Circle Theorem Investigations
Table of Contents
- Investigate
- Inscribed Angle Theorem (V1)
- Inscribed Angle Theorem: Corollary 1
- Copy of Animation 20 (Inscribed Angle Dance)
- Copy of Thales' Theorem (VA)
- Prove
- Proof Exercise: Inscribed Angle Theorem (Case 1)
- Proof Exercise: Inscribed Angle Theorem (Case 2)
- Proof Exercise: Inscribed Angle Theorem (Case 3)
- Secants and Tangents
- Copy of Angles from Secants & Tangents (V1)
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
Proof Exercise: Inscribed Angle Theorem (Case 1)
Case 1: Center of Circle Lies ON Inscribed Angle
[b][color=#0000ff][url=https://docs.google.com/document/d/14ZVo3KYR9NqVMFagPwZmucnSIdrqnM892AXSnvCAy8A/edit?usp=sharing]Link to Proof Activity[/url][/color][/b]
Describe
What is the 'How to Prove?' slider showing? How do the added details show how to prove the theorem?
Copy of Angles from Secants & 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]
Questions:
1) Suppose the pink arc measures 200 degrees and the green arc measures 50 degrees. What would the measure of the blue angle be?
2) Move the pink points so that only 1 of the rays becomes tangent to the circle (while keeping the other ray a secant ray.) Answer question #1 again within this context.
3) Move the pink points so that both rays become tangent to the circle. Suppose, in this case, the entire pink arc measures 210 degrees. What would the measure of the green arc be? What would the measure of the blue angle be?
4) Next, move the blue point as close to the circle as possible so that the green arc almost disappears. (It won't disappear entirely). Keep the blue point on the circle. Now slowly re-slide the slider again. What previously learned theorem do these transformations reveal?
5) Suppose 2 secant rays (drawn from a point outside a circle) intersect the circle above so that the blue angle measures 60 degrees and the entire pink arc measures 200 degrees. If this is the case, what would the measure of the entire green arc be? [br]
6) Answer question number 5 again, this time in the case of a secant and a tangent.