Unit 10: Circles

Jamie Nordstrom, Aug 16, 2017

Table of Contents

  1. 11.1 Lines that intersect circles
    1. Circle Terminology
    2. Circle Tangent-Types Illustrator
    3. Tangents to Circles: Investigation
    4. Properties of Tangents Drawn to Circles (A)
    5. Animation 3
    6. Circumference = ? (Animation)
  2. 11.2 Arcs and Chords
    1. Congruent Chords: Quick Investigation
    2. Congruent Chords (I)
    3. Diameters & Chords I (VA)
    4. Diameters & Chords I (VB)
    5. Congruent Chords Action (A)
    6. Equidistant Chord Action
    7. Radius Unwrapper v2.0
    8. 1 Radian: Clear Definition
    9. Movie: Radians to Revs
    10. Arc length: Quick Investigation
  3. 11.3 Sector Area and Arc Length
    1. Area of a Sector
    2. TRUE MEANING of π
  4. 11.4 Inscribed Angles
    1. Inscribed Angle Theorem: Take 2!
    2. Inscribed Angle Theorem (V1)
    3. Inscribed Angle Theorem: Corollary 1
    4. Animation 20 (Inscribed Angle Dance!)
    5. Thales' Theorem (VA)
    6. Animation 12
    7. Where to Sit? (I)
    8. Proof Exercise: Inscribed Angle Theorem (Case 1)
    9. Where to Sit (II)?
    10. Proof Exercise: Inscribed Angle Theorem (Case 2)
    11. Animation 116
  5. 11.5 Angle Relationships in circles
    1. Angle Formed by Chord & Tangent (I)
    2. Angle Formed by Chord & Tangent (II)
  6. 11.6 Segment Relationships in Circles
    1. Angle Formed by 2 Chords (I)
    2. Angle Formed by 2 Chords (II)
    3. Angles from Secants and Tangents (V1)
    4. Angle From 2 Secants (V2)
    5. Crossing Chords: Proof Hint
    6. Crossing Chords Property & Proof Start (II)
    7. Crossing Chords Property & Proof Start
    8. Secants: Proof Hint
  7. 11.7 Circles in the Coordinate Plane
    1. Circle Equation: Center (0,0)
    2. Circle Equation: Center (h,k)
    3. Equation & Graph of a Circle
    4. Circle Equation: Center NOT (0,0)
  8. More stuff from Tim's Circles
    1. Circle Terminology
    2. Circle Tangent-Types Illustrator
    3. Properties of Tangents Drawn to Circles (A)
    4. Properties of Tangents Drawn to Circles (B)
    5. Tangents to Circles: Investigation
    6. Where to Sit? (I)
    7. Where to Sit (II)?
    8. Inscribed Angle Theorem (V1)
    9. Inscribed Angle Theorem: Take 2!
    10. Inscribed Angle Theorems: Take 3!
    11. Proof Exercise: Inscribed Angle Theorem (Case 3)
    12. Inscribed Angle Theorems: Take 4!
    13. Inscribed Angle Theorem: Corollary 1
    14. Animation 20 (Inscribed Angle Dance!)
    15. Thales' Theorem (VA)
    16. Thales' Theorem (VB)
    17. Animation 54
    18. Angle Formed by Chord & Tangent (I)
    19. Animation 50
    20. Angle Formed by Chord & Tangent (II)
    21. Animation 78
    22. Congruent Chords: Quick Investigation
    23. Congruent Chords (I)
    24. Animation 76
    25. Diameters & Chords I (VA)
    26. Diameters & Chords I (VB)
    27. Animation 24
    28. Congruent Chords Action (A)
    29. Animation 96
    30. Equidistant Chord Action
    31. Animation 101
    32. Angle Formed by 2 Chords (I)
    33. Animation 32
    34. Angle Formed by 2 Chords (II)
    35. Animation 33
    36. Angles from Secants and Tangents (V1)
    37. Animation 61
    38. Angle From 2 Secants (V2)
    39. Animation 45
    40. Crossing Chords: Proof Hint
    41. Animation 79
    42. Crossing Chords Property & Proof Start
    43. Crossing Chords Property & Proof Start (II)
    44. Secants: Proof Hint
    45. Animation 88
    46. Animation 92
  9. HSG.C.B.5

1.

11.1 Lines that intersect circles

  • 1. Circle Terminology
  • 2. Circle Tangent-Types Illustrator
  • 3. Tangents to Circles: Investigation
  • 4. Properties of Tangents Drawn to Circles (A)
  • 5. Animation 3
  • 6. Circumference = ? (Animation)

1.1.

Circle Terminology

[color=#000000]There are many vocabulary terms we use when talking about a circle. [br]The following app was designed to help you clearly see and interact with each term. [br][br]Explore this app for a few minutes. Then answer the questions that follow. [/color]
Note: LARGE POINTS are moveable.
How would you describe or define a [b]CIRCLE[/b] as a locus (set of points that meets specified criteria)?
How would you describe the term [b][color=#38761d]RADIUS[/color] [/b][i]without using the words "half" or "diameter" [/i]in your description?
What does the term [b][color=#9900ff]CHORD[/color] [/b]mean here in the context of a circle?
How would you describe the term [b][color=#ff7700]DIAMETER[/color] [/b][i]without using the words "two", "double", or "diameter" [/i]in your description?
How would you describe/define the term [b][color=#cc0000]SECANT[/color][/b]?
What does it mean for a line to be [b][color=#1e84cc]TANGENT [/color][/b]to a circle?
– Tim Brzezinski

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1.3.

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1.4.

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1.5.

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1.6.

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2.

11.2 Arcs and Chords

  • 1. Congruent Chords: Quick Investigation
  • 2. Congruent Chords (I)
  • 3. Diameters & Chords I (VA)
  • 4. Diameters & Chords I (VB)
  • 5. Congruent Chords Action (A)
  • 6. Equidistant Chord Action
  • 7. Radius Unwrapper v2.0
  • 8. 1 Radian: Clear Definition
  • 9. Movie: Radians to Revs
  • 10. Arc length: Quick Investigation

2.1.

Congruent Chords: Quick Investigation

Interact with the applet below for a minute or two. [br][br]If [b]two chords of a circle are congruent[/b], what can we conclude about the [b][color=#980000]arcs[/color][/b] these chords determine (i.e. "cut off")?
Quick Demo: 0:00 sec - 0:58 sec (BGM: Andy Hunter)
– Tim Brzezinski

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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3.

11.3 Sector Area and Arc Length

  • 1. Area of a Sector
  • 2. TRUE MEANING of π

3.1.

Area of a Sector

[color=#000000]The following applet was designed to help you discover how to to calculate the [/color][color=#ff00ff][b]area of a sector[/b][/color][color=#000000]. [br]Use this applet as a "check", so to speak, for the examples you work out on today's warm-up. [br][br]Then answer the questions that appear below the applet. [/color]
[color=#000000][b]Questions: [/b] [br][br]1) How would you answer the question that appears on the right side of the applet? [br]2) What is the formula for the area of the polygon name you gave as a response to this question?[br][br]3) Substitute appropriate values (seen in the [/color][color=#ff00ff][b]sector[/b][/color][color=#000000] on the right) into this formula to help derive[br] the formula for the area of a sector. What do you get?[br][br]4) Does using this formula work to obtain the areas of the sectors asked in the questions on your[br] warm-up sheet? Try it out! [/color]
– Tim Brzezinski

3.2.

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4.

11.4 Inscribed Angles

  • 1. Inscribed Angle Theorem: Take 2!
  • 2. Inscribed Angle Theorem (V1)
  • 3. Inscribed Angle Theorem: Corollary 1
  • 4. Animation 20 (Inscribed Angle Dance!)
  • 5. Thales' Theorem (VA)
  • 6. Animation 12
  • 7. Where to Sit? (I)
  • 8. Proof Exercise: Inscribed Angle Theorem (Case 1)
  • 9. Where to Sit (II)?
  • 10. Proof Exercise: Inscribed Angle Theorem (Case 2)
  • 11. Animation 116

4.1.

Inscribed Angle Theorem: Take 2!

The [color=#ff00ff][b]pink angle[/b][/color] is said to be an [color=#ff00ff][b]inscribed angle[/b][/color] within the circle below. [br]This [color=#ff00ff][b]inscribed angle[/b][/color] intercepts the [color=#1e84cc][b]thick blue arc[/b][/color] of the circle. [br]Because of this, this [color=#1e84cc][b]thick blue arc[/b][/color] is said to be the [color=#ff00ff][b]inscribed angle[/b][/color]'s [color=#1e84cc][b]intercepted arc[/b][/color]. [br][br]Notice how the [color=#1e84cc]blue central angle[/color] also intercepts this same [color=#1e84cc][b]thick blue arc[/b][/color]. [br][br][b]To start:[/b][br]1) Move [color=#1e84cc][b]point [i]D[/i][/b][/color] wherever you'd like.[br]2) Adjust the size of the [b][color=#1e84cc]thick blue intercepted arc[/color] [/b]by moving the other 2 [b][color=#1e84cc]blue points [/color][/b](if you wish.) [br]3) Click the checkbox to lock [color=#1e84cc][b]point [i]D[/i][/b][/color]. [br]4) Follow the interactive prompts that will appear in the applet. [br][br]Interact with the following applet for a few minutes. [br]Then, answer the questions that follow.
1.
How many [color=#ff00ff][b]pink inscribed angles[/b][/color] fill a [color=#1e84cc]central angle[/color] that intercepts the [color=#1e84cc][b]same arc[/b][/color]?
2.
How does the [color=#1e84cc]measure of an central angle[/color] (of a circle) compare with the [color=#1e84cc][b]measure of the arc it intercepts[/b][/color]?
3.
Given your responses to (1) and (2) above, how would you describe the [color=#ff00ff][b]measure of an inscribed angle [/b][/color](of a circle) with respect to the [color=#1e84cc][b]measure of its intercepted arc[/b][/color]?
Quick (Silent) Demo
– Tim Brzezinski

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4.5.

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4.7.

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4.8.

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4.9.

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4.10.

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4.11.

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5.

11.5 Angle Relationships in circles

  • 1. Angle Formed by Chord & Tangent (I)
  • 2. Angle Formed by Chord & Tangent (II)

5.1.

Angle Formed by Chord & Tangent (I)

[color=#000000]In the circle in the applet below, the [/color][b][color=#ff00ff]angle[/color][/b][color=#000000] is said to be an [/color][b][color=#ff00ff]angle formed by a chord and a tangent[/color][/b][color=#000000]. [/color]
Be sure to drag the LARGE POINTS each time before you decide to re-slide the slider (lower left)!
What do you notice? Describe!
– Tim Brzezinski

5.2.

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6.

11.6 Segment Relationships in Circles

  • 1. Angle Formed by 2 Chords (I)
  • 2. Angle Formed by 2 Chords (II)
  • 3. Angles from Secants and Tangents (V1)
  • 4. Angle From 2 Secants (V2)
  • 5. Crossing Chords: Proof Hint
  • 6. Crossing Chords Property & Proof Start (II)
  • 7. Crossing Chords Property & Proof Start
  • 8. Secants: Proof Hint

6.1.

Angle Formed by 2 Chords (I)

[color=#000000]In the applet below, the [/color][b]gray angle[/b][color=#000000] is said to be an [/color][b]angle formed by 2 chords of a circle[/b][color=#000000]. [br][br][/color][color=#000000]This applet accompanies the [/color][i][color=#0000ff]Angle Formed by 2 Chords[/color][/i][color=#000000] activity sheet you received at the beginning of class. [br][br]Before answering the questions on this sheet, interact with this applet for a few minutes. As you do, be sure to change the locations of the BIG POINTS each time [/color][i][color=#000000]before[/color][/i][color=#000000] re-sliding the slider! [/color]
Slide the vertical slider (to the right) slowly. Carefully observe what happens.
Quick (Silent) Demo
– Tim Brzezinski

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6.3.

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6.4.

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6.5.

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6.6.

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6.7.

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6.8.

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7.

11.7 Circles in the Coordinate Plane

  • 1. Circle Equation: Center (0,0)
  • 2. Circle Equation: Center (h,k)
  • 3. Equation & Graph of a Circle
  • 4. Circle Equation: Center NOT (0,0)

7.1.

Circle Equation: Center (0,0)

For the questions below, be sure to zoom out if you need to!
1.
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius 5. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
2.
What is the equation of a circle with center (0,0) and radius [i]r[/i] = 9?
3.
Suppose another circle has center (0,0). Suppose this circle also passes through the point (12, -5).[br]Write the equation of this circle. [br]
4. FINAL QUESTION:
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius [i]r[/i], where [i]r[/i] > 0. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
Quick (Silent) Demo
– Tim Brzezinski

7.2.

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7.3.

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7.4.

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8.

More stuff from Tim's Circles

  • 1. Circle Terminology
  • 2. Circle Tangent-Types Illustrator
  • 3. Properties of Tangents Drawn to Circles (A)
  • 4. Properties of Tangents Drawn to Circles (B)
  • 5. Tangents to Circles: Investigation
  • 6. Where to Sit? (I)
  • 7. Where to Sit (II)?
  • 8. Inscribed Angle Theorem (V1)
  • 9. Inscribed Angle Theorem: Take 2!
  • 10. Inscribed Angle Theorems: Take 3!
  • 11. Proof Exercise: Inscribed Angle Theorem (Case 3)
  • 12. Inscribed Angle Theorems: Take 4!
  • 13. Inscribed Angle Theorem: Corollary 1
  • 14. Animation 20 (Inscribed Angle Dance!)
  • 15. Thales' Theorem (VA)
  • 16. Thales' Theorem (VB)
  • 17. Animation 54
  • 18. Angle Formed by Chord & Tangent (I)
  • 19. Animation 50
  • 20. Angle Formed by Chord & Tangent (II)
  • 21. Animation 78
  • 22. Congruent Chords: Quick Investigation
  • 23. Congruent Chords (I)
  • 24. Animation 76
  • 25. Diameters & Chords I (VA)
  • 26. Diameters & Chords I (VB)
  • 27. Animation 24
  • 28. Congruent Chords Action (A)
  • 29. Animation 96
  • 30. Equidistant Chord Action
  • 31. Animation 101
  • 32. Angle Formed by 2 Chords (I)
  • 33. Animation 32
  • 34. Angle Formed by 2 Chords (II)
  • 35. Animation 33
  • 36. Angles from Secants and Tangents (V1)
  • 37. Animation 61
  • 38. Angle From 2 Secants (V2)
  • 39. Animation 45
  • 40. Crossing Chords: Proof Hint
  • 41. Animation 79
  • 42. Crossing Chords Property & Proof Start
  • 43. Crossing Chords Property & Proof Start (II)
  • 44. Secants: Proof Hint
  • 45. Animation 88
  • 46. Animation 92

8.1.

– Tim Brzezinski

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8.17.

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8.18.

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8.19.

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8.20.

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8.21.

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8.22.

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8.23.

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8.24.

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8.25.

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8.26.

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8.27.

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8.28.

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8.29.

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8.30.

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8.31.

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8.32.

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8.33.

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8.34.

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8.35.

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8.36.

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8.37.

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8.38.

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8.39.

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8.40.

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8.41.

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8.42.

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8.43.

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8.44.

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8.45.

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8.46.

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9.

HSG.C.B.5


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