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Honors Geometry
Review Exercises for Cycle 10
Due on Day 1, Thursday, 18 February 2016
Due on Day 2, Friday, 19 February 2016
This set of Cycle Problems is not a review of previous material, as most of the other sets have
been. Instead, this is an investigation of some properties about circles. So the purpose of this
set is to see whether you can work on new material and gain some new knowledge
CIRCLES: An Investigation
Defn:
A circle is the set of all points in a plane which are equidistant from one point,
called the center of the circle. A circle is named by its center, such as A
Defn:
The radius of a circle is the line segment from the center to any loint on the
circle.
Defn:
A tangent line to a circle is a line (or line segment) which intersects a circle at
only one point. (The rest of the tangent line lies on the outside of the circle.)
The point of tangency is the one point where the tangent meets the circle
Tangent
Honors Geometry Cycle 10 Problems
1.
page 2
Show that a tangent line is perpendicular to the radius drawn to the point of tangency.
(Proof shown below.)
Hint 1: Use indirect proof.
Hint 2: If a segment is drawn from O to
a point on TQ , how long is it
compared to TO ?
Given:
TQ is tangent to circle O.
Prove:
TO  TQ
Proof:
Assume that TO is not  TQ . Then there is another segment PO  TQ .
Thus TO is the hypotenuse of right  TOP. Then TO > PO. But since P is
outside the circle, PO>TO. This is a contradiction. Thus TO  TQ .
Defn:
A central angle of a circle is an angle between two radii (such as < AOB above).
Defn:
A chord is a line segment whose endpoints lie on a circle.
Defn:
An inscribed angle of a circle is an angle between two chords. (such as < ACB
above)
Defn:
A diameter of a circle is a chord that passes through the center of the circle.
Honors Geometry Cycle 10 Problems
2.
What is the name for the longest chord in a circle?
3.
If the arc AB is one quarter of the circle,
what is its measure (in degrees)?
4.
page 3
Measure < AOB and < ACB with a protractor.
a) m < AOB = _____
b) m < ACB = ________
5.
Measure with a protractor again here.
m < FOG = _______
Think about the portion of the circle that the arc occupies, and think about the measure
all the way around a circle. (Note: Each tick mark is 50.)
m FG = ________
6.
Make a conjecture about the measure of a central angle and the measure of its arc.
Honors Geometry Cycle 10 Problems
page 4
Use what you have just conjectured together with the geometry of triangles to investigate the
following. (Note: “O” is the center of each circle.)
The following clips from Sketchpad drawings look at the measure of a central angle, an
inscribed angle, and the arc that both the central and inscribed angles intercept (or “subtend”).
B
mBOC = 58.09°
mBAC = 29.04°
A
C
O
m BC on
OE = 58.09°
B
E
mBOC = 131.26°
mBAC = 65.63°
A
O
C
m BC on
OE = 131.26°
E
mBOC = 30.30°
B
mBAC = 15.15°
A
O
C
m BC on
OE = 30.30°
E
WATCH THE VIDEO that I have provided at this point. I will ask you to use the conclusions
from the video, and I will ask you to repeat some of what you saw.
7.
8.
m < 1 = ______
m < 3 = _______
m < 2 = ______
m < 4 = ______
Honors Geometry Cycle 10 Problems
9.
page 5
Derive a relationship which must be true about m < 1 and m < 2. (You can do an
informal explanation here.) Think about where < 1 is with respect to the triangle, and
what kind of triangle is here.
Another placement of angle which can be calculated is between two chords as they intersect
inside a circle. Look at these examples, and see what conclusion you can make. You will want
to watch the angle measure and the arc measures.
G
L
mGKF = 58.13°
m GF on
m GLH on
O
F
K
H
OL = 62.84°
OL = 162.53°
m HI on
OL = 53.42°
m IF on
OL = 81.21°
I
Let me remove some of the information which is distracting to a conclusion. See the next
Sketchpad drawings.
Honors Geometry Cycle 10 Problems
G
L
page 6
mGKF = 58.13°
m GF on
m HI on
O
F
K
OL = 62.84°
OL = 53.42°
H
I
L
H
G
F
O
K
mGKF = 90.02°
m GF on
OL = 51.96°
m HI on
OL = 128.09°
I
H
G
K
O
F
L
mGKF = 120.00°
m GF on
OL = 51.96°
m HI on
OL = 171.96°
I
10.
Don’t let a little bit of roundoff bother you. How does the measure of an angle between
two chords seem to related to the arc that the angle intercepts together with the arc that
its vertical angle intercepts? Write your conjecture.
Honors Geometry Cycle 10 Problems
11.
page 7
Try to verify your conjecture by considering
ΔGKH and the angle, <GKF, that you have seen
measured in Sketchpad. Think again about
where < GKF is with respect to the triangle, and
use a theorem you already know to develop new
idea about the angle in the circle here.
L
H
G
O
K
I
F
Now, here is one last set of examples. We are still investigating angles associated with circles.
This time, the angle is on the outside (or exterior) of the circle. It is the angle between two
secant lines (or perhaps two tangent lines. or one tangent lien and one secant line).
Take a look at the Sketchpad drawings again to see a relationship between the angle and the arc
measures.
E
D
O
C
G
B
A
mECA = 28.08°
m EA on
OG = 99.12°
m DB on
OG = 42.96°
Honors Geometry Cycle 10 Problems
page 8
E
D
mECA = 38.68°
O
C
G
B
m EA on
OG = 146.11°
m DB on
OG = 68.74°
A
E
mECA = 64.01°
D
O
G
C
B
m EA on
OG = 161.85°
m DB on
OG = 33.83°
A
12.
Do you see a relationship between the angle and the two arc measures that have been
provided? Look at a generic drawing, and consider what you know about angles
associated with triangles to see if this helps.
Look specifically at ΔCBE.
E
D
O
G
B
A
C
Honors Geometry Cycle 10 Problems
page 9
Write a summary of the measures of the angles associated with circles:





The angle between a radius and a tangent line drawn to the point of tangency has
measure ________.
The measure of a central angle of a circle is measured by _______________________.
The measure of an inscribed angle is measured by __________________________.
The measure of an angle between two chords which intersect inside a circle is measured
by ___________________________.
The measure of an angle between two secants which intersect outside a circle is
measured by _______________________________.
Stated in another way:
 You use the measure of an intercepted arc to measure which kind of angle?

You use one-half of the measure of the intercepted arc to measure what kind of angle?

You use one half of the sum of the intercepted arc and the arc of the vertical angle to
measure what kind of angle?

You use one-half of the difference of the two intercepted arcs to measure what kind of
angle?