A Groups: Relationships between measures of arcs and central angles, inscribed angles.
Hint: use a protractor to measure
each central angle.
Note: the measure of an arc is
different from the length of an arc.
Investigate the following diagram (1 on GSP)
-Determine the lengths of KM and PN .
-Draw in radii to points K, M, P, N.
-Use a protractor to measure the central
angles.
K
-Determine measures of arcs KM and PN.
-Write a sentence to describe the arcs of
congruent segments.
B
N
M
P
Use point B as a center of dilation. Dilate
the figure using several different scale
factors. Repeat your measurements and
decide whether or not your theory holds.
Given that BC  DE .
Investigate the following diagram (2 on GSP)
Determine the measures of the following:
- BAC
- BC
- BDC
- BEC
- BFC
B
D
E
A
F
Write a sentence to describe a relationship
between the measure of an inscribed angle
and its intercepted arc.
C
Draw another point on the circle between E and F and label it G. Draw in chords from G
to B and G to C. Determine the measure of BGC . Does your theory still hold?
Is it possible for different inscribed angles to have the same measures?
Investigate the following figure. (3 on GSP)
H
What is the measure of LHM ?
How do you know?
L
D
Q
Draw in chords from D to L and D to M,
From E to L and E to M, from F to L and
From F to M, from G to L and from G to M.
M
E
What are the measures of the inscribed angles
LDM , LEM , LFM , LGM ? How do you know?
F
G
Note: When you have a 90 degree inscribed angle, you also have a right triangle. The
Pythagorean theorem may be useful in these situations!
Things to share with your classmates:
-Define central angle
-Relationship of measure of central angle and measure of arc
-Congruent segments have congruent arcs
-Define inscribed angle
-Relationship between inscribed angles and their intercepted arcs
-Inscribed angles that are 90 degrees
-Examples of different homework type problems
B Groups: Relationships between measures of segments (chords, chord parts, secant
segments, tangent segments)
Describe why you think each of these segments/lines was named as they were.
You will be investigating segment lengths formed by these segments and lines.
In the following figure, describe lines CE and CG. (1 on GSP)
Determine the lengths of CD, CE , CF , CG .
E
D
C
F
G
Triangles CDG and CFE are similar (CFE is reflected over CD , rotated, and dilated).
The following figure shows the result after these transformations.
Determine the scale factor
using two different ratios:
CD
and
CF '
CG
.
CE '
These scale factors should be the
same!
D
F'
C
E'
Rewrite the equation
G
CD CG
=
by first multiplying both sides by CF’.
CF ' CE '
Multiply both sides of the equation by CE’.
Write the result of your manipulation.
This result is known as the secant segment rule.
If two secant segments share the same endpoint outside a circle, then the product
of the length of the secant and its external segment is the same as the product of the
length of the other secant and its external segment.
Write the secant segment relationship for the following diagram.
Determine the value of the variable in each problem.
C
B
A
AC=20
BC=2
EC=10
DC=x
D
E
Investigate the following figure. (2 on GSP)
-Determine a list of transformations that
would map triangle DEF onto triangle CBF.
C
B
F
A
D
-If triangles DEF and CBF are similar, the
secant segment relationship applies here, as
well.
-Determine the lengths CF, BF, DF, and EF.
E
Use these lengths to check whether or not
the secant segment relationship holds.
Use the following figure to write out the chord segment relationship.
If two chords intersect in the interior of a circle,
then the product of the lengths of the segments of
one chord is equal to the product of the lengths of
the segments of the other chord.
14
7
4
x
Investigate the following figure. (3 on GSP)
A
Determine the lengths AB, BC, CD.
B
C
D
Triangles ADC and DBC are similar. (reflection, rotation, and dilation). The following
figure shows the result after these transformations.
A
D''
C
B
To determine the scale factor we could use
CD "
CB '
the ratios
or
. They should be the
CA
CD
same.
B'
D
CD " CB '

by first
CA CD
multiplying both sides by CA.
Rewrite the equation
Multiply both sides of the equation by CD.
Write the result of your manipulation.
This result is known as the secant tangent segment rule. If a secant and a tangent
intersect at a point outside the circle, then the product of the length of the secant and its
external segment is the same as the product of the length of the tangent segment and
itself.
Write the secant tangent segment rule for the
given figure.
Determine the value of the variable in each problem.
x
5
4
Write a conjecture about two segments that are tangent to a circle from the same external
point, as shown.
Things to share with your classmates:
-definitions of chord, secant, and tangent and where their names came from
-secant segment rule and where it comes from
-chord segment rule
-secant and tangent segment rule
-tangent segments from same external point
-example homework problems
C Groups: Relationships between measures of angles formed by secants, tangents, radii
Describe why you think each of these segments/lines was named as they were.
Using a compass and a straightedge (or GSP) draw 3 different sized circles and 1 tangent
on each of the circle.
Connect the center of each circle to its point of tangency on the circle. (A radius)
Measure the angle formed between the radius and the tangent line segment.
Write a sentence to describe these measurements. Do you believe these measurements
will always hold? Why?
Investigate the following figure. (1 on GSP)
BC is tangent to circle A.
C
B
Determine the measure of CBD .
Determine the measure of BAD .
A
D
The measure of BAD is also the
measure of BD .
Describe the relationship between CBD and BD .
In general, if a tangent and chord intersect, the measure of the angle formed is equal to
half the measure of the arc it intercepts.
Determine the measure of CBE .
Determine the measure of PSR
Determine the measure of CBD .
Investigate the following figure. (2 on GSP)
Determine the measures of ABC and DBE .
A
mAC=134
What do you notice about these angle
measures?
B
C
Find the sum of the two arc measures.
E
D
Do you notice any relationship between the
angle measure and the sum of the arc
measures?
mED=110
Repeat the same investigation with the following figure.
H
26
I
J
K
40
L
The measure of an angle formed by two chords intersecting inside the circle is equal to
the average of the two chords it intersects.
Determine the value of x in the following problem.
Investigate the following figure. (3 on GSP)
F
Determine the measure of AFB .
C
Find the sum of the two arc measures.
36
E
A
98
B
Find the difference of the two arc measures.
Do you notice any relationship between the
angle measure and the sum of the arc
measures? The difference?
Write a description of the relationship between an angle formed by two secants (or a
secant and a tangent) outside the circle and the arcs it intercepts.
Determine the value of x in the problem.
Things to share with your classmates
-measure of angle formed by a tangent and radius. Extension to using Pythagorean
theorem.
-relationship between angle formed with a tangent and chord and the arc it intercepts.
-relationship between angle formed by chords and the arcs it intercepts.
-relationship between angle formed by secants (or a secant and tangent) and the arcs it
intercepts.