In the applet below, note the [b]circle[/b] with [color=#ff7700][b]center [/b][/color][i][color=#ff7700][b]A[/b][/color]. [br][/i]You can alter the radius by dragging the [b][color=#ff7700]unlabeled point[/color][/b] on the circle. [br][br][color=#0000ff][b]Point [i]B[/i] is a point on the circle.[/b] [/color][br][b][color=#0000ff]Point [i]C[/i] is a point that lies outside the circle.[/color] [/b] [br]Feel free to alter the locations of any of these 4 points at any time. [br][br]1. Select the TANGENTS [icon]/images/ggb/toolbar/mode_tangent.png[/icon] tool. Then select point [i]B[/i] and then select the circle. This will construct a line through [i]B[/i] that is tangent to the circle. [br][br]2. Use the SEGMENT [icon]/images/ggb/toolbar/mode_segment.png[/icon]tool to construct radius [math]\overline{AB}[/math]. [br][br]3. Use the POINT ON OBJECT [icon]/images/ggb/toolbar/mode_pointonobject.png[/icon] tool to plot a point [i]D[/i] on this tangent line. [br] Use the ANGLE [icon]/images/ggb/toolbar/mode_angle.png[/icon]tool to measure [math]\angle DBA[/math]. (Be sure to select the points in this order.)
4. Select the MOVE [icon]/images/ggb/toolbar/mode_move.png[/icon]tool. Move point [i]B[/i] around the circle. What do you notice?
5. Use the TANGENTS [icon]https://www.geogebra.org/images/ggb/toolbar/mode_tangent.png[/icon]tool to construct a line passing through [i]C[/i] that is tangent to the circle. [br]How many tangents can be drawn from point [i]C[/i] to the circle?
6. Use the INTERSECT [icon]https://www.geogebra.org/images/ggb/toolbar/mode_intersect.png[/icon] tool to plot the point(s) at which these tangents intersect the circle.[br] GeoGebra should name these points [i]E[/i] and [i]F[/i].
7. Use the DISTANCE OR LENGTH [icon]https://www.geogebra.org/images/ggb/toolbar/mode_distance.png[/icon] tool to measure the distance from [i]C[/i] to [i]E[/i] and the distance from [i]C[/i] to [i]F[/i]. What do you notice? [br]
8. What two new properties can we conclude about tangents drawn to circles? [i]Describe. [/i]