Circles (Theorems)
Circle Theorems, Circle Properties, etc...
Table of Contents
- Circle Vocabulary
- Circle Terminology
- Tangents
- Properties of Tangents (B)
- Inscribed Angle Theorems & Corollaries
- Inscribed Angle Theorem (V1)
- Inscribed Angle Theorem: Take 2!
- Where to Sit? (I)
- Inscribed Angle Intercepts Semicircle
- Cyclic Quadrilaterals (IAT: Corollary 3)
- Theorems Involving Chords
- Congruent Chords: Quick Investigation
- Diameters & Chords I (VB)
- 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
- Crossing Chords: Proof Hint
- Secant & Tangent Theorems
- Angles from Secants and Tangents (V1)
- Angle From 2 Secants (V2)
- Secants: Proof Hint
- Older (Earlier) Applets
- Inscribed Angle Theorem (Proof without Words)
- Inscribed Angle Theorem (Corollary 1) (Proof without Words)
- Inscribed Angle Theorem (Corollary 2) (Proof without Words)
- Inscribed Angle Theorem (Corollary 3) Cyclic Quadrilaterals
- Angle formed by a Secant & Tangent: What Do You See? (V1)
- Angle Formed by a Secant & Tangent: What Do You See? (V2)
- Angle Formed by 2 Tangents (V2): What Do You See?
- Cyclic Quadrilaterals
- Congruent Chords in the Same Circle
- Inscribed Angle Theorem (Intro): What Do You See Here?
- Inscribed Angle Action !!!
- Inscribed Angle Action! (2 Theorems)
- Angle Formed by 2 Chords (V1): What Do You See?
- Angle Formed by 2 Chords (V2): What Do You See?
- Chords Equidistant from Circle's Center
- Crossing Chords Property & Proof Start
- Crossing Chords Property & Proof Start (II)
Circle Terminology
[color=#000000]There are many vocabulary terms we use when talking about a circle. [br]The following applet was designed to help you clearly see (and interact) with each term. [br]Mess around with this applet for a few minutes. [br][br][/color][i][b][color=#000000]As you do, be sure to change the locations of the BIG POINTS displayed! [/color][br][br][/b][/i][b][color=#000000]Use this applet to help you author sentence definitions for each of these terms on your [/color][i][color=#0000ff]Circle Terminology [/color][/i][color=#000000]activity sheet given to you at the beginning of class. [/color][/b]
Properties of Tangents (B)
[color=#000000]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, then answer the questions that follow. [/color]
[color=#980000][b]Questions:[/b][/color][color=#000000][br][br]1) How would you describe the intersection of a radius drawn to either point of tangency? [br][br]2) What can you conclude about the lengths [i]EB[/i] and [i]EA[/i]? [br][br][/color]
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.
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)
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]?
Inscribed Angle Theorem (Proof without Words)
Recall that the measure of an [color=#0a971e]arc[/color] of a circle is the same as the measure of its corresponding [color=#0a971e]central angle[/color]. (See applet.) [br][br][b]Definition[/b]: An [color=#b20ea8][b]INSCRIBED ANGLE[/b][/color] of a circle is an angle whose vertex lies on the circle and has each of its rays intersect the circle at one other point. (Click checkbox to show [color=#b20ea8]inscribed angle[/color].) [br][br]Notice how both the [color=#b20ea8]inscribed angle[/color] and [color=#0a971e]central angle[/color] both intercept the same [color=#0a971e]arc[/color]. [br][br]Click on the [color=#c51414]CHECK THIS OUT !!![/color] checkbox that appears afterwards. Be sure to move points [color=#1551b5]A[/color], [color=#1551b5]B[/color], and the [color=#b20ea8]pink vertex[/color] of the [color=#b20ea8]inscribed angle[/color] around. (You can also change the radius of the circle if you wish.)[br][br][b]In a circle, what is the relationship between the measure of an [color=#b20ea8]inscribed angle[/color] with respect to the measure of its [color=#0a971e]intercepted arc[/color]? [/b]
Inscribed Angle Theorem (Proof without Words)
Key directions and question are located above the applet.