Recall that in a circle, an INSCRIBED ANGLE is an angle whose vertex lies on the circle, and its sides are chords that interact the circle at two distinct points.[br][br]In the activity below, the [color=#0000ff]blue angle [/color]is an [color=#0000ff]INSCRIBED ANGLE [/color]that intercepts the[color=#ff0000] red arc[/color].[br][br]The[color=#ff0000] red angle[/color] is a [color=#ff0000]central angle[/color] (of the circle) that also intercepts the same[color=#ff0000] red arc[/color].[br]In fact, this applet was designed so that both the i[color=#0000ff]nscribed angle[/color] and [color=#ff0000]central angl[/color]e always intercept the same[color=#ff0000] red arc[/color].[br]1) [color=#38761d]Drag the green slider[/color] all the way to the right in the applet below and watch what happens.[br]2) Now drag the [color=#38761d]green slider [/color]all the way back.[br][br]Move any one or more of the [color=#0000ff]blue [/color]and/or[color=#ff0000] red[/color] points around and repeat step (1).
Answer each one of the questions below so they can be submitted in the final activity. [br]Use either the PDf or a piece of paper. [color=#9900ff] Answer in complete sentences[/color]: [br][br]1) How does the measure of any [color=#ff0000]central angle of a circle[/color] compare with the measure of its[color=#0000ff] intercepted arc[/color]? [br][br]2) According to what you've observed in the activity above, how does the [color=#0000ff]measure of the inscribed angle [/color]compare with the measure of the [color=#ff0000]central angle[/color] (that intercepts the same arc?)[br][br] 3) Use your results from (1) and (2) to describe how one could find the measure of an[color=#0000ff] inscribed angle [/color]given the measure of the [color=#ff0000]arc it intercepts.[/color]
In this activity, the [color=#38761d][b]central angle always remains a straight angle (180 degrees)[/b][/color][b].[/b] Therefore the [color=#38761d][b]intercepted arc [/b][/color]is a [color=#38761d][b]semicircle[/b][/color]. [br]Click on the[color=#ff00ff] [b]pink checkbox[/b][/color] to show the[color=#ff00ff] [b]inscribed angle[/b][/color][b].[/b] [br]Notice how the [color=#ff00ff][b]inscribed angle[/b][/color] and [b][color=#38761d]central angle[/color][/b] both intercept the same arc. [br]Use the i[color=#9900ff]nscribed angle theorem[/color] (activity 1) you've just learned to make a conjecture, an educated guess, as to what the measure of the [color=#ff00ff][b]inscribed angle[/b] i[/color]n this applet should be. Be sure to move points B, C, and the[color=#ff00ff][b] pink vertex of the inscribed angle[/b][/color] around as well. (You can also change the radius of the circle if you wish.)