Mathematics 20
Module 3
Lesson 19
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Circles
Lesson 19
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Lesson 19
Circles
Introduction
In this lesson you will study properties of circles and lines or line segments related to the
circle.
The circle is a geometric figure, examples of which are found everywhere. The wheel, sun,
gears, tree trunks, pipes, clocks, and waves in water radiating from the point where a
stone is dropped are all examples of circular shapes.
Many properties of circles have interesting applications. For example, by the end of this
lesson you will be able to solve the following problem using your knowledge of circles.
A small piece of a ceramic plate was found by an archaeologist. Fortunately it was a piece
from the edge of the plate and one side was in the shape of an arc or part of a circle. By
the use of a ruler and compass determine the diameter of the plate.
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Objectives
After completing this lesson, you will be able to
•
define the measure of a minor arc and calculate the measure of a central angle.
•
determine the relationship that exists between
•
•
the tangent to a circle and the radius of a circle drawn to the point of
tangency.
•
two tangents drawn to a circle from the same point.
•
chords and arcs in the same circle or in congruent circles.
•
diameter and a chord bisected by the diameter.
•
two chords that intersect in a circle.
solve problems based on the above relationships.
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19.1 Vocabulary for Circles
Some of the vocabulary that is associated with circles is given in this lesson. Before
reading further, be sure that you have a straight edge and a compass with you for some of
the activities and exercises that follow.
A circle is the set of all points in a plane that are a fixed distance from a given point P in
the plane called the center of the circle. (See circle Q.)
The radius of a circle is the line segment from the center to any point on the circle. The
length of the radius is also called the radius. All radii (plural for radius) of a given circle
are congruent segments .
•
The points enclosed by the circle, or whose distance from the center is less than the
radius, form the interior of the circle.
•
The points outside the circle, or whose distance from the center is greater than the
radius, form the exterior of the circle.
•
The circle combined with its interior points is called a disc.
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Two different circles are concentric if they share the same center.
If two circles have the same radius (same length of radius), they are called congruent
circles.
A line on the same plane as the circle is tangent to the circle if and only if its
intersection with the circle is exactly one point. The point of intersection is called the
point of tangency.
A line which intersects a circle in exactly two points is called a secant line.
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Activity 19.1
Materials Needed: Paper, straight edge, compass
Object: To construct a circle inscribed in a triangle and to
construct a circle which circumscribes a triangle.
Discussion: The diagram of the circles is shown below.
A circle inscribed in a triangle intersects each side of the triangle at exactly one point.
A circle which circumscribes a triangle intersects each vertex.
Perform the following procedure.
To inscribe a circle in a given triangle
1.
On any triangle you construct, draw the bisectors of each angle of the triangle.
(See Answers to Exercises on bisecting angles.)
2.
From the point of intersection P of the three bisectors, draw the perpendicular to
any one of the sides to intersect at a point X. (See Answers to Exercises on
perpendicular lines.)
3.
Draw a circle with PX as radius and center at P.
This is the required inscribed circle.
To circumscribe a given triangle by a circle
1.
2.
On any triangle you construct, draw the perpendicular bisectors of each of the
three sides so that they intersect at a single point P. (See Answers to Exercises on
perpendicular bisectors.)
With center at P and radius the same as the distance from P to anyone
of the vertices, draw a circle.
This is the required circle which circumscribes the triangle.
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Activity 19.2
Materials Needed: straight edge, compass
Object: Formally construct a tangent to a given circle from a given
external point.
Discussion:
With only a straight edge and no compass it is possible to draw the approximate tangent
to the circle from X by positioning the straight edge so that it passes through X and
touches the circle. It is usually difficult to locate the precise point of tangency this way.
The following formal method solves this problem.
To draw a tangent to a circle with center P from a given external point X
1.
draw the line segment XP ,
2.
bisect XP at a point Y,
3.
with center at Y and radius XY, draw a circle which intersects the given circle
at two points A and B,
4.
construct a line containing A and X and one containing B and X.
Lines
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are the two possible required tangent lines.
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If a line is tangent to two circles, it is a common tangent of the circles.
•
A common tangent which intersects the segment joining the two centers is called a
common internal tangent.
•
A common tangent which does not intersect the segment joining the two centers is
a common external tangent.
Exercise 19.1
1.
a.
Draw the two different ways that two circles can have no points of
intersection.
b.
Draw the two different ways that two circles can have exactly one point of
intersection.
c.
Draw one way that two circles can have exactly two points of intersection.
2.
True or False? Two circles can intersect in more than two points.
3.
True or False? Two non-congruent circles can intersect in more than two
points.
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4.
Use the figure to give a name to each of the following notations.
c.
K
G
A
d.
BL
a.
b.
e.
f.
g.
h.
Circles B and C
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19.2
Tangents to Circles and their Properties
Activity 19.3
Material: compass, straight edge
Object: Determine the relationship between the radius of a circle and a
tangent to a circle.
Part 1:
1. Draw a radius to any point X on the circle.
2. Draw a perpendicular to PX passing through X. (See Answers to Exercises on
perpendicular lines.)
3. On the perpendicular line mark any point Y that does not coincide with X.
4. Follow the method in Activity 19.2 to draw the tangents to the circle from Y.
Discussion: With an accurate construction a point of tangency should be X. From this
you may conclude that the tangent is perpendicular to the radius if the radius is drawn to
the point of tangency. The Tangent-Radius Theorem summarizes this observation.
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Tangent-Radius Theorem
If a line is tangent to a circle, it is perpendicular to the radius which is drawn to the
point of tangency.
Conversely, if a line is perpendicular to the radius, and passes through the end point of
the radius on the circle, then it is tangent to the circle.
Example 1
Find the length of the unknown side if it is known that RQ is a tangent to
the circle P.
a.
b.
Solution:
a.
Since line QR is a tangent and PRQ is a right angle, the Pythagorean Theorem
may be used.
2
2
PQ  PR 2  RQ
60 2  11 2  RQ 2
RQ
2
 60 2  11 2
RQ
2
 3600  121
RQ
2
 3479
RQ
2
 58 .98
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b.
PQ 2  PR 2  RQ 2
2
2
2
13  PR  12
2
2
2
PR  13  12
2
PR  169  144
PR 2  25
PR  5
Example 2
Determine if RQ is a tangent to the circle P or not.
Solution:
PR 2  RQ 2
42 72
16  49
65
 PQ 2
 82
 64
 64
Therefore, PRQ is not a right angle.
Therefore, PR is not perpendicular to RQ .
Therefore, RQ is not a tangent to the circle P.
For any exterior point Q of a circle there are always two lines passing through Q which
are tangent to the circle. The following theorem states that the two segments from Q to
the points of tangency are congruent.
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Theorem
If Q is exterior to circle P, the two segments from Q to the points of tangency are
congruent.
Diagram:
Given: RQ, SQ are tangent to circle P with R and S being points of tangency.
Prove: RQ  SQ .
Proof:
Statement
Reason
1.
2.
PR  RQ , PS  SQ
PRQ  PSQ
1.
2.
3.
4.
5.
PR  PS
PQ  PQ
PRQ  PSQ
3.
4.
5.
6.
RQ  SQ
6.
(Solutions can be found in Answers to Exercises.)
•
In Section 19.1 a formal method using a compass and straight edge was used to
construct a circle inscribed in a given triangle and to construct a circle
circumscribing a given triangle.
•
A similar construction can be done with regular polygons.
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A circle is inscribed in a polygon if each side of the polygon is a tangent to the circle.
A circle circumscribes a polygon if each vertex is a point on the circle.
Example 3
Decide in each case if the circle is inscribed in, circumscribes a polygon or neither.
a.
b.
c.
d.
e.
f.
g.
Solution:
a.
b.
c.
d.
e.
f.
g.
neither
inscribed
circumscribes
neither
neither
circumscribes
neither
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Example 4
For the given regular pentagon,
a.
b.
inscribe a circle.
circumscribe the pentagon.
Solution:
a.
To inscribe a circle in a pentagon draw perpendicular
bisectors of any two sides so that they intersect at a point
P. With center P and radius the same as the length of
the bisector from P to the side of the pentagon that was
bisected, draw a circle. This is the required inscribed
circle.
b.
To circumscribe the pentagon, draw bisectors of any two of the angles so that they
intersect at point Q. With center at Q and radius the length of the bisector from the
vertex to Q, draw a circle. This is the required circle.
Are circles P and Q concentric?
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Exercise 19.2
1.
Determine if RQ is tangent to the circle or not.
a.
b.
2.
Determine the radius if RQ is a tangent line.
3.
Determine RQ and SQ.
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4.
Circle P is inscribed in the regular pentagon.
Prove that CPD  EPD, and DP bisects CDE .
5.
Find the value of x , where RQ is a tangent.
19.3 Arcs and Central Angles of Circles
A central angle of a circle is an angle in the
plane of the circle with its vertex coinciding
with the center of the circle.
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•
Notice that no mention is made of the radius in the definition of a central angle. In
the example below, APB is a central angle for all concentric circles with center P.
If the measure of APB is less than 180 , (m APB < 180 ) , then the circumference from
X to S to Y is called a minor arc of the circle. It is denoted by
or simply
if it is
understood which part of the circle is meant. In other words, the minor arc is the part of
the circle in the interior of the central angle. To name a particular arc it is best to use
three letters. These are the end points of the arc and one other point between the end
points.
The measure of the minor arc
, denoted by
is defined to be the measure of the
central angle which marks off the arc. Note that the measure of an arc is in degrees and is
not the length of the arc. In circle P,
= 71°.
•
Two different arcs in concentric circles can have the same measure but are of
different lengths.
The portion of the circle in the exterior of the central angle is called the major arc of the
circle. The measure of the major arc is 360  minus the measure of the minor arc. Using
 360   71   289  .
circle P,
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An arc whose measure is 180 is called a semicircle. In this case the arms of the central
angle form the diameter.
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In the same circle two arcs are adjacent if they intersect in exactly one point. This
intersection point is a common end point of each arc.
Arc Addition Postulate
In the same circle or in congruent circles two different arcs are called congruent if and
only if they have the same measure.
(since the arcs are
not in the same
or in  circles)
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Example 1
Example 2
The sum of the measures of all arcs in a circle graph is 360  .
40
 360  144  . Similarly, the arc
100
covering 10% of the circle has a measure of 36 , the arc covering 15% of the circle has a
measure of 54% and the measure of the arc covering 25% is 90 .
144   36   54   36   90   360 
The measure of the arc covering 40% of the circle is
Example 3
Find the value of x if
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and
where AD , CB are diameters.
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Lesson 19
Solution:
Since vertically opposite angles are congruent,
m APC  m BPC and m APB  m CPD .
2(6 x)  2(10 x  20 )  360 
12 x  20 x  40  360
32 x  320
x  10
 m APB  6 x  6(10 )  60 
m APC  10 x  20  10 (10 )  20  120
Therefore,
(measures of all arcs add up to 360°)
Exercise 19.3
1.
Name all the arcs using three letters and state if each is a minor arc, major arc, or
semicircle. Also, give the measure of each arc based on the given measure.
2.
In the given diagram, find the measure of each based on the given measures.
,
a.
b.
c.
d.
e.
f.
3.
m FPE
Two adjacent arcs form a semicircle and have measures 3 x and 7 x  10 . Find the
measure of each arc.
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4.
Given that
and
,
,
a.
d.
b.
e.
c.
f.
are diameters, find the indicated measures.
19.4 Arcs and Chords of Circles
A chord of a circle is a line segment whose end points
are on the circle. (chord AB )
The diameter of a circle is a chord which passes through
the center of the circle. (chord AC )
Example 1
Find the measure of the minor arc corresponding to the chord AB in the
above diagram (circle P).
Solution:
After constructing APB , measure APB with a protractor. Therefore,
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= ____°.
Lesson 19
Activity 19.4
Materials:
ruler, compass, protractor
Object:
Determine the relationship between congruent chords and
1.
2.
1.
the minor arcs whose end points are the endpoints of the chord, and
the shortest distance from the chord to the center of the circle.
Use a compass to draw two congruent arcs in each circle.
Method:
•
•
•
•
•
Place the compass point on any point on the circle and with a suitable compass
radius cut the circle at another point. These will be the end points of one chord.
Join the points from the first chord.
With the same compass radius but from a different point on the circle repeat the
process to form another chord.
Label and measure the chords in each circle and enter the data into the table.
With a protractor measure the minor arc corresponding to each chord and enter the
measurement into the table.
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Circle
Chord
Length
Arc
Length
Distance
to Chord
A
B
C
2.
Draw the perpendicular bisector of each of the chords in Part 1.
•
•
Does each bisector pass through the center of the circle?
Measure the length of each bisector from the center to the chord and enter
the data into the table.
From the data in the table make 3 statements about what appears to be true about chords.
A.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
B.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
C.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Contact your teacher to confirm your statements.
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The previous observations may be summarized in three theorems.
Congruent Arc-Chord Theorem
In the same or in congruent circles if two chords are congruent, then the two minor arcs
corresponding to the chords are congruent.
Given:
AB  CD
Prove:
Plan:
Proof:
If it is shown that CPD  APB , then the central angles CPD, APB are
congruent.
Statement
Reason
1.
CD  AB
1.
Given
2.
DP  CP , AP  BP
2.
______________________
3.
___________________
CPD  APB
3.
SSS
4.
______________________
5.
______________________
4.
5.
Perpendicular Bisector Theorem
The perpendicular bisector of a chord passes through the center of the circle.
Hint: Using this theorem the introduction question can be solved.
Congruent Chords Theorem
In the same circle or in congruent circles if chords are congruent, then they are
equidistant from the center of the circle.
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Lesson 19
Exercise 19.4
1.
2.
In the diagram, prove that if BP bisects APC , then
.
Prove that in the same circle if two arcs are congruent, then the chords joining the
end points of the arcs are congruent.
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Summary
The following is a list of concepts that you have learned in this lesson:

•
Definitions
Circle
• radius
• interior/exterior
• disc
• concentric
• congruent
• tangent (interior/exterior)
• point of tangency
• secant
• semi-circle
• chord
• diameter
Arcs and Central Angles
• minor/major arc
• measurement
• adjacent arcs
• congruent

•
•
•

•
Constructions
inscribed circle in a triangle/polygon
circumscribed circle around a triangle/polygon
tangents to a circle from a point outside the circle
Theorems
Tangent-Radius Theorem
If a line is tangent to a circle, it is perpendicular to the radius which is drawn to the
point of tangency. Conversely, if a line is perpendicular to the radius, and passes
through the end point of the radius on the circle, then it is tangent to the circle.
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• Theorem
If Q is exterior to circle P, the two segments from Q to the points of tangency are
congruent.

Arc Addition Postulate

Congruent Arc-Chord Theorem
In the same or in congruent circles if two chords are congruent, then the two minor
arcs corresponding to the chords are congruent.

Perpendicular Bisector Theorem
The perpendicular bisector of a chord passes through the center of the circle.

Congruent Chords Theorem
In the same circle or in congruent circles if chords are congruent, then they are
equidistant from the center of the circle.
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Answers to Exercises
Activity 19.1
Recall the steps for bisecting an angle formally.
•
•
After completing arc 1, put the point of the compass on B to make arc 2 and with
the same measure put the point of the compass on C to make arc 3.
Join A to the intersection of the two arcs.
Recall the steps for constructing a perpendicular line to line AB passing through P.
•
•
•
Begin by placing the point of your compass on P and make arcs 1 and 2.
Put the point of the compass on 1 and then 2 (with the same measure), make arcs 3
and 4.
Join P with the intersection of the two arcs.
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Recall the steps for constructing a perpendicular bisector of line segment AB .
•
•
•
Begin by placing the point of your compass on A and draw arcs 1 and 2.
With the same measure, place your compass on point B and draw arcs 3 and 4.
Join the intersections.
Exercise 19.1
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1.
a)
b)
2.
True. The circles can be the same circles.
3.
False.
4.
a.
internal point
b.
external point
c.
center
d.
radius
e.
secant line
f.
external tangent
g.
internal tangent
h.
congruent circles
309
c)
Lesson 19
Activity 19.3
Recall the steps for constructing a perpendicular line through a point on the line.
•
•
•
Begin by placing the point of your compass on P and make arcs 1 and 2.
Put the point of the compass on A and make arcs 3 and 4.
Put the point of the compass on B and make arcs 5 and 6.
Theorem - Reason:
1.
2.
3.
4.
5.
6.
Tangent-radius theorem
All right angles are congruent
Radii of the same circle are congruent
Reflexive property of congruence on segments
HL theorem for right triangles
CPCTC
Exercise 19.2
1.
Use the Pythagorean theorem to see if PR 2  QR 2  PQ 2 .
a.
2
2
2
9  12  15
225  225
Therefore, PRQ is a right angle and RQ is a tangent.
b.
2
PQ  10 2  100
2
2
2
2
PR  QR  (7 .5)  (12 .5)  212 .5
Since 100  212 .5 , PRQ is not a right triangle and
RQ is not a tangent line.
2.
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2
2
2
r  17  13 . 6  104 .4
r  10 .2
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Lesson 19
3.
2
2
RQ  RP 2  PQ
2
RQ  3 2  5 2
2
RQ  25  9  16
RQ  4
RQ  SQ
Therefore, SQ  4 .
4.
Given: Circle P with radii AP , BP , CP , DP , EP
Prove: CPD  EPD and DP bisects CDE
Proof:
5.
1.
2.
3.
4.
Statements
CP and EP are radii.
CP  EP
DP  DP
m DCP  m DEP  90 
5.
6.
CPD  EPD
CPD  EPD
By Pythagoras Theorem
Reasons
1.
Given
2.
Radii are  .
3.
Reflexive property
4.
Tangent-radius
theorem
5.
HL theorem
6.
CPCTC
2
2
PQ  PR 2 + QR
2
PQ  3 2  7 2  58
PQ  7.6
Therefore, x  7 .6  3  4 .6
Exercise 19.3
1.
Major arcs
Minor arcs
Semicircles
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Lesson 19
100 
160 
260 
200 
300 
100 
2.
a.
b.
c.
d.
e.
f.
3.
3 x  (7 x  10 )  180
10 x  170
x  17
 arcs are 3(17 )  51  and 7 x  10  7(17 )  10  129 
Exercise 19.4
65 
65 
95 
265 
85 
245 
4.
a.
b.
c.
d.
e.
f.
1.
Given: BP bisects APC
Prove:
Proof:
1.
Statements
APB  CPB
2.
Reasons
1.
Definition of angle
bisector
2.
Arc measure definition
2.
Given: Circle with center 0 and
Prove: AB  CD
Proof: Construct OA , OB , OC , OD
Plan: Prove first that AOB  COD
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Assignment 19
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Optional insert: Assignment #19 frontal sheet here.
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Assignment 19
Values
(40)
A.
Multiple Choice: Select the best answer for each of the following and place a
check () beside it.
1.
The two circles are ***.
____
____
____
____
2.
on the circle
in the interior
in the exterior
coincident with the center
a.
b.
c.
d.
the same center point
the same center and radius
the same set of points
the same radius
To circumscribe a given triangle you start by ***.
____
____
____
____
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a.
b.
c.
d.
For two circles to be congruent they must have ***.
____
____
____
____
4.
the same
congruent
concentric
non congruent
In circle P, the point Q is ***.
____
____
____
____
3.
a.
b.
c.
d.
a.
b.
c.
d.
bisecting two angles of the triangle
bisecting each of the three angles of the triangle
drawing the perpendicular bisectors of any two sides
drawing a circle with center at one vertex and radius the
same as the largest side
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Lesson 19
5.
6.
The one true statement is ***.
____
a.
____
b.
____
____
c.
d.
The perpendicular bisectors of any two chords of a circle ***.
____
____
____
____
7.
8.
a.
b.
c.
d.
intersect at the center
do not always intersect
do not always intersect at the center
are tangents to the circle
For the given diagram ***.
____
a.
____
____
____
b.
c.
d.
Q is a point of tangency
PQR is a right angle
Q is not a point of tangency
Q
3
P
5
4
R
If a circle is to circumscribe a regular hexagon, then the arcs
corresponding to each side have a measure of ***.
____
____
____
____
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A tangent line contains exactly one interior point of the
circle
From an exterior point of a circle there is exactly one
tangent line to the circle
A circle and a disc can intersect at infinitely many points
Every tangent line is a secant line
a.
b.
c.
d.
120 
80 
70 
60 
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Lesson 19
9.
10.
The one false statement about the circles is ***.
____
a.
____
b.
____
c.
____
d.
The measure of an arc of a sector which covers 35% of the circle in a
circle graph is ***.
____
____
____
____
11.
126°
100°
63°
35°
a.
b.
c.
d.
The radii of the two concentric circles are 3 and 7. The length of the
tangent chord AB is ***.
____
____
____
____
Mathematics 20
a.
b.
c.
d.
The one true statement about the two minor arcs is ***.
____
____
____
____
12.
m APB  m CQD
a.
b.
c.
d.
2 10
4 10
58
2 58
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Lesson 19
13.
If PX  QY and CX  3 .5 , then AB is ***.
____
____
____
____
14.
its length
its distance from the center
the fraction of the area of the circle it covers
the measure of the central angle
a.
b.
c.
d.
similar
congruent
scalene
equilateral
The reason that the triangles are congruent is ***.
____
____
____
____
Mathematics 20
a.
b.
c.
d.
If two arcs of different circles have the same measures, then the
triangles formed by the corresponding chords and the radii are ***.
____
____
____
____
16.
3.5
7
9.5
unknown
The measure of an arc is defined to be ***.
____
____
____
____
15.
a.
b.
c.
d.
a.
b.
c.
d.
LL
HL
SSS
SAS
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Lesson 19
17.
If a circle of radius 6 circumscribes an equilateral triangle, then the
perpendicular distance from any side of the triangle to the center is
***.
(Hint: AOB is isosceles).
____
____
____
____
18.
19.
____
____
a.
b.
____
c.
____
d.
c sin A
b cos A
c
sin A
b
cos A
A 2 m ladder is leaned against a vertical wall so that it makes an angle
of 70 with the ground. The ladder reaches to a height of
approximately ***.
a.
b.
c.
d.
1.6 m
1.7 m
1.8 m
1.9 m
A chord of length 2 m is 1.6 m from the center. The angle between the
chord and a radius to the end point of the chord has measure ***.
____
____
____
____
Mathematics 20
2
3
4
5
For the given triangle, the length of side a is ***.
____
____
____
____
20.
a.
b.
c.
d.
a.
b.
c.
d.
58 
59 
60 
61 
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Lesson 19
Mathematics 20
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Lesson 19
Part B can be answered in the space provided. You also have the option to do
the remaining questions in this assignment on separate lined paper. If you
choose this option, please complete all of the questions on separate paper.
Evaluation of your solution to each problem will be based on the following.
(40)
B.

A correct mathematical method for solving the problem is shown.

The final answer is accurate and a check of the answer is shown where
asked for by the question.

The solution is written in a style that is clear, logical, well organized,
uses proper terms, and states a conclusion.
1.
Mathematics 20
Use a compass and a straight edge to construct all the tangents to a
circle from a given external point. Create your own circle and point.
Be sure to leave all the compass marks showing so as to indicate all
your construction.
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Lesson 19
2.
Given three points in the plane, construct a circle which contains these
three non-collinear points. Show all compass arcs so that your method
of construction is evident.
3.
With a compass and a straight edge locate the center of the circle of
which the arc is a part.
Mathematics 20
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Lesson 19
4.
State and prove the converse of the Congruent Arc-Chord Theorem.
5.
Find AC and write a proof for your answer.
Mathematics 20
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Lesson 19
(20)
C.
When doing the following problems you are encouraged to discuss them with
your Technology Supported Learning teacher or refer to any resources that
you have.
1.
Suppose that two chords of a circle intersect at an interior point shown
in the diagram. The point P creates 4 line segments. The object of this
problem is to find a relationship between the lengths of the 4
segments.
Create 5 different circles and arbitrarily draw two chords in each circle
which intersect. Label the chords as in the diagram, measure each
segment and enter the data into the table. Include the circles with
your solution.
Circle
AP
PB
DP
PC
1
2
3
4
5
Hint: To find a relationship experiment with proportions.
When you think you have found a relation write it as a general statement
"If two chords intersect at an interior point of a circle, then ------."
Mathematics 20
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Lesson 19
2.
Do an investigation similar to Question 1 but for secant lines that
intersect at a point in the exterior of the circle.
Measure BP , AP , CP , DP .
In your solution include circles, table, and a general statement of your
conclusion.
100
Mathematics 20
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Lesson 19
Mathematics 20
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Lesson 19