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In this chapter you learned some new circle vocabulary and solved
real-world application problems involving circles. You discovered
the relationship between a radius and a tangent line. You discovered
special relationships between angles and their intercepted arcs. And
you learned about the special ratio and how to use it to calculate
the circumference of a circle and the length of an arc.
You should be able to sketch these terms from memory: chord,
tangent, central angle, inscribed angle, and intercepted arc. And you
should be able to explain the difference between arc measure and
arc length.
EXERCISES
You will need
1. What do you think is the most important or useful circle property
you learned in this chapter? Why?
2. How can you find the center of a circle with a compass and a
straightedge? With patty paper? With the right-angled corner of a
carpenter’s square?
3. What does the path of a satellite have to do with the Tangent Conjecture?
4. Explain the difference between the degree measure of an arc and its arc length.
Solve Exercises 5–19. If the exercise uses the “=” sign, answer in terms of . If the
exercise uses the “ ” sign, give your answer accurate to one decimal place.
5. b =
6. a =
7. c =
8. e =
9. d =
10. f =
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11. circumference
12. circumference = 132 cm
d
13. r = 27 cm. The length
of AB is .
14. r = 36 ft. The length
of CD is .
15. Developing Proof What’s
wrong with this picture?
16. Developing Proof What’s wrong
with this picture?
17. Developing Proof Explain
why KE || YL.
18. Developing Proof Explain
why JIM is isosceles.
19. Developing Proof Explain why
KIM is isosceles.
20. On her latest archaeological dig,
Ertha Diggs has unearthed a portion
of a cylindrical column. All she has
with her is a pad of paper. How can
she use it to locate the diameter of
the column?
21. Construction Construct a scalene
obtuse triangle. Construct the
circumscribed circle.
22. Construction Construct a scalene acute
triangle. Construct the inscribed
circle.
23. Construction Construct a rectangle. Is it possible to construct the circumscribed circle,
the inscribed circle, neither, or both?
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24. Construction Construct a rhombus. Is it possible to construct the circumscribed
circle, the inscribed circle, neither, or both?
25. Find the equation of the line tangent to circle S centered at (1, 1) if the point
of tangency is (5, 4).
26. Find the center of the circle passing through the points (−7, 5), (0, 6), and
(1, −1).
27. Rashid is an apprentice on a road crew for a civil engineer. He needs to find
a trundle wheel similar to but larger than the one shown at right. If each
rotation is to be 1 m, what should be the diameter of the trundle wheel?
28. Melanie rides the merry-go-round on her favorite horse on the outer edge,
8 meters from the center of the merry-go-round. Her sister, Melody, sits
in the inner ring of horses, 3 meters in from Melanie. In 10 minutes,
they go around 30 times. What is the average speed of each sister?
29. Read the Geography Connection below. Given that the polar radius
of Earth is 6357 kilometers and that the equatorial radius of Earth
is 6378 kilometers, use the original definition to calculate one
nautical mile near a pole and one nautical mile near the equator.
Show that the international nautical mile is between both values.
Geography
One nautical mile was originally defined to be the length of one minute
of arc of a great circle of Earth. (A great circle is the intersection of the
sphere and a plane that cuts through its center. There are 60 minutes of
arc in each degree.) But Earth is not a perfect sphere. It is wider at the
great circle of the equator than it is at the great circle through the poles.
So defined as one minute of arc, one nautical mile could take on a range
of values. To remedy this, an international nautical mile was defined as
1.852 kilometers (about 1.15 miles).
30. While talking to his friend Tara on the phone, Dmitri
sees a lightning flash, and 5 seconds later he hears
thunder. Two seconds after that, Tara, who lives 1 mile
away, hears it. Sound travels at 1100 feet per second.
Draw and label a diagram showing the possible
locations of the lightning strike.
31. King Arthur wishes to seat all his knights at a round
table. He instructs Merlin to design and create an oak
table large enough to seat 100 people. Each knight is
to have 2 ft along the edge of the table. Help Merlin
calculate the diameter of the table.
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32. If the circular moat should have been a circle of
radius 10 meters instead of radius 6 meters, how
much greater should the larger moat’s circumference
have been?
33. The part of a circle enclosed by a central angle and
the arc it intercepts is called a sector. The sector of
a circle shown below can be curled into a cone by
bringing the two straight 45-cm edges together.
What will be the diameter of the base of the cone?
MIXED REVIEW
In Exercises 34–56, identify the statement as true or
false. For each false statement, explain why it is false or
sketch a counterexample.
34. If a triangle has two angles of equal measure, then the third angle is acute.
35. If two sides of a triangle measure 45 cm and 36 cm, then the third side must be
greater than 9 cm and less than 81 cm.
36. The diagonals of a parallelogram are congruent.
37. The measure of each angle of a regular dodecagon is 150°.
38. The perpendicular bisector of a chord of a circle passes through the center of
the circle.
39. If CD is the midsegment of trapezoid PLYR with PL one of the bases, then
CD = (PL + YR).
40. In BOY, BO = 36 cm, m B = 42°, and m O = 28°. In GRL, GR = 36 cm,
m R = 28°, and m L = 110°. Therefore, BOY
GRL.
41. If the sum of the measures of the interior angles of a polygon is less than 1000°,
then the polygon has fewer than seven sides.
42. The sum of the measures of the three angles of an obtuse triangle is greater than the
sum of the measures of the three angles of an acute triangle.
43. The sum of the measures of one set of exterior angles of a polygon is always less
than the sum of the measures of interior angles.
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44. Both pairs of base angles of an isosceles trapezoid are supplementary.
45. If the base angles of an isosceles triangle each measure 48°, then the vertex angle has
a measure of 132°.
46. Inscribed angles that intercept the same arc are supplementary.
47. The measure of an inscribed angle in a circle is equal
to the measure of the arc it intercepts.
48. The diagonals of a rhombus bisect the angles of the
rhombus.
49. The diagonals of a rectangle are perpendicular bisectors
of each other.
50. If a triangle has two angles of equal measure, then the
triangle is equilateral.
51. If a quadrilateral has three congruent angles, then it is a
rectangle.
52. In two different circles, arcs with the same measure are
congruent.
53. The ratio of the diameter to the circumference of a
circle is .
54. If the sum of the lengths of two consecutive sides of a
kite is 48 cm, then the perimeter of the kite is 96 cm.
55. If the vertex angles of a kite measure 48° and 36°, then
the nonvertex angles each measure 138°.
56. All but seven statements in Exercises 34–56 are false.
57. Find the measure of each lettered angle in the diagram below.
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Developing Proof In Exercises 58–60, from the information given, determine which triangles, if any, are
congruent. State the congruence conjecture that supports your congruence statement.
58. STARY is a regular pentagon.
59. FLYT is a kite.
60. PART is an isosceles
trapezoid.
61. Adventurer Dakota Davis has uncovered a
piece of triangular tile from a mosaic. A
corner is broken off. Wishing to repair the
mosaic, he lays the broken tile piece down
on paper and traces the straight edges. With
a ruler he then extends the unbroken sides
until they meet. What triangle congruence
shortcut guarantees that the tracing reveals
the original shape?
62. Circle O has a radius of 24 inches. Find the
measure and the length of AC .
63. Developing Proof EC and ED are tangent to
the circle, and AB = CD. Find the measure
of each lettered angle. Explain.
64. Use your protractor to draw and label a pair of supplementary angles that is not a
linear pair.
65. Find the function rule f (n) of this sequence and find the 20th term.
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66. The design at right shows three hares joined by three ears,
although each hare appears to have two ears of its own.
a. Does the design have rotational symmetry?
b. Does the design have reflectional symmetry?
67. Construction Construct a rectangle whose length is twice
its width.
68. If AB = 15 cm, C is the midpoint of AB , D is the
midpoint of AC, and E is the midpoint of DC, what
is the length of EB?
69. Draw the next shape in this pattern.
70. Construction Construct any triangle. Then construct its centroid.
1. Developing Proof Show how the Tangent Segments Conjecture follows logically from the
Tangent Conjecture and the converse of the Angle Bisector Conjecture.
2. Developing Proof Investigate the quadrilateral formed by two
tangent segments to a circle and the two radii to the points of
tangency. State a conjecture. Explain why your conjecture is true,
based on the properties of radii and tangents.
3. Developing Proof State the Cyclic Quadrilateral Conjecture in
“if-then” form. Then state the converse of the conjecture in
“if-then” form. Is the converse also true?
4. Developing Proof A quadrilateral that can be inscribed in a circle is also called a cyclic
quadrilateral. Which of these quadrilaterals are always cyclic: parallelograms, kites,
isosceles trapezoids, rhombuses, rectangles, or squares? Which ones are never cyclic?
Explain why each is or is not always cyclic.
5. Use graph paper or a graphing calculator to graph the data collected from the
investigation in Lesson 6.5. Graph the diameter on the x-axis and the circumference
on the y-axis. What is the slope of the best-fit line through the data points? Does
this confirm the Circumference Conjecture? Explain.
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6. Construction Construct a tangent to a circle from a point external to the circle. For
this to be a construction, you need to determine the precise point of tangency using
your construction tools, not just by placing your ruler on the point and circle so
that it appears to be tangent. (Hint: Look at the diagram on page 357.)
7. Developing Proof In Lesson 6.6, Exercise 10, you proved the Intersecting
Secants Conjecture using auxiliary line SA and the Triangle Exterior Angle
Conjecture. Another proof uses an auxiliary line parallel to secant SE
through point A and the Parallel Lines Intercepted Arcs Conjecture.
Write this proof.
With the different assessment methods you’ve used so far, you should be getting
the idea that assessment means more than a teacher giving you a grade. All the
methods presented so far could be described as self-assessment techniques. Many
are also good study habits. Being aware of your own learning and progress is the
best way to stay on top of what you’re doing and to achieve the best results.
WRITE IN YOUR JOURNAL
You may be at or near the end of your school year’s first semester. Look
back over the first semester and write about your strengths and needs. What
grade would you have given yourself for the semester? How would you justify
that grade?
Set new goals for the new semester or for the remainder of the year. Write them
in your journal and compare them to goals you set at the beginning of the year.
How have your goals changed? Why?
ORGANIZE YOUR NOTEBOOK Review your notebook and conjectures list to be sure
they are complete and well organized. Write a one-page chapter summary.
UPDATE YOUR PORTFOLIO Choose a piece of work from this chapter to add to
your portfolio. Document the work according to your teacher’s instructions.
PERFORMANCE ASSESSMENT While a classmate, a friend, a family member, or a
teacher observes, carry out one of the investigations or Take Another Look
activities from this chapter. Explain what you’re doing at each step, including how
you arrived at the conjecture.
WRITE TEST ITEMS Divide the lessons from this chapter among group members
and write at least one test item per lesson. Try out the test questions written by
your classmates and discuss them.
GIVE A PRESENTATION Give a presentation on an investigation, exploration, Take
Another Look project, or puzzle. Work with your group, or try giving a
presentation on your own.
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