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MATH: GEOMETRY
UNIT 5A: CIRCLES WITH AND WITHOUT COORDINATES - - PART I (3 weeks) 2013-2014
Synopsis: Students will become familiar with different types of angles formed by chords, secants, and tangents
and methods of finding their measures. They will also construct inscribed and circumscribed triangles and
quadrilateral and discover theorems concerning these polygons. Towards the end of this unit, students for the first
time, will be exposed to radian measure when referring to angles and will work with arc measure and area of a sector
with angles in terms of degrees as well as radians.
STANDARDS
G.C.1 Prove that all circles are similar.
G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and
circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius
intersects the circle.
G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle.
G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure
of the angle as the constant of proportionality; derive the formula for the area of a sector.
G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).*
MATH PRACTICES
1. Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.1 Learn to read mathematical text (including textbooks, articles, problems, problem explanations)
L.2 Communicate using correct mathematical terminology
L.4 Listen to and critique peer explanations of reasoning
L.5 Justify orally and in writing mathematical reasoning
L.7 Research mathematics topics or related problems
The following web sites may be helpful in this unit:
http://www.geogebra.org/en/upload/files/english/Guy/Circles_and_angles/Angles_Circles_Lesson.pdf deals with different angles –
shows proofs – student discovery worksheets.
http://www.mathsisfun.com/geometry/circle-theorems.html theorems about angles
MOTIVATION
TEACHER NOTES
1. Students work on star polygon activities using Circular Geoboards star polygons, or the link:
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YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 1
MOTIVATION
TEACHER NOTES
http://starpolygons.com/ (When using the link make number of points 12, connect every 5 points, inradius
0, drawing mode-framework, display mode - - line, inner stars - - increment connecting points.). Copy
attached to unit on page 6. Students can make their own designs by using the circular Geoboard paper
on page 558 in your text resource packet, or using the links:
https://mathaction.wikispaces.com/file/view/Circular+Geoboard+Paper.doc/32622829/Circular%20GeoboardPaper.doc
http://www.mathedpage.org/angles/paper-circles.html
2. Preview expectations for end of Unit
3. Have students set both personal and academic goals for this Unit or grading period.
TEACHING-LEARNING
Vocabulary
Central Angles
Secant
Intercepted Arc
Inscribed
Circumcenter
Radian
TEACHER NOTES
Inscribed Angles
Tangent
Minor Arc
Circumscribed
Orthocenter
Sector
Semicircle
Chord
Major Arc
Incenter
Centroid
1. To introduce students to circles go to geometry sketchpad or an equivalent software, create several
circles, show a translation (moving smaller circle on top of the other) and dilation (enlarging the smaller
circle to be exactly like the larger circle) to illustrate similarity. Have students state a theorem that is
illustrated by this and then prove it. To verify their proof, present the formal proof on the link
http://www.cpm.org/pdfs/state_supplements/Similar_Circles.pdf also attached to the unit on page 7-8.
(G.C.1, MP.2, MP.3, MP.4, MP.5, MP.6, MP.8, L.1, L.2, L.5)
2. The Glencoe Geometry Ohio Edition 2005 edition text provides the bulk of the material for the standards
in Chapter 10. (Note: There are three concepts listed below marked with a* that are not included in the
textbook. There is material online and in various texts to be used to develop these. It is the teacher’s
discretion as to the method of teaching, presenting proofs, working problems and/or discovery activities.
A few discovery activities are listed and can be used. To illustrate many of these theorems, discovery
activities can be developed with students measuring angles, chords, secants, parts of chords, parts of
secants and parts of tangents with protractors and rulers and then making conclusions from these. )
Use the text, its resources and any additional resources as needed to cover the following:
a) central angles, page 529-530
b) inscribed angles, page 544-551 (section 10-4). An activity to illustrate this would be to give
students several drawings of a central angle and an inscribed angle intercepting the same arc.
Have them measure the inscribed angle and the central angle to show the inscribed angle is half
the central angle, concluding an inscribed angle is half its intercepted arc.
c) special case – angle inscribed in a semicircle is a right angle page 547, use dynamic geometry
to show this by moving the vertex of the angle around the semicircle showing that it is always a
right angle
http://www.learnalberta.ca/content/memg/Division03/Central%20Angle%20Property/index.html
applet that allows movement of the inscribed angle and also movement of the radii to discover
measure of an inscribed angle in relation to the arc and central angle.
d) angles formed by secant and tangent intersecting on the circle, page 562
e) angles formed by two secants intersecting inside the circle, page 561-562
f) angles formed by tangent and secant, two secants or two tangents intersecting outside the
circle, page 563-564
g) radius perpendicular to chord bisects it, page 537. The following discovery activity can be
used with students: Give students several drawings of circles with the center marked. Have
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YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 2
TEACHING-LEARNING
TEACHER NOTES
them draw a chord and then a radius perpendicular to the chord using a protractor. Ask them to
measure the two sections of the chord, repeat several times. Then ask them to state and explain
their conclusion.
h) if radius bisects a chord, it is perpendicular to it, The following discovery activity can be used
with students: Give students several drawings of circles with the center marked. Have them
draw a chord and bisect it, then draw a radius from the center to the midpoint of the chord. Using
a protractor, ask them to measure the angle where the radius and chord meet. Repeat several
times. Then ask them to state and explain their conclusion.*
i) perpendicular bisector of chord passes through the center. The following discovery activity
can be used with students: Give students several drawings of circles with the center marked.
Have them draw a chord and then using a protractor draw the perpendicular bisector of it. Ask
the students to explain the relationship between the perpendicular bisector and the circle.*
j) Two central angles are congruent if and only if their intercepted arcs are congruent; page
530
k) two central angles are congruent if and only if their intercepted chords are congruent; page
530
l) two arcs are congruent if and only if their intercepted chords are congruent; page 536
m) Two chords intersecting inside a circle – the product of the segments of one chord equal
the product of the segments of the other chord. Page 569
n) A radius is perpendicular to a line at its outer endpoint if and only if it is tangent to the
circle at the point of intersection. Page 553
o) Two tangents drawn from an outside point are congruent. At this time have students
construct a tangent line from a point outside a circle to the circle. Have them construct another
tangent through the same outside point, measure the two tangents and arrive at a conclusion.
From that formulate the theorem: Two tangents from an outside point are congruent.
Construction on page 554 and problems on page 555- 557
p) The sum of the measures of a tangent-tangent angle and its minor arc is 180°
q) If a tangent and secant intersect, then the square of the tangent segment equals the
product of the outside part of the secant times the whole secant, page 571
r) If two secants intersect outside a circle, then the product of the outside part times the
whole secant of one equals the product of the outside part times the whole secant of the
other. Page 570
(G.C.2, G.MG.1, G.C.4, MP.1, MP.2, MP.3, MP.4, MP.5, MP.6, MP.7, MP.8, L.1, L.2, L.4, L.5)
3. After discussing angles, radii, chords, tangents and secants, take a break and work on inscribed and
circumscribed triangle constructions found on pages 559-560. Prior to the constructions, have students
research the terms: incenter, circumcenter, orthocenter, and centroid; discuss them as a class and
formulate their own definitions. (G.C.3, MP.4, MP.5, MP.6, MP.7, L.1, L.2, L.4, L.7)
4. Have students construct a quadrilateral inscribed in a circle, measure angles and form a conjecture.
Have students discuss their findings and critique. (Opposite angles are supplementary). Then have
students draw a parallelogram inscribed in a circle and make a conjecture about the parallelogram. (It is
a rectangle). Work on problems on pages 548-550. (G.C.3, MP.1, MP.2, MP.3, MP.4, MP.5, MP.6, L.2,
L.4, L.5) The following link may be helpful:
http://www.math.uakron.edu/amc/Geometry/HS_MSGeometryLessons/6.3inscribedquadsandllogram.pdf
5. Students have been working with degrees and it is now time to introduce them to radians. A central
angle whose measure is one radian is one that is formed by subtending an arc whose measure is equal
to the radius of the circle. Since all circles are similar, it can be shown that the central angles whose
measure is equal to one radian are similar in circles of different radii.
An activity that helps students understand this is to have them cut out sectors of circles with varying
radii whose arc length is equal to the radius and conclude that the angles are the same size or use
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YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 3
TEACHING-LEARNING
TEACHER NOTES
concentric circles and mark off central angles whose arc length is equal to the radii. Once students
understand the meaning of a radian then proceed to discuss the relationship between radians and
degrees starting with 2πr = 3600 which leads to πr = 1800, dividing both sides by π gives 1 radian =
or dividing by 180 gives
. Discuss when dividing the circle into 4ths, 6ths, 8ths, 12ths, what is
the measure of the central angles formed in radians. Practice changing degrees to radians and radians
to degrees. Worksheet is attached on page 9-10. (G.C.5, MP.2, MP.4, MP.5, MP.6, MP.7, L.2)
The following link may be helpful:
http://learni.st/users/S33572/boards/3056-arc-length-and-radians-common-core-standard-9-12-g-c-5
6. Have students arrive at an intuitive definition of arc length which is a fractional part of a circle
times the circumference of a circle. From this definition the formula for arc length with central
angle in degrees would be:
. For radians: s = rè where s is the arc
length, r is the radius, and θ is the central angle subtending the arc. Reinforce with additional
problems. Worksheet is attached on page 11. (G.C.5, MP.2, MP.3, MP.4, MP.7, L.2, L.5)
7. Have students arrive at an intuitive definition of area of a sector, which is a fractional part of a
circle times the area of a circle: A =
for degrees and A = r2 è for radians.
Reinforce with additional problems. Worksheet is attached on page 12-13. (G.C.5, MP.2,
MP.3, MP.4, MP.7, L.2, L.5) The following link contains problems and explanations for T/L #5,
#6 and #7. http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-radians-2009-1.pdf contains
problems involving radians, degrees, arc length, and area of a sector.
TEACHER NOTES
TEACHER CLASSROOM ASSESSMENT
1. Quizzes
2. In class participation and practice problems for each concept
3. 2- and 4-point questions
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Paper-pencil test with M-C questions and 2- and 4-point questions
AUTHENTIC ASSESSMENT
1. Students evaluate goals for the unit. Or on a weekly basis
TEACHER NOTES
2. A company has decided to use the design below as an architectural pattern on building. They would
like you to assign a measure to the radius and tangent and then find the perimeter and area of the figure
with your given radius and tangent. Name the circle P. To find the perimeter and area, first find the
measure of <PAC, measure of <APC (show work), measure of <APB (show work). Show work for finding
the perimeter and area. Create a scale drawing of your figure and label the points.
(G.C.2, G.C.4, G.MG.1, MP.2, MP.4, MP.5, MP.7, L-2)
A
C
B
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YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 4
RUBRIC
ELEMENTS OF THE
0
PROJECT
State radius and
Did not attempt
tangent length
1
2
3
State either radius or
tangent length
State both radius and
tangent length without
dimensions
Angle measure
correct
Used correct trig
function – error in
calculations
Used an acceptable
method but made
errors
Made scale drawing
without labels
Found arc length or
tangents correctly and
added them
Found area of sector
or triangle correctly
and subtracted them
State both radius and
tangent length with
dimensions
N/A
Measure of <PAC
Did not attempt
Measure of <APC
Did not attempt
Measure of <APB
Did not attempt
Used wrong method
for finding angle
Scale drawing
Did not attempt
Drawing not to scale
Perimeter of figure
Did not attempt
Found arc length or
tangents
Area of figure
Did not attempt
Found area of sector
or triangle
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Angle measure
incorrect
Used incorrect trig
function
Found angle correctly
Found angle correctly
Made scale drawing
with labels
Found arc length and
tangents correctly and
added them
Found area of sector
and triangle correctly,
subtracted them,
doubled that result
and added it to the
area of the circle
YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 5
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YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 8
T/L/ #5: Worksheet 1: Changing degrees to radians and radians to degrees
Change the following to radians, leave answers in terms of π:
1. 600
2. 1800
3. 1350
4. 2700
5. 3300
6. 2250
7. 1500
8. 3150
9. 300
10. 2400
11. 3000
12. 450
Change the following to radians, leave answers in decimal form
1. 750
2. 350
3. 1400
4. 3120
5. 2560
6. 1000
7. 2700
8. 1300
9. 2000
10. 2800
11. 3000
12. 3500
Change the following to degrees:
1.
2.
3.
4.
7. 2.4r
5.
8. 0.25 r
6.
9. 1.6 r
10. 1.2 r
11. 3.4 r
12. 5.8 r
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YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 9
T/L/ #5: Worksheet 1: ANSWERS: Changing degrees to radians and radians to degrees
Change the following to radians, leave answers in terms of π:
Answers:
2. π
1.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Change the following to radians, leave answers in decimal form
Answers:
1. 1.309
2. 0.611
3. 2.444
4. 5.445
5. 4.468
6. 1.745
7. 4.712
8. 2.269
9. 3.491
10. 4.887
11. 5.236
12. 6.109
Change the following to degrees:
Answers:
1. 60
4. 135
7. 137.5
10. 68.8
2. 150
5. 270
8. 14.3
11. 194.8
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3. 120
6. 330
9. 91.7
12. 332.3
YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 10
T/L #6: Worksheet #2: Find the length of the following arcs:
1. In a circle of radius 10″, find the length of an arc whose central angle is 750.
2. In a circle of radius 4 cm, find the length of an arc whose central angle is 1750.
3. In a circle of radius 8″, find the length of an arc whose inscribed angle is 250.
4. In a circle of radius 5′, find the length of an arc whose central angle 2.3r.
5. In a circle of radius 12 m, find the length of an arc whose central angle 1.5r.
6. In a circle of radius 25″, find the measure of the central angle if the length of an arc is 50″.
_________________________________________________________________________________________________
Worksheet #2: Answers:
1. 13.1″
4. 11.5 ′
2. 12.2 cm
5. 18 m
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3. 3.5 ″
6. 2r
YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 11
T/L #7: Worksheet #3: Find the area of the following sectors:
1. A vegetable patch which is in the shape of a sector has a radius of 5m and an inside angle of 76⁰. Work out the area of
the vegetable patch.
2. A stretched out paper fan forms a sector with a radius of 18 cm and an angle of 175⁰. Calculate the area of the stretched out fan
using the formula A = x/360 × π × r². Give your answer to 1 decimal place.
3. A window wiper of length 20 inches goes through an angle of 160⁰. Work out the area of window covered by the window wiper.
4. A silver pendant is made in the form of a sector of a circle. If the radius is 3 cm, what is the angle, in radians so that the area
is 6 cm?
5. Find the area of a piece of a 16″ pizza if the central angle is 1.2r.
6. A manufacturer that makes discs used to move furniture has decided that the discs should be in the shape of a sector. If the
company has specified a disc of radius 15cm. and the central angle is 1.4r, find the area of the sector shaped disc.
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Worksheet #3: Answers:
1. 16.6 m2
4. 1.3r
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2. 494.5 cm2
5. 153.6 sq. in.
3. 559 sq. in.
6. 157.5 sq. cm
YCS Geometry: Unit 5A: Circles With and Without Coordinates Part I 2013-2014 13