Circles (Theorems)
Circle Theorems, Circle Properties, etc...
Table of Contents
- Circle Vocabulary
- Circle Terminology
- Circle Tangent-Types Illustrator
- Tangents
- Properties of Tangents Drawn to Circles (A)
- Properties of Tangents Drawn to Circles (B)
- Properties of Tangents
- Inscribed Angle Theorems & Corollaries
- Inscribed Angle Theorem (V1)
- Inscribed Angle Theorem: Take 2!
- Inscribed Angle Theorems: Take 3!
- Where to Sit? (I)
- Where to Sit (II)?
- Inscribed Angle Theorem: Corollary 1
- Inscribed Angle Theorem Dance: Take 2!
- Animation 20 (Inscribed Angle Dance!)
- Inscribed Angle Intercepts Semicircle
- Cyclic Quadrilaterals (IAT: Corollary 3)
- Inscribed Angle Theorems: Take 4!
- Theorems Involving Chords
- Diameters & Chords I (VA)
- Diameters & Chords I (VB)
- Angle Formed by 2 Chords (I)
- Angle Formed by 2 Chords (II)
- Angle Formed by Chord & Tangent (I)
- Angle Formed by Chord & Tangent (II)
- Congruent Chords Action (A)
- Equidistant Chord Action
- Secant & Tangent Theorems
- Angles from Secants and Tangents (V1)
- Angle From 2 Secants (V2)
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?
Properties of Tangents Drawn to Circles (A)
In the applet below, 2 tangent rays are drawn to a circle from a point outside that circle. [br][br]Interact with the applet below for a few minutes. As you do, be sure to change the locations of the [b][color=#3c78d8]blue points[/color][/b] (i.e. alter the size of the circle) and [b][color=#bf9000]yellow point[/color][/b]. [br] [br]After doing so, please answer the questions that follow.
1.
What is the measure of each [b][color=#ff00ff]pink angle[/color][/b] displayed? Explain how you know this.
2.
What can you conclude about the [b]distances[/b] from the [b][color=#bf9000]yellow point (outside the circle)[/color][/b] to each of the smaller white points on the circle?
Quick (Silent) Demo
3.
Given your response for (1), how could we prove that your response to (2) is true? (Assume we didn't see the rotation at the very end of the animation).
Inscribed Angle Theorem (V1)
[b][color=#ff00ff]The PINK ANGLE is said to be an INSCRIBED ANGLE[/color][/b] of a circle. [br][br]You can move the pink point anywhere on the NON-BLUE arc of the circle. [br][color=#0000ff][b]You can change the size of the BLUE intercepted arc[/b][/color] by moving either of the white points. [br]You can also adjust the circle's radius using the [color=#666666][b]GRAY POINT[/b][/color]. [br][br]Answer the questions that follow.
1.
Without looking up the definition on another tab in your internet browser, [b][color=#ff00ff]how would you describe (define) the concept of an inscribed angle of a circle? [/color][/b]
2.
[b][color=#ff00ff]How many inscribed angles[/color][/b] fit inside the [b][color=#0000ff]blue central angle[/color][/b] that intercepts (cuts off) the [b][color=#0000ff]same arc[/color][/b]?
3.
Given your result for (2), how does the [b][color=#ff00ff]measure of the pink inscribed angle[/color][/b] compare with the [color=#0000ff][b]measure of the blue intercepted arc? [/b][/color]
4.
Try testing your informal conclusions for (responses to) (2) and (3) a few times by dragging the slider back to its starting position, [b][color=#ff00ff]changing the location of the pink inscribed angle[/color][/b], and [b][color=#0000ff]changing the size of the blue intercepted arc[/color][/b]. [br][br]Then slide the slider again. [br][b][br]Do your conclusions for (2) and (3) ALWAYS hold true? [/b]
Quick (Silent) Demo
Diameters & Chords I (VA)
[color=#000000]Once you slide the slider in the applet below, you'll immediately notice a chord drawn and a[/color] [b][color=#6d9eeb]diameter drawn to this chord. [/color][/b][br][br][color=#000000]Interact with this applet for a few minutes, then answer the questions that follow. As always, be sure to change the locations of the BIG POINTS at any time before re-sliding the slider. [/color]
[color=#980000][b]Questions:[/b][/color][br][br][color=#000000]1) What does the movement of the[/color] [color=#ff00ff]pink angle[/color] [color=#000000]imply about [b]the intersection[/b] of the diameter and this chord? [/color][br][br][color=#000000]2) Fill in the blank: [br][br]In essence, the action you observed from the[/color] [color=#ff00ff]pink angle[/color] [color=#000000]implies that this[/color] [color=#3d85c6]diameter[/color] [color=#000000]and this chord are __________________. [/color][br][br][color=#000000]3) If a[/color] [color=#3d85c6]diameter[/color] [color=#000000]intersects a chord in the manner you've described in your response to (2) above, what does this[/color] [color=#3d85c6]diameter[/color] [color=#000000]do to this chord? How was this evidenced in the applet above? [br][/color][br][color=#000000]4) If a[/color] [color=#3d85c6]diameter[/color] [color=#000000]intersects a chord in the manner you've described in your response to (2) above, what does this[/color] [color=#3d85c6]diameter[/color] [color=#000000]do to[/color] [b][color=#000000]the entire arc[/color][/b] [color=#000000]this chord intercepts? How was this evidenced in the applet above? [br][br]5) Formally prove your responses to exercises (3) and (4) above using a 2-column format. [/color]
Angles from Secants and Tangents (V1)
[color=#000000]Interact with the applet below for a few minutes. Then answer the questions that follow. [br][br][/color][color=#0000ff]Be sure to change the locations of the[/color][color=#000000] [b]BIG POINTS[/b] [/color][color=#0000ff]each time [i]before[/i] you slide the slider. [/color]
1.
Suppose the [color=#ff00ff][b]entire pink arc measures 200 degrees[/b][/color] and the [color=#1e84cc][b]entire blue arc measures 50 degrees[/b][/color]. [br][b][color=#bf9000]What would the measure of the manila angle be? [br][/color][/b]
2.
[b][color=#ff00ff]Move ANY ONE (just ONE -- doesn't matter which) of the PINK POINTS[/color][/b] so the secant segment (for which this pink point is an endpoint) becomes TANGENT to the circle. [br][br]Answer question #1 again within THIS CONTEXT.
3.
Now [color=#ff00ff][b]m[/b][b]ove the pink points[/b][/color] so that BOTH secant segments become TANGENT SEGMENTS.[br]Suppose, in this case, the entire pink arc measures 200 degrees. [br][br][b][color=#1e84cc]What would the measure of the blue arc be? [/color][/b][br][color=#bf9000][b]What would the measure of the manila angle be? [/b][/color]
4.
Next, move the [b][color=#bf9000]MANILA POINT[/color][/b] (outside the circle) as close to the circle as possible so that the [b][color=#1e84cc]blue arc[/color][/b] almost disappears. (It won't disappear entirely). Keep the [b][color=#bf9000]MANILA POINT[/color][/b] on the circle. Now slowly re-slide the slider again. [br][br]What previously learned theorem do these transformations reveal? [br]
5.
Suppose the 2 secant segments (drawn from the [b][color=#bf9000]manila point[/color][/b] outside the circle) intersect the circle above so that the [b][color=#bf9000]manila angle measures 60 degrees[/color][/b] and the [b][color=#ff00ff]entire pink arc measures 200 degrees[/color][/b]. If this is the case, [color=#1e84cc][b]what would the measure of the entire blue arc be[/b][/color]?