Name ________________________
Worksheet 6.1
Circles
Before embarking on our study of the circle, we define the following:
Circle: the set of all points equidistant from a common point (the center of the circle)
Radius: a line segment connecting the center point to a point on the circle
Chord: a line segment connecting two points on the circle
We also present a postulate regarding radii of a circle: all radii of a circle are congruent
Using these definitions and postulate, prove the following theorem in paragraph form:
Theorem: If a radius is perpendicular to a chord of a circle, then it bisects the chord.
Q: Is the converse of this theorem true? ________
Here is an additional definition regarding circles:
Tangent: a line that intersects a circle only once (a tangent is always
perpendicular to a radius drawn to the point of tangency)
Using this definition of a tangent, prove the following, also in paragraph form:
Theorem: If two tangent segments share a common endpoint outside the circle, then they are
congruent.
Measures of Arcs
By angle: An arc can be measured by its central angle,
for example, a 70º arc.
By arc length: The length of an arc can be calculated
simply by understanding it as a fraction of a
circumference.
Arc length = _______ 2R
Exercises:
1. Find the arc length.
6
82º
82º
2. This circle has radius 10 and arc AB has length 15.
Find the measure of arc AB in degrees.
A
AB=15
B
A
Xº
R
Xo
B
Inscribed Angle Theorem
2X
An inscribed angle is half the size of the central angle to the same arc.
X
Inscribed Angle and Its Intercepted Arc
Our work with the circle leads us to encounter and work with the inscribed angle quite often.
While we have been exposed to the relationship that exists between this angle and its intercepted
arc, we have yet to fully understand why this relationship exists. Let us now take the time to
prove the following theorem:
Theorem: If an angle is an inscribed angle, then it has a measure equal to
one-half that of its intercepted arc.
Since an intercepted arc may present itself in many forms (scenarios), we must consider proving
the theorem in all these scenarios.
Scenario #1:
Given: AP is a diameter
Prove algebraically (with proper discussion) that mAPB 
1
m AB
2
(Hint: draw radius BO)
P
O
B
A
Scenario #2:
Prove algebraically (with proper discussion) that mAPB 
1
m AB
2
(Hint: draw diameter through P & O)
P
A
O
B
Scenario #3:
Prove algebraically (with proper discussion) that mAPB 
1
m AB
2
(Hint: draw diameter through P & O)
P
O
A
B
Exercises:
3. Find the arcs and angles labeled (note the center
is not part of the figure in the circle).
100º
w
58º
y
z
v
x
120º
4. This quadrilateral circumscribes
the circle. Find the missing side.
21
32
25
Useful Corollaries to the Inscribed Angle Theorem
An inscribed triangle with one
Opposite angles in an inscribed An inscribed "butterfly" like
side a diameter is a right triangle. quadrilateral are
this is made up of similar
supplementary.
triangles.
107º
92º
88º
73º
Exercises:
5. Label all missing angles and arcs.
68º
65º
72º
6. Find angles X and Y.
Then find length A.
80º
3
2
Y
5
A
36º
X
7. Find angle X (hint: draw in AB).
X
47º
A
149º
B
8. Explain why this is a true theorem. If two chords are equal then they have equal arcs and
central angles and vice versa.
8.
9.
9. The circles have radii of 5 and 11. AB and CX are tangents. Find AX , XB , and XC .
B
X
A
C
10. One piece of a locomotive gear recovered in a train accident must be used to make a
replacement. AB is carefully measured to be 81.9 cm. CD, the perpendicular bisector of AB, is
measured to be 24.1 cm. Find the radius of the original gear. Maintain enough accuracy to be
sure your final value is accurate to 0.1 cm.
C
A
D
B
11. Find the missing side of the circumscribed quadrilateral.
64
83
57
12. Two tangent circles of radius 7 and 17 have a common tangent AB. Find the length of AB.
A
B
13. Find X. (Hint: You will need to draw in one chord).
17º
X
114º
14. Find all missing arcs and angles.
110º
24º
156º