Download Acquisition Lesson – Segments of

Acquisition Lesson Planning Form
Key Standards addressed in this Lesson: MM2G3a
Time allotted for this Lesson: 10 Hours
Essential Question: LESSON 3 – Segments of Circles
How do we use triangle similarity to understand properties of intersecting chords, tangents,
and secants of circles?
Activating Strategies: (Learners Mentally Active)
Session 1
 Cut terms from the page “Activating Strategy - Parts of a Circle”. Give each student a
card with a term for a part of a circle. In collaborative pairs, students will discuss the
part of the circle that is on his/her card. Draw a large circle on board. Each student
will draw, label, and identify the part of the circle that is on his/her card.
Session 2
 Complete and answer questions for “Activating Strategy – Similar Triangles”.
Subsequent Sessions
 Use the activating strategy “Chords and Similar Triangles” with collaborative pairs.
Think-Pair-Share
.
Acceleration/Previewing: (Key Vocabulary)
Chord, secant, tangent, point of tangency, central angle, inscribed angle, intercepted arc,
similar triangles, AA similarity
Teaching Strategies: (Collaborative Pairs; Distributed Guided Practice; Distributed
Summarizing; Graphic Organizers)
Session 1

Students will work in collaborative pairs. Using a ruler, students will complete “Chords
of a Circles” worksheet. Whole class will discuss results.

Practice problems. (supplemental material) – radius drawn perpendicular to chord,
radius bisects chord and arc, equal chords intercept equal arcs.

Students will work in collaborative pairs. Using a ruler, students will complete
“Tangents of Circles” worksheet. Whole class will discuss results.

Practice problems. (supplemental material) – tangents to circles
Session 2

Discuss similar triangles from activating strategy. Ask pairs to go to board and draw 2
non-similar triangles with 2 pairs of congruent angles. As a whole group, discuss how
this is not possible because If two angles of one triangle are congruent to two
angles of another triangle, then the two triangles are similar (AA similarity).
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 1

In pairs, students should complete GO “What relationship exists among segments of
two intersecting chords in a circle?” As a whole group, discuss results.

Practice problems. (supplemental material)
Session 3

In whole group, discuss activating strategy.

In collaborative pairs, students will complete GO “What relationship exists among the
segments of two secants drawn from a common point outside the circle?” and “What
relationship exists among the segments of a secant and a tangent drawn from a
common point outside the circle?” In whole group, discuss results.
Distributed Guided Practice/Summarizing Prompts: (Prompts Designed to Initiate
Periodic Practice or Summarizing)
“If two angles in one triangle are congruent to two angles in another triangle, what can be
determined about the third angles? Are the two triangles congruent? Are the two triangles
similar?”
“What is the relationship between two inscribed angles that intercept the same arc?”
“What relationship exists among segments of two intersecting chords in a circle?”
“What relationship exists among the segments of two secants drawn from a common point
outside the circle?”
“What relationship exists among the segments of a secant and a tangent drawn from a
common point outside the circle?”
Extending/Refining Strategies:
Students will create a foldable graphic organizer that describes angles and segments that are
on, inside, and outside circle. Fold paper hotdog style and divide into three parts: On the
circle, Inside the circle, and Outside the circle. On inside of foldable for each category, put
angle information on top half and segment information on bottom half. Using the foldable
GO, students will work practice problems. (supplemental material)
Summarizing Strategies: Learners Summarize & Answer Essential Question
Session 2
 As the students enter the room, they will be given worksheet “Summarizing - Similar
Triangles?”. Students will decide if each set of triangles is similar. Patty paper may be
used to help students.
 Additional Tasks/Activities on pages 13 – 15 may be used as needed.
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 2
Activating Strategy - Parts of a Circle
CENTER
MINOR ARC
MAJOR ARC
CHORD
INSCRIBED ANGLE
CENTRAL ANGLE
TANGENT
SECANT
RADIUS
DIAMETER
SEMICIRCLE
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 3
Activating Strategy – Similar Triangles
Use the triangles at the right to answer each question. ABC  DEF.
1.
Measure each segment and angle of the triangles.
E
AB = _____
DE = _____
BC = _____
EF = _____
B
AC = _____
DF = _____
mA = _____
mD = _____
mB = _____
mE = _____
mC = _____
mF = _____
F
C
A
2.
What is the scale factor of ABC to DEF?
3.
AB
4.
What angle relationship do you observe?
=
_
=
DF
D
BC
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 4
Activating Strategy
Inscribed Angles and Similar Triangles
A
B
1
E
2
D
C
Using the figure above, complete the following:
1. B 
1
arc ______
2
C 
1
arc ______
2
What can you conclude about B and C ?__________________________________
2. What relation exists between 1 and 2 ? Explain how you know.________________
_________________________________________________________________________
3. From the results in steps 1 and 2, what must be true about A and D ? Explain your
answer.__________________________________________________________________
_________________________________________________________________________
4. What conclusion can be reached about ∆ ABE and ∆ DCE? Explain your reasoning.
__________________________________________________________________________
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 5
Chords of a Circle
O
1.
Draw the following chords: AB, CD, EF, and GH.
2.
Measure the length of each chord.
AB= _____, CD = _____, EF = _____, GH = _____
3.
Measure the distance from the chord to the center of the circle.
The distance from the center of the circle to:
AB = _____, CD = _____, EF = _____, GH = _____
4.
Mary made the following conjecture: If two chords are the same distance from the
center of the circle, the chords are congruent.
a.
Do you agree or disagree?
b.
Support your answer mathematically.
c.
State the converse of this conjecture.
d.
Explain whether or not the converse is true.
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 6
Tangents of Circles
1.
Measure the lengths of the following line segments.
AB = ____ and BC = ____
DE = ____ and EF = ____
GH = ____ and HI = ____
2.
What conjectures can you make?
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 7
3.
Use a protractor to find the measure of the following angles.
mJKL = ____
mMNO = ____
mPQR = ____
4.
What conjectures can you make?
5.
Using the figure at the right,
which line segments are the
same length?
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 8
What relationship exists among segments of two
intersecting chords in a circle?
C
A
Draw chords AC and BD.
Can you show CAB  BDC? _____
X
Why? ________________________
Can you show ACD  ABD? _____
Why? ________________________
Is ACX ~ DBX? _____
Why? ________________________
D
Complete the following proportion.
B
AX ?

? XB
So
 AX XB  ? ?
When two chords intersect, the ___________ of the segments of one chord equals the
__________ of the segments of the other chord.
Try this one.
3
4
n
n = ____
10
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 9
What relationship exists among the segments of two
secants drawn from a common point outside the circle?
B
Draw chords XC and BY.
Can you show ABY  ACX? _____
X
Why? ____________________
Is ABY ~ ACX? _____
A
Why? ____________________
Y
Complete the following proportion.
C
AX AC

?
?
SO
AX ?  ? AC
When two secants are drawn from a common point, the ___________ of the outside
part and the whole secant equals the __________ of the outside part and the whole
other secant.
Try this one.
n = ____
n
4
2
8
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 10
What relationship exists among the segments of a
secant and a tangent drawn from a common point
outside the circle?
Draw chords BC and BD.
B
Can you show ABD  ACB? _____
Why? ____________________
A
Is ABD ~ ACB? _____
Why? ____________________
Complete the following proportion.
D
AD ?

? AC
C
SO
 ?2  AD AC
When a secant and a tangent are drawn from a common point, the ___________ of
the tangent equals the __________ of the outside part and the whole secant.
Try this one.
n
4
12
Math 2 Unit 3 Lesson 3
n = _____
Segments of Circles
Page 11
Summarizing - Similar or Not?
1.
Is ABC ~ DEF?
2.
Is ABD ~ ACE?
3.
Is ABC ~ DFE?
B
A
60°
D
30°
30°
E
90°
C
F
4.
Is ABD ~  ACB?
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 12
Additional Tasks/Activities
Lines and Line Segments of a Circle Learning Task
http://www.geogebra.org/en/wiki/index.php/Circles_%28Angles%29 :refer to the perpendicular
and center of circle investigations ONLY
Alternate version of center of circles:
Finding the Center Again
Show the students a broken plate or some circular object that has only part showing. Ask
them if they have any ideas about how to find the entire circle. Anthropologists find artifacts
that are only parts of the complete item and must work to discover what the item might be –
depending on its size. Do astronomers see an entire crate on the moon or must they use
mathematics to determine the size of the crater?
After the discussion the following activity can be done with MIRAs. Have the students
construct a circle on a sheet of paper and construct two nonparallel and noncongruent chords.
Using a MIRA construct the perpendicular bisector of each chord. Do these two perpendicular
bisectors intersect? _______________ Do you notice anything about the intersection?
__________________ Compare with your neighbors and try to fill in the blanks of the
following statements. The perpendicular bisectors of chords of a circle
_________________________________________.
The perpendicular from the center of a
circle to a chord is the ___________________ of the chord. (And the _____________ of the
arc.) If two chords of a circle are congruent then they determine two central angles of the
circle that are ______________.
The following link investigates the length of intersecting chords:
http://www.geogebra.org/en/upload/files/UC_MAT/chords_in_a_circle.html
The following link investigates the length of secant segments and tangent segments:
http://www.geogebra.org/en/upload/files/UC_MAT/chords_outside_a_circle.html
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 13
After the students have written the relationships for the sides, they should think about how to
prove what they have learned through experimentation.
This theorem states that a×b is always equal to c×d no matter where the chords are
placed. By adding two segments in the picture, you can create two triangles:
c
a
b
d
Figure 1
Challenge the students to prove this relationship is true by proving these triangles are
similar. What condition for the similarity do you use to prove it? By establishing proportional
equations for corresponding sides of the similar triangles, you can make the same conclusion
as the theorem.
Again, using the using the same principle, have the students prove that the secants formula
they discovered works:
B
A
P
C
D
Figure 2
Prove this theorem by adding segments BC and AD to the above figure.
Find pairs of angles of the same size, and determine which triangles are similar to each other.
Write a proportion using ratios of the corresponding sides of these triangles
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 14
Another Way to Find the Center and Facts about Tangents
Learning Task
Draw a circle using a compass and mark the center O
Put a pencil point on the circle and then put the straight edge up next to the pencil so that it
touches the circle in one and only one point. Label that point P (point of tangency). Use the
straight edge to draw the tangent line through P and mark point A on the line. Construct the
radius OP . Measure angle APO using a protractor.
Repeat the above activity using the same circle. Put another point S on the circle. This time
construct a line through the radius OS and either construct a perpendicular at S with compass
and straight edge or use a MIRA to construct the perpendicular at S  OS . Mark point B on
the perpendicular line.
Are both APO and BSO both right angles? _______________ Check with your neighbors
to see if they got the same results and complete the following statements. A tangent to a circle
____________________ to the radius drawn to the point of tangency.
B
D
A
C
BA  DB
and
AC  DC
With this information, what can you conclude about two tangents drawn from the same point
outside the circle?
What do these tangent conjectures have to do with space travel?
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 15
Summary of Angles and Segments:
Location of Vertex
Relationship of Angle and Arcs
On the center of circle
On the circle
Outside the circle
Inside the circle but not the
center of the circle
Segments
Relationship of Lengths of Segments
Intersecting chords
Chord and Tangent at Point
of Tangency
Two secants
Two tangents
Intersecting secant and
tangent
Math 2 Unit 3 Lesson 3
Segments of Circles
Page 16