Download ASSIGNMENT SHEET FOR PACKET 1 OF UNIT 7

M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
ASSIGNMENT SHEET FOR PACKET 1 OF UNIT 7
This packet includes sections 10-1 TO 10-5 from our textbook and review for those sections.
Date Due
Number
7A
7B
Assignment
Topics
p. 701 – 702
# 5, 6, 10-16 all, 24, 28, 32
p. 711 – 712
# 16-21 all, 36, 40, 45, 46, 47
7C
p. 719
# 7-10 all, 16-19 all
7D
p. 727
# 5, 6, 8-13 all
7E
p. 736 – 737
# 4, 5, 6, 8, 14, 15, 24-27 all
10-1 Circles & Circumference (p. 2-4)
Vocabulary: circle, center, radius,
chord, diameter, concentric circles,
circumference, pi, inscribed,
circumscribed
Name and identify parts of a circle.
Solve problems involving circumference
of a circle.
10-2 Measuring Angles & Arcs (p. 5-7)
Vocabulary: central angle, arc, minor
arc, major arc, semicircle, congruent
arcs, adjacent arcs, arc length
Use algebra to solve problems involving
arcs of a circle.
10-3 Arcs & Chords (p. 8-9)
Use algebra to solve problems involving
chords.
10-4 Inscribed Angles (p. 10-12)
Vocabulary: inscribed angle,
intercepted arc
Use algebra to solve problems involving
inscribed angles.
10-5 Tangents (p. 13-14)
Vocabulary: tangent, point of
tangency, common tangent
Use algebra to solve problems involving
tangents to a circle.
Quiz on 10-1 to 10-5
For all problems in this packet, label all answers with appropriate units, when given.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
SECTION 10-1 CIRCLES & CIRCUMFERENCE
Circle: all points in a plane that are the same distance (radius) from a given point (center)
Radius: (plural: radii) a segment with endpoints at the center of a circle and on the circle
Chord: a segment with both endpoints on the circle
Diameter: a chord that passes through the center of a circle
Name circles using a circle symbol  and a single capital letter (the center). For example, a
circle with center O would be named O . Name radii, chords, and diameters as you would
name any other segments, using 2 capital letters with a bar over the top.
Use the circle with center R shown below to answer questions 1 through 7:
1. Name the circle.
2. Name all radii shown in the diagram.
3. Name all chords shown.
4. Name all diameters shown.
5. If AB = 18 mm, find AR.
6. If RY = 10 in., find AR and AB.
7. Is AB  XY ? Explain.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
Recall that circumference is the distance around a circle. C   d or C  2 r
8. Find the diameter and radius of the circle with given circumference. Round answers to
the nearest hundredth.
a. C = 40 in.
b. C = 15.62 m
9. Use your answers above to find the ratio
a. Use (a) above.
c. C = 79.5 yd
circumference
. Round to the nearest hundredth.
diameter
b. Use (b) above.
c. Use (c) above.
10. (a) Similar figures have the same shape and lengths that are in the same proportion. Are
the circles in #8-9 similar to each other? Explain.
(b) Are all circles similar to each other? Explain.
To find exact circumference, leave  as a symbol instead of typing it in your calculator.
11. Find the exact circumference of each circle. You may need to use the Pythagorean
Theorem or other geometric rules to find the diameter first.
a.
b.
c.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
12. Fill in the blanks. The words in parts (d) and (g) have not come up yet. You’ll need to
Google the definitions to find the appropriate terms. You can also check against the
vocabulary list on page 1 of this packet.
a. __________________________ is the distance around a circle.
b. A segment with one endpoint at the center of a circle and the other on the circle is
known as a ____________.
c. A ____________ is the set of all points in a plane that are equidistant from a given point.
d. ____________________ circles are coplanar and have the same center. (Google this
one. It hasn’t come up in any problems yet.)
e. A ________________ is chord that passes through the center of the circle.
f. ____ is an irrational number that is found by the ratio of the circumference to the
diameter of any circle.
g. A polygon drawn inside a circle so that all of its vertices lie in the circle is said to be
__________________ in the circle. That circle is said to be __________________________
around the polygon. (Google these.)
h. A __________ is any segment with both endpoints on the circle.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
SECTION 10-2 MEASURING ANGLES & ARCS
Part 1: Vocabulary and Naming
1. A central angle is an angle with the center of the circle as its
vertex and two radii of the circle as its sides. Name 3 different
central angles shown in  E at right. Use the angle  symbol
in front of each name.
________
________
________
2. The diagram of  E shows 4 non-overlapping central angles.
What is the sum of the measures of these 4 angles?
An arc is a portion of the circumference of a circle. A central angle Here are some different
types of arcs:
A minor arc is the shortest arc between two points on a circle. Name a minor arc using
its endpoints in any order with a curved arc over both letters. For example,
is the
shortest arc formed by CEH in  E . This arc can also be named
.
3. Give 2 different names for the minor arc formed by HEF .
A major arc is the longest arc between two points on a circle. The name of a major
arc must contain 3 letters, the endpoints with any other point on the arc between
them. For example, the major arc formed by CEH in  E can be named
, or
.
4. Give as many names as you can for the major arc formed
by HEF in  E at right, and emphasize this arc by
making it bolder in the diagram.
A semicircle is an arc with endpoints on a diameter of a circle.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
Part 2: Arc Measure
Arcs have both length and measure. The measure of a minor arc is expressed in degrees and
is equal to the measure of its central angle. The measure of a major arc is found by
calculating 360 minus the measure of its minor arc.
5. (a) What is the measure of any semicircle?
(b) The measure of any minor arc is between _______  and _______  .
(c) Can the measure of a major arc be less than 180 ? Explain.
6. In  R at right, AC is a diameter, and mARB  42 . Find the
measure of each arc:
______
______
______
______
______
Part 3: Arc Length
The length of an arc can be found using the following proportion:
Part to whole for
distances around
the circle
arc length
central angle

circumference
360
Part to whole
for angles in
the circle
7. Write a proportion, and solve to find the missing measure in O at right. Round to the
nearest hundredth.
a. Length of
b. Length of
if r = 2 m.
if d = 7 in.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
arc length
central angle

circumference
360
8. Follow the steps below to solve this problem: The arc length between
the blades of the wind turbine shown at right was calculated to be
309.8 feet. To the nearest whole number, what is the length of each
blade of the turbine?
a. Assuming the blades are set at equal distances around the circle,
what is the degree measure of the angle between two blades? (This
is the central angle.)
b. Use the formula at the top of this page to write a proportion using the arc length given
at the beginning of this problem, the central angle you found in part (a), and C as the
circumference.
c. Solve your proportion for C.
d. Substitute the value of C above into the circumference formula: C  2 r . Then divide
to solve for r, and answer the problem.
9. Follow the steps below to solve this problem: A stage is in the shape of a semicircle with
radius 18 feet and the curved edge facing the audience. If a banner is hung all around
the front of the state, how long will the banner need to be? Round to the nearest tenth.
a. Calculate the exact circumference of a circle with radius 18 ft.
b. What is the degree measure for the central angle of a semicircle?
c. Use the formula at the top of this page to write a proportion using the circumference
you found in part (a), the central angle you found in part (b), and x as the arc length.
d. Answer the problem.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
SECTION 10-3 ARCS & CHORDS
Part 1: Congruent Chords in a Circle
1. In  E at right, chords AB and CD are congruent.
a. Draw AEB and CED in the diagram.
b. Since AEB  CED by the ______ triangle congruence postulate, AEB   ______. This
shows that central angles formed by congruent chords in the same circle are
__________________. Also, arcs formed by congruent chords are __________________.
2. In the same diagram above, draw a segment from E, perpendicular to AB , and another
segment from E, perpendicular to CD .
a. Explain why these 2 new segments must be congruent.
b. Congruent chords in the same circle are ______________________ from the center.
Conversely, if chords are ______________________ from the center of a circle, then they
are congruent.
3. Find the value of x in each circle.
a.
b.
c.
d. CD  CB , GQ  x  5 , EQ  3 x  6
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
Part 2: Perpendicular Bisector of a Chord
4. Open the applet at https://www.geogebra.org/o/SteKWsKP.
5. In the applet, drag point E until it is the midpoint of chord CD . If you have difficulty
making this work, you can also move point C or D.
6. Drag point F until the measure of FED is approximately 90 .
7. The perpendicular bisector of a chord passes through the ____________ of the circle and is
a ________________ of the circle. Conversely, if a ________________ of a circle is
perpendicular to a chord, then it also ______________ the chord.
8. In O , CD  OE , OD = 15, and CD = 24.
a. OE is the __________________________ ________________ of CD .
b. Find ED.
c. Find OE.
9. In  P , the radius is 13, and RS = 24.
a. Find RT.
b. Find PT. (HINT: Draw another radius of  P to form a right triangle.)
c. Find TQ.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
SECTION 10-4 INSCRIBED ANGLES
Part 1: Inscribed Angles and Intercepted Arcs
An angle is inscribed in a circle if its vertex lies on the circle and both of its sides contain
chords of the circle. The intercepted arc is the minor arc cut off by the sides of the angle.
1. In each circle, name a central angle, an inscribed angle, and the intercepted arc
formed by both angles.
a.
b.
Central angle:  ______
Central angle:  ______
Inscribed angle:  ______
Inscribed angle:  ______
Intercepted arc: ______
Intercepted arc: ______
Part 2: The Measure of an Inscribed Angles and its Intercepted Arc
2. Open the applet at https://www.geogebra.org/o/Kh756zzu.
3. Drag point B, C, or D to change the angle measures, and use a calculator to compute
the ratio shown on the screen, rounded to the nearest tenth. (Do not drag any of these
points past each other around the circle.)
Repeat until you’ve calculated the ratio at least 5 different times.
4. The measure of an inscribed angle is _________________ the measure of the central angle
that intercepts the same arc. This means the measure of the inscribed angle is also
_________________ the measure of its intercepted arc. Thus, if two inscribed angles
intercept the same arc, the angles must be __________________.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
5. Find the requested measure.
a.
______
b.
∠
= ______
c.
6. Use the circle at right to answer the following:
a. Which angle intercepts the same arc as ∠ ?
b. Which angle intercepts the same arc as ∠ ?
c. Find the measure of ∠ .
d. Find the measure of ∠ .
7. In  A at right, BC is a diameter.
a. What is the measure of
?
b. What is the measure of D ?
c. An inscribed angle that intercepts a semicircle measures ________  .
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
Part 2: Angles of Inscribed Polygons
In an inscribed polygon, all sides are chords, and all vertices are points on the circle.
8. In quadrilateral ABCD, diagonal BD is a diameter of  R .
a. mA  ______  ;
______ 
b. mC  ______  ;
______ 
c. mA  mC  ______  ;
d.
+
______ 
= ______  ; mABC  mADC  ______ 
e. If a quadrilateral is inscribed in a circle, then its opposite angles are
_________________________.
9. Find the value of x in each.
a.
b.
c.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
SECTION 10-5 TANGENTS
A tangent is a line, ray, or segment that intersects a circle in exactly one point, called the
point of tangency. A common tangent is tangent to two circles in the same plane.
Part 1: Relationship Between Tangent and Radius
1. Open the applet at https://www.geogebra.org/m/e4qx3dFC.
2. Drag point C until ⃖ ⃗ appears to be tangent to the circle.
3. A line tangent to a circle is __________________________ to a radius at a point of tangency.
4. In each diagram below, assume that segments that appear to be tangent are tangent.
Find the value of x in each:
a.
b. AD = 9, BC = 8
Part 2: Relationship Between Two Tangents
5. Open the applet at https://www.geogebra.org/m/ZwjDjfqG.
6. Right-click the pink and orange segments, and check Show Label to show the length of
each segment.
7. Drag point A around the screen, observing the lengths of the pink and orange segments.
8. If two segments from the same point are tangent to a circle, then they are
__________________.
9. Find x in each. Assume that segments that appear to be tangent are tangent.
a.
b.
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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES
Part 3: Relationship Between Two Tangents
When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent
to the circle.
10. Find x. Then find the perimeter of the polygon.
a.
b.
c.
d.
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