cm, how long is CD?
17.If a radius = 13 ft and CD = 24 ft, how long is AB?
C
A
B
D
18.In the figure at the right, sphere O with radius 13 cm is intersected by a plane
5 cm from center O. Find the radius of the cross section }A.
19.A plane intersects a sphere that has radius 10 in., forming the cross section }B
5 cm O 13 cm
with radius 8 in. How far is the plane from the center of the hsm11gmse_1202_t06858.ai
sphere?
20.Connect Mathematical Ideas (1)(F) Two concentric circles have radii of
4 cm and 8 cm. A segment tangent to the smaller circle is a chord of the larger
circle. What is the length of the segment to the nearest tenth?
21.Display Mathematical Ideas (1)(G) Use Theorem 12-5 to construct a regular
octagon.
A
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497
22.In the diagram at the right, the endpoints of the chord are the
points where the line x = 2 intersects the circle x2 + y 2 = 25.
What is the length of the chord? Round your answer to the
nearest tenth.
23.Explain Mathematical Ideas (1)(G) Theorems 12-4 and 12-5
both begin with the phrase “within a circle or in congruent
circles.” Explain why the word congruent is essential for both
theorems.
y
x2
3
x
O
3
3
3
Prove each of the following.
Proof
24.Converse of Theorem 12-4: Within a circle or in congruent
circles, congruent arcs have congruent central angles.
25.Converse of Theorem 12-5: Within a circle or in congruent circles, congruent
chords have congruent central angles.
hsm11gmse_1202_t06859.ai
26.Converse of Theorem 12-6: Within a circle or in congruent circles, congruent arcs
have congruent chords.
27.Converse of Theorem 12-7: Within a circle or congruent circles, congruent chords
are equidistant from the center (or centers).
28.If two circles are concentric and a chord of the larger circle is tangent to the
Proof smaller circle, prove that the point of tangency is the midpoint of the chord.
C
A
29.Analyze Mathematical Relationships (1)(F) In }O, AB is a diameter of the
circle and AB # CD. What conclusions can you make?
TEXAS Test Practice
hsm11gmse_1202_t06842.ai
A.9.0 cm
C.18.0 cm
B.9.6 cm
D.19.2 cm
31.From the top of a building you look down at an object on the ground. Your eyes are
50 ft above the ground, and the angle of depression is 50°. Which distance is the
best estimate of how far the object is from the base of the building?
F.
42 ft
H.65 ft
G.60 ft
J.78 ft
32.A bicycle tire has a diameter of 17 in. How many revolutions of the tire are
necessary to travel 800 ft? Show your work.
Lesson 12-2 Chords and Arcs
B
D
30.The diameter of a circle is 25 cm and a chord of the same circle is 16 cm. To the
nearest tenth, what is the distance of the chord from the center of the circle?
498
E
O
12-3 Inscribed Angles
TEKS FOCUS
VOCABULARY
TEKS (12)(A) Apply theorems about circles,
including relationships among angles, radii,
chords, tangents, and secants, to solve noncontextual problems.
TEKS (1)(G) Display, explain, and justify
mathematical ideas and arguments using precise
mathematical language in written or oral
communication.
•Inscribed angle – an angle whose vertex is on the circle and
whose sides are chords of the circle
•Justify – explain with logical reasoning. You can justify a
mathematical argument.
•Argument – a set of statements put forth to show the truth or
falsehood of a mathematical claim
Additional TEKS (1)(D), (5)(A)
ESSENTIAL UNDERSTANDING
Angles formed by intersecting lines have a special relationship to the arcs the intersecting
lines intercept. In this lesson, you will study arcs formed by inscribed angles.
Theorem 12-11 Inscribed Angle Theorem
The measure of an inscribed angle is half the measure
of its intercepted arc.
¬
m∠B = 12 mAC
A
B
C
You will prove Theorem 12-11 in Exericises 9 and 10.
Corollaries to Theorem 12-11: hsm11gmse_1203_t06876
The Inscribed Angle Theorem
Corollary 1
Two inscribed angles that
intercept the same arc are
congruent.
Corollary 2
An angle inscribed in a
semicircle is a right angle.
Corollary 3
The opposite angles of a
quadrilateral inscribed in a
circle are supplementary.
B
A
A
B
D
C
C
You will prove these corollaries in Exercises 14–16.
hsm11gmse_1203_t08254.ai
hsm11gmse_1203_t08255.ai
hsm11gmse_1203_t08256.ai
PearsonTEXAS.com
499
Theorem 12-12
The measure of an angle formed by a tangent and a chord is half the measure of the
intercepted arc.
B
D
B
D
C
¬
m∠C = 12 m BDC
C
You will prove Theorem 12-12 in Exercise 17.
hsm11gmse_1203_t06899.ai
Problem 1
Using the Inscribed Angle Theorem
Which variable should
you solve for first?
You know the inscribed
¬
angle that intercepts PT ,
which has the measure a.
You need a to find b. So
find a first.
What are the values of a and b?
¬
m∠PQT = 12 m PT
Inscribed Angle Theorem
Substitute.
60 = 12 a
¬
m∠PRS = 12 m PS
¬
¬
m∠PRS = 12 (m PT + m TS )
120 = a
P
a
T
Q 60
Multiply each side by 2.
b
30
S
R
Inscribed Angle Theorem
Arc Addition Postulate
b = 12 (120 + 30)
Substitute.
b = 75
Simplify.
hsm11gmse_1203_t06881
Problem 2
TEKS Process Standard (1)(D)
Using Corollaries to Find Angle Measures
What is the measure of each numbered angle?
Is there too much
information?
Each diagram has more
information than you
need. Focus on what you
need to find.
A 40
1
500
70
B 70
∠1 is inscribed in a semicircle.
By Corollary 2, ∠1 is a right angle, so
m∠1 = 90.
hsm11gmse_1203_t06892
Lesson 12-3 Inscribed Angles
2
38
∠2 and the 38° angle intercept the
same arc. By Corollary 1, the angles
are congruent, so m∠2 = 38.
hsm11gmse_1203_t06893
Problem 3
TEKS Process Standard (1)(G)
Using Arc Measure
< >
¬
In the diagram, SR is tangent to the circle at Q. If mPMQ = 212,
what is mjPQR?
S
Q
< >
M
• SR is tangent to the circle at Q
¬
•m PMQ = 212
R
P
m∠PQS + m∠PQR = 180. So
¬
first find m∠PQS using PMQ .
m∠PQR
NLINE
HO
ME
RK
O
How can you check
the answer?
One way is to use
m∠PQR to find
¬
m PQ . Confirm that
¬
¬
m PQ + m PMQ = 360.
WO
1 ¬
2 mPMQ = m∠PQS
1
2 (212) = m∠PQS
106 = m∠PQS
The measure of an ∠ formed by a hsm11gmse_1203_t08257.ai
tangent and a
chord is 12 the measure of the intercepted arc.
Substitute.
Simplify.
m∠PQS + m∠PQR = 180
Linear Pair Postulate
106 + m∠PQR = 180
Substitute.
Simplify.
m∠PQR = 74
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Find the value of each variable. For each circle, the dot represents the center.
95
1.
2.
3.
4.
a
For additional support when
completing your homework,
go to PearsonTEXAS.com.
c
116
p
25
b
b
a
c
q
58
a
Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Find the value of each variable. Lines that appear to be tangent are tangent.
246
5.
6.
7.
hsm11gmse_1203_t06913
hsm11gmse_1203_t06915
hsm11gmse_1203_t06916
hsm11gmse_1203_t06906
y
w
x
230
115
e
f
8.
Justify Mathematical Arguments (1)(G) Can a rhombus that is not a square be
inscribed in a circle? Justify your answer.
hsm11gmse_1203_t06919.ai
hsm11gmse_1203_t06918.ai
hsm11gmse_1203_t06917.ai
PearsonTEXAS.com
501
To prove Theorem 12-11, there are three cases to consider. In Case I, the
center of the circle is on a side of the inscribed angle. In Case II, the center
is inside the inscribed angle. In Case III, the center is outside the inscribed
angle. Below is a proof of Case I.
Proof Given: }O with inscribed j B and diameter BC
¬
Prove: mjB = 12 mAC
C
A
O
Draw radius OA to form isosceles △AOB with OA = OB. So
B
mjA = mjB by the Isosceles Triangle Theorem. Then, by the Triangle
¬
Exterior Angle Theorem, mjAOC = mjA + mjB. So mAC = mjAOC
¬
since j AOC is a central angle. This means that mAC = mjA + mjB, and
¬
¬
therefore mAC = 2mjB. By the Division Property of Equality, mjB = 12 mAC .
hsm11gmse_1203_t06880
Write a proof of Case II and Case III for Exercises 9 and 10.
9.Inscribed Angle Theorem, Case II
Proof
Given: }O with inscribed ∠ABC
¬
Prove: m∠ABC = 12 mAC 10.Inscribed Angle Theorem, Case III
Proof
Given: }S with inscribed ∠PQR
¬
Prove: m∠PQR = 12 m PR
P
R
T
C
P
A
S
O
Q
B
11.Explain Mathematical Ideas (1)(G) The director of a
telecast wants the option of showing the same scene
from three different views.
hsm11gmse_1203_t06934.ai
hsm11gmse_1203_t06935.ai
a.Explain why cameras in the positions shown in the
diagram will transmit the same scene.
b.Will the scenes look the same when the director
views them on the control room monitors?
Explain.
Scene
Find the value of each variable. For each circle, the dot represents
the center.
12.
13.
c
44
b
Lesson 12-3 Inscribed
Angles
hsm11gmse_1203_t06921.ai
Camera
3
c
a
160
e
502
Camera
2
d
120
b
a
Camera
1
56
hsm11gmse_1203_t06929.ai
hsm11gmse_1203_t06922.ai
Write a proof for Exercises 14–17.
14.Inscribed Angle Theorem, Corollary 1 15.Inscribed Angle Theorem, Corollary 2
Proof
Proof
¬
Given: }O, ∠A intercepts BC , Given: }O with ∠CAB inscribed
¬
∠D intercepts BC .
in a semicircle
Prove: ∠A ≅ ∠D
Prove: ∠CAB is a right angle.
B
C
A
D
O
B
O
C
D
A
16.Inscribed Angle Theorem, Corollary 3
Proof
17.Theorem 12-12
Proof
Given: Quadrilateral ABCD Given: GH and tangent /
inscribed in }Ohsm11gmse_1203_t06938.ai
intersecting }E at H
hsm11gmse_1203_t06937.ai
¬
Prove: ∠A and ∠C are supplementary. Prove: m∠GHI = 12 m GFH
∠B and ∠D are supplementary.
D
A
B
F
G
I
E
O
C
H
TEXAS Test Practice
hsm11gmse_1203_t06939.ai
hsm11gmse_1203_t06940.ai
For Exercises 18 and 19, what is the value of each variable in }O?
18.
A.25
O
130
x
B.35
C.45
D.65
19.
y
O
60 F.
20
G.30
H.50
J.60
20.Is the following proof valid? If not, explain why, and then write a valid proof.
B
C
hsm11gmse_1203_t06942.ai
∠ABC
Given: Quadrilateral ABCD, ∠A ≅ ∠C, BD bisects
hsm11gmse_1203_t06941.ai
Prove: ∠ADB ≅ ∠CDB
BD ≅ BD by the Reflexive Property. Since BD bisects ∠ABC, it also
bisects ∠ADC. So ∠ADB ≅ ∠CDB.
A
D
hsm11gmse_1203_t08261
503
PearsonTEXAS.com
Technology Lab
Use With Lesson 12-4
Exploring Chords and Secants
teks (5)(A), (1)(E)
1
Construct }A and two chords BC and DE that intersect at F.
E
C
1.Measure BF, FC, EF, and FD.
2.Use the calculator program of your software to find BF
# FC and EF # FD.
F
A
B
3.Manipulate the lines. What pattern do you observe in the products?
4.What appears to be true for two intersecting chords?
D
2
A secant is a line that intersects a circle in two points. A secant segment is
a segment that contains a chord of the circle and has only one endpoint
<
>
<
>
Ghsm11gmse_1204a_t06963.ai
outside the circle. Construct a new circle and two secants DG and DE
that intersect outside the circle at point D. Label the intersections with
the circle as shown.
F
A
5.Measure DG, DF, DE, and DB.
6.Calculate the products DG # DF and DE # DB.
D
B
E
7.Manipulate the lines. What pattern do you observe in the products?
8.What appears to be true for two intersecting secants?
hsm11gmse_1204a_t06964.ai
3
Construct }A with tangent DG perpendicular to radius AG and secant DE
that intersects the circle at B and E.
9.Measure DG, DE, and DB.
10. Calculate the products (DG)2 and DE
G
# DB.
A
11. Manipulate the lines. What pattern do you observe in the products?
12. What appears to be true for the tangent segment and secant segment?
D
B
E
hsm11gmse_1204a_t06965.ai
504
Technology Lab Exploring Chords and Secants
12-4 Angle Measures and Segment Lengths
TEKS FOCUS
VOCABULARY
TEKS (12)(A) Apply theorems about circles, including relationships among
angles, radii, chords, tangents, and secants, to solve non-contextual problems.
•Secant – a line that intersects a
TEKS (1)(C) Select tools, including real objects, manipulatives, paper and
pencil, and technology as appropriate, and techniques, including mental math,
estimation, and number sense as appropriate, to solve problems.
•Number sense – the understanding
circle at two points
of what numbers mean and how
they are related
Additional TEKS (1)(D), (1)(F), (5)(A), (6)(A), (6)(D)
ESSENTIAL UNDERSTANDING
• Angles formed by intersecting lines have a special
• There is a special relationship between two intersecting
relationship to the related arcs formed when the lines
intersect a circle.
chords, two intersecting secants, or a secant that intersects a tangent. This relationship allows you to find the lengths of unknown segments.
Theorem 12-13
The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs.
m∠1 = 12(x + y)
x
y
1
For a proof of Theorem 12-13, see the Reference section on page 683.
Theorem 12-14
hsm11gmse_1204_t06968 .ai
The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs.
x°
y°
1
I: angle formed by two secants
x°
y°
x°
1
II: angle formed by a
secant and a tangent
y°
1
III: angle formed by
two tangents
m∠1 = 12(x - y)
You will prove Theorem 12-14 in Exercises 24 and 25.
PearsonTEXAS.com
505
Theorem 12-15
For a given point and circle, the product of the lengths of the two segments from
the point to the circle is constant along any line through the point and the circle.
I. II.
III.
a
d P
a
c
x
w
b
y
t
P
P
y
z
z
# b = c # d
(w + x)w = ( y + z)y
( y + z) y = t 2
You will prove Theorem 12-13 in Exercises 13 and 14.
hsm11gmse_1204_t06982hsm11gmse_1204_t06983
hsm11gmse_1204_t06984
Problem 1
TEKS Process Standard (1)(C)
Investigating Special Angles of Circles
Choose from a variety of tools (such as a protractor, a compass, or geometry
software) to investigate angles formed by chords intersecting inside a circle.
Explain why you chose that tool. Construct a circle and several pairs of
intersecting chords. Then make a conjecture about the measure of an angle
formed by two chords intersecting inside a circle.
Geometry software could help you investigate angles created by chords. You can create
a circle and two chords and then drag the chords around the circle to investigate angle
measures and make a conjecture.
Which angles are
formed by the
intersection of
the chords in the
diagram?
∠CFD, ∠EFB, ∠CFE, and
∠DFB
Step 1Draw a circle and two chords that intersect in the
interior of the circle.
Step 2Measure angles formed by the intersection of the
chords and their intercepted arcs. Record your
findings in the table below.
E
C
F
A
Step 3Drag the chords to form different angles. Record
your findings in a table.
Vertical angle measure
65°
41°
81°
Intercepted arc
44°
59°
30°
Intercepted arc
86°
23°
132°
130°
82°
162°
Sum of arc measures
D
hsm11gmse_1204a_t06963.ai
Conjecture: The measure of an angle formed by two chords that intersect inside a
circle is half the sum of the measures of its intercepted arcs.
506
B
Lesson 12-4 Angle Measures and Segment Lengths
Problem 2
Finding Angle Measures
Algebra What is the value of each variable?
A Remember to add
arc measures for arcs
intercepted by lines
that intersect inside
a circle and subtract
arc measures for arcs
intercepted by lines that
intersect outside a circle.
B 95
x
46
z
90
20
x = 12(46 + 90)
Theorem 12-13
20 = 12(95 - z) Theorem 12-14
x = 68
Simplify.
40 = 95 - z
hsm11gmse_1204_t06975.ai
z = 55
Solve for z.
hsm11gmse_1204_t06974.ai
Multiply each side by 2.
Problem 3
Finding an Arc Measure
Satellite A satellite in a geostationary orbit
above Earth’s equator has a viewing angle
of Earth formed by the two tangents to the
equator. The viewing angle is about 17.5°.
What is the measure of the arc of Earth
that is viewed from the satellite?
A
Earth
E
How can you
represent the
measures of the arcs?
The sum of the measures
of the arcs is 360°. If the
measure of one arc is x,
then the measure of the
other is 360 - x.
Satellite
17.5
B
¬
Let mAB = x.
¬
Then m AEB = 360 - x.
hsm11gmse_1204_t08584.ai
¬
¬
17.5 = 12(m AEB - mAB ) Theorem 12-14
17.5 = 12[(360 - x) - x]
Substitute.
17.5 = 12(360 - 2x)
Simplify.
17.5 = 180 - x
Distributive Property
x = 162.5
Solve for x.
A 162.5° arc can be viewed from the satellite.
PearsonTEXAS.com
507
Problem 4
TEKS Process Standard (1)(F)
Finding Segment Lengths
How can you identify
the segments needed
to use Theorem 12-15?
Find where segments
intersect each other
relative to the circle. The
lengths of segments that
are part of one line will
be on the same side of
an equation.
Algebra Find the value of the variable in }N.
A N
84 = 49 + 7y 35 = 7y
5=y
HO
RK
O
ME
WO
Thm. 12-15, Case II
8
16
(8 + 16)8 = z 2 Thm. 12-15, Case III
192 = z 2 Distributive Property
hsm11gmse_1204_t06986
NLINE
N
(6 + 8)6 = (7 + y)7
z
y
7
B 8
6
Simplify.
hsm11gmse_1204_t06987
13.9 ≈ z
Solve for z.
Solve for y.
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Find the value of each variable using the given chord, secant, and tangent
lengths. If the answer is not a whole number, round to the nearest tenth.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
15
26
2.
x
20
3.
20
11
13
x
y
5
7
15
c
In the diagram at the right, CA and CB are tangents to }O. Write an expression
for each arc or angle in terms of the given variable.
¬
¬
4.
m ADB using x
5.m∠C using x
6.m AB using y
A
D
O
y
x
C
hsm11gmse_1204_t06999
hsm11gmse_1204_t07002B
hsm11gmse_1204_t07000
Find the value of each variable.
7.
8.
160
53
60
y
x 68
x
9.
y
x
hsm11gmse_1204_t07004
70
10.Analyze Mathematical Relationships (1)(F) You focus
your camera on a circular fountain. Your camera is at the
hsm11gmse_1204_t06997
vertex
of the angle formed byhsm11gmse_1204_t06996
tangents to the fountain.
hsm11gmse_1204_t06992
You estimate that this angle is 40°. What is the measure of
the arc of the circular basin of the fountain that will be in
the photograph?
508
Lesson 12-4 Angle Measures and Segment Lengths
STEM 11.Apply Mathematics (1)(A) Arc
The basis for the design x
of the Wankel rotary engine
8 in.
is an equilateral triangle.
Each side of the triangle is
a chord to an arc of a circle.
The opposite vertex of the
triangle is the center of the
circle that forms the arc. In
Center
the diagram at the right, each
side of the equilateral triangle is 8 in. long.
Wankel engine
a.Use what you know about equilateral triangles
and find the value of x.
hsm11gmse_1204_t06991
b.Justify Mathematical Arguments (1)(G) Copy the diagram and complete
the circle with the given center. Then use Theorem 12-15 to find the value
of x. Show that your answers to parts (a) and (b) are equal.
12.A circle is inscribed in a quadrilateral whose four angles have measures 85, 76,
94, and 105. Find the measures of the four arcs between consecutive points of
tangency.
Recall that there are three cases to prove Theorem 12-15. In Case I, the product of
the chord segments are equal. In Case II, the products of the secants and their outer
segments are equal. In Case III, the product of a secant and its outer segment
equals the square of the tangent. Below is a proof of Case I.
Proof Given: a circle with chords AB and CD intersecting at P
Prove: a
# b=c # d
A
Draw AC and BD. jA ≅ jD and jC ≅ jB because each pair intercepts
the same arc, and angles that intercept the same arc are congruent.
△APC = △DPB by the Angle-Angle Similarity Postulate. The lengths of
a
corresponding sides of similar triangles are proportional, so d
= bc .
Therefore, a b = c d.
#
C
a
d
c
P
b
B
D
#
Use similar triangles to prove Case II and Case III.
13.Prove Theorem 12-15, Case II.
Proof
14.Prove Theorem 12-15, Case III.
Proof
hsm11gmse_1204_t06985
15.The diagram at the right shows a unit circle, a circle with radius 1.
E
a.What triangle is similar to △ABE?
b.Describe the connection between the tangent ratio for ∠A and the segment that is tangent to }A.
D
1
A
B C
hypotenuse
lengthoflegadjacenttoanangle . Describe the connection c.The secant ratio is
between the secant ratio for ∠A and the segment that is the secant in }A.
hsm11gmse_1204_t08590 .a
PearsonTEXAS.com
509
For Exercises 16 and 17, use the diagram at the right. Prove each statement.
¬
¬
16.m∠1 + m PQ = 180
17.m∠1 + m∠2 = m QR
Proof
Q
Proof
18.a. Select Tools to Solve Problems (1)(C) What tool(s) can you use to
investigate the relationships between an angle formed by two perpendicular
tangents to a circle and the arcs intercepted by the angle? Explain your choice.
R
1
2
P
b.Construct a circle with a several pairs of tangents that intersect at a right
angle. Make a conjecture about the relationship between the right angle and
hsm11gmse_1204_t07013
the minor arc intercepted by the angle.
19.In the diagram at the right, the circles are concentric. What is a formula
you could use to find the value of c in terms of a and b?
20.△PQR is inscribed in a circle with m∠P = 70, m∠Q = 50, and
¬ ¬
¬
m∠R = 60. What are the measures of PQ , QR , and PR ?
c
b
a
Find the values of x and y using the given chord, secant, and tangent
lengths. If your answer is not a whole number, round it to the nearest tenth.
21.
y
6
10
x
22.
8 x
4
16
23.
y
24.Prove Case II of Theorem 12-14.
12
y
5
hsm11gmse_1204_t08588 .ai
x
A
B
Given: }O with secants CA and CE
O
C
¬ hsm11gmse_1204_t07009
¬
1
hsm11gmse_1204_t07010
D
Prove: m∠ACE = 2 (m AE - m BD )
hsm11gmse_1204_t07008
E
b
P
25.Prove Case I and Case III of Theorem 12-14.
Proof
Proof
26.Use the diagram at the right and the theorems of this lesson to prove the
Proof Pythagorean Theorem.
a
a
Q
R
c
O
hsm11gmse_1204_t07012
a
27.If an equilateral triangle is inscribed in a circle, prove that the tangents
Proof
S
to the circle at the vertices form an equilateral triangle.
TEXAS Test Practice
For Exercises 28 and 29, use the diagram at the right.
hsm11gmse_1204_t07014
A
28.If BC = 6, DC = 5, and CE = 12, find AC.
¬
¬
29.If m∠C = 14 and m AE = 140, find m BD .
B
C
E
D
hsm11gmse_1204_t07015
510
Lesson 12-4 Angle Measures and Segment Lengths
Topic 12 Review
TOPIC VOCABULARY
• chord, p. 492
• point of tangency, p. 486
• inscribed angle, p. 499
• secant, p. 505
• tangent to a circle, p. 486
Check Your Understanding
Use the figure to choose the correct term to complete each sentence.
A
D
C
X
F
E
B
< >
1. EF is (a secant of, tangent to) }X .
2.DF is a (chord, secant) of }X .
hsm11gmse_12cr_t09116.ai
3.△ABC is made of (chords
in, tangents to) }X .
4.∠DEF is an (intercepted arc, inscribed angle) of }X .
12-1 Tangent Lines
Quick Review
Exercises
A tangent to a circle is a line that intersects the circle at
exactly one point. The radius to that point is perpendicular
to the tangent. From any point outside a circle, you can
draw two segments tangent to a circle. Those segments are
congruent.
Use }O for Exercises 5–7.
5
A
O
2
Example
>
B
x
>
PA and PB are tangents. Find x.
The radii are perpendicular to the
tangents. Add the angle measures
of the quadrilateral:
x + 90 + 90 + 40 = 360
x + 220 = 360
x = 140
A
O x
40
B
P
60
3
C
5.What is the perimeter of △ABC?
6.OB = 128. What is the radius?
hsm11gmse_12cr_t09053.ai
7.
What is the value of x?
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12-2 Chords and Arcs
Quick Review
Exercises
A chord is a segment whose endpoints
are on a circle. Congruent chords are
equidistant from the center. A diameter
that bisects a chord that is not a
diameter is perpendicular to the chord.
The perpendicular bisector of a chord
contains the center of the circle.
Use the figure at the right for Exercises 8–10.
Example
What is the value of d?
8.If AB is a diameter and CE = ED,
then m∠AEC = ? .
9.If AB is a diameter and is at right
angles to CD, what is the ratio of
CD to DE?
+
122
=
C
D
E
10. If CE = 12 CD and m∠DEB = 90,
what is true of AB?
hsm11gmse_12cr_t09054.ai
B
Use the circle below for Exercises 11 and 12.
Since the chord is bisected, m∠ACB = 90.
The radius is 13 units. So an auxiliary
segment from A to B is 13 units. Use the
Pythagorean Theorem.
d2
A
9
13
B
12
132
d
C 12
x
hsm11gmse_12cr_t09056.ai
5
y
5
A
d 2 = 25
11. What is the value of x?
d=5
12. What is the value of y?
hsm11gmse_12cr_t09055.ai
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12-3 Inscribed Angles
Quick Review
Exercises
Find the value of each variable. Line O is a tangent.
An inscribed angle has its
Intercepted A
arc
vertex on a circle, and its sides
13. 14.
C
B
b are chords. An intercepted arc
40
Inscribed angle
has its endpoints on the sides
a
a
of an inscribed angle, and its
d
20 c other points in the interior of the
c
b
angle. The measure of an inscribed
angle is half the measure of its
hsm11gmse_1203_t06873
intercepted arc.
b
15. c 16.
¬
What is m PS ? What is mjR?
¬
m∠Q = 60 is half m PS , so
¬
m PS = 120. ∠R intercepts the
same arc as ∠Q, so m∠R = 60.
a
59 b hsm11gmse_12cr_t09058.ai
72
Example
P
S
d
a
hsm11gmse_12cr_t07097.ai
140
45
d
c
60
Q
R
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Topic 12 Review
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12-4 Angle Measures and Segment Lengths
Quick Review
Exercises
A secant is a line that intersects a circle at two points.
The following relationships are true:
Find the value of each variable.
c
a
x
O
d
1
17. x
y
26
100
b
abcd
m1 1 (x y)
18. 145
2
hsm11gmse_12cr_t07106.ai
a b
x
w
O
b
B
ahsm11gmse_12cr_t07102.ai
z
y
45
6
19. (w x)w ( y z)y
5
x
mB 1 (a b)
hsm11gmse_12cr_t07105.ai
2
10
a
t
O
b
hsm11gmse_12cr_t07103.ai
z
B
y
20.
( y z)y t 2
5 10
hsm11gmse_12cr_t09059.ai
x 8
mB 1 (a b)
2
Example
What is the value of x?
hsm11gmse_12cr_t07104.ai
19
(x + 10)10 = (19 + 9)9
9
10x + 100 = 252
x
10
x = 15.2
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Topic 12 TEKS Cumulative Practice
Multiple Choice
4.What is the value of x in the figure shown below?
Read each question. Then write the letter of the correct
answer on your paper.
1.A vertical mast is on top of building and is positioned
6 ft from the front edge. The mast casts a shadow
perpendicular to the front of the building, and the tip of
the shadow is 90 ft from the front of the building. At the
same time, the 24-ft building casts a 64-ft shadow. What
is the height of the mast?
B.
9 ft 9 in.D.
33 ft
2.Javier leans a 20-ft-long ladder against a wall. If the base
of the ladder is positioned 5 feet from the wall, how
high up the wall does the ladder reach? Round to the
nearest tenth.
3.△FGH has the vertices shown below. If the triangle is
rotated 90° counterclockwise about the origin, what are
the coordinates of the rotated point F′?
y
F
x
4
2
4
5.A photographic negative is 3 cm by 2 cm. A similar
print from the geom12_gm_c12_csr_t0005.ai
negative is 9 cm long on its shorter
side. What is the length of the longer side?
A.
1.5 cmC.
12 cm
B.
6 cmD.
13.5 cm
G.
60°
H.
75°
45
x
75
J.
105°
7.One side of a triangle has length 6 in. and another
side has length 3 in. Which is the greatest possible
whole-number value for the length of the third side?
4
2
F.
112H.
102
45°
F.
G.
18.8 ft J.
15 ft
4 2
76
6.What is the value of x?
F.
19.4 ftH.
17.5 ft
G
x
G.
104J.
89
A.
7 ft 6 in.C.
12 ft
102
H
A.
3 in.C.
8 in.
geom12_gm_c12_csr_t0004.ai
B.
6 in.D.
9 in.
¬
8.What is m RT in the figure at the right?
F.
162°
G.
146°
( -1, 2)
A.
(1, -3)C.
B.
(3, -1)D.
(1, -2)
T
R
55
H.
110°
J.
73°
104
S
geom12_gm_c12_csr_t0002.ai
514
Topic 12 TEKS Cumulative Practice
geom12_gm_c12_csr_t0003.ai
Gridded Response
Constructed Response
9.A ski ramp on a lake has the dimensions shown below.
To the nearest hundredth of a meter, what is the height
h of the ramp?
15. △DEF has vertices D(1, 1), E( -2, 4), and F(4, 7). What
is the perimeter of △DEF ? Show your work.
h
4.8 m
75
16. In }A below, AE = 13.1 and AC # BD. If BC = 6.8,
what is AC, to the nearest tenth? Show your work.
B
D
A
10. What is the value of n in the trapezoid shown below?
C
(23n)
E
geom12_gm_c12_csr_t0008.ai
(14n5)
11. In the figure shown below, MN } OP, LM = 12,
MN = 15, and MO = 6. What is OP?
L
geom12_gm_c12_csr_t0009.ai
M
18. The endpoints of HK are H( -3, 7) and K(6, 1), and
the endpoints of MN are M( -5, -8) and N(7, 10).
What are the slopes of the line segments? Are the
line segments parallel, perpendicular, or neither?
Explaingeom12_gm_c12_csr_t0011.ai
how you know and show your work.
19. Suppose a square is inscribed in a circle such
as square HIJK shown below.
N
O
17. What is the measure of an exterior angle of a regular
dodecagon (12-sided polygon)? Show your work.
P
12. In the figure below, XY and XZ are tangent to }O at
points Y and Z, respectively. What is m∠YOZ in degrees?
Y
geom12_gm_c12_csr_t0010.ai
50
X
O
H
K
I
J
a.Show that if you form a new figure by connecting
the tangents to the circle at H, I, J, and K, the
new figure is also a square.
13. In parallelogram ABCD below, DB is 15. What is DE?
b.The inscribed square and the square formed
by the tangents are similar. What is the scale
geom12_gm_c12_csr_t0013.ai
factor of the similar
figures?
Z
B
A
E
geom12_gm_c12_csr_t0011.ai
C
D
c.Let a regular polygon with n sides be inscribed
in a circle. Do the tangent lines at the vertices
of the polygon form another regular polygon
with n sides? Explain.
14. The measure of the vertex angle of an isosceles triangle
is 112. What is the measure of a base angle?
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