An angle with its [u][b]vertex at the center of a circle[/b][/u] is called a [b][color=#ff00ff]central angle[/color][/b]. An angle [u][b]whose sides are chords[/b][/u] of a circle and [u][b]whose vertex is on the circle[/b][/u] is called an [color=#00ff00][b]inscribed angle[/b][/color]. In this activity you'll investigate relationships among central angles, inscribed angles, and the arcs they intercept.
[size=50][size=85][size=100]Points B & C divide Circle A into two arcs. The shorter arc is called a [color=#1e84cc]minor arc [/color]and the larger one is called a [color=#1e84cc]major arc[/color]. A minor arc is named after its endpoints.[/size][br][br][size=100]You can drag points B & C to change the measure of the arc. (Because point B was used to construct the circle; moving it will also change the size of the circle.)[/size][/size][br][size=100][br][color=#9900ff]1) [icon]/images/ggb/toolbar/mode_segment.png[/icon] Construct segment AB & segment AC. [icon]/images/ggb/toolbar/mode_angle.png[/icon] Measure [math]\angle[/math]BAC [/color][color=#9900ff][br]2) [icon]/images/ggb/toolbar/mode_move.png[/icon] [/color][/size][size=100][color=#9900ff]Drag point C around the circle and observe the measures. Pay attention to the differences when the arc is a minor arc & when it is a major arc.[br][br][/color][/size][size=85][size=100][color=#ff0000][b]Q1[/b] Write a conjecture about the measure of the central angle & the measure of the minor arc it intercepts. [br][/color][/size][br][/size][size=100][color=#ff0000][b]Q2[/b] Write a conjecture about the measure of the central angle and the measure of the major arc it intercepts.[/color][/size][/size][br][color=#9900ff]3)[icon]/images/ggb/toolbar/mode_point.png[/icon] Place point D on the circle. Open the Style menu. [icon]/images/ggb/toolbar/mode_segment.png[/icon]Change the from dashed to solid line. Construct segment DB & segment DC [icon]/images/ggb/toolbar/mode_angle.png[/icon]Measure [math]\angle[/math]CDB.[br][br][/color]
[color=#9900ff]4) Drag point C and observe the measures of the arc angle & [math]\angle[/math]CDB.[br][br][/color][color=#ff0000][b]Q3[/b] Write a conjecture about the measure of an inscribed angle & the arc it intercepts.[br][/color][br][color=#9900ff]5) Drag point D (without crossing points B or C), and observe the measure of [math]\angle[/math]CDB.[br][br][/color][color=#ff0000][b]Q4[/b] Write a conjecture about all inscribed angles and their intercepted arcs. [br][br][/color][color=#9900ff]6) Drag point C so that the measure of the central angle is 180[math]^\circ[/math][/color][br][br][color=#ff0000][b]Q5[/b] Write a conjecture about angles inscribed in a semicircle.[/color]