Experiment with the following tools below. [br][br]Move: [icon]/images/ggb/toolbar/mode_move.png[/icon][br][br]Point: [icon]/images/ggb/toolbar/mode_point.png[/icon], Intersect: [icon]/images/ggb/toolbar/mode_intersect.png[/icon], Midpoint or Center: [icon]/images/ggb/toolbar/mode_midpoint.png[/icon][br][br]Line: [icon]/images/ggb/toolbar/mode_join.png[/icon], Segment: [icon]/images/ggb/toolbar/mode_segment.png[/icon], Ray: [icon]/images/ggb/toolbar/mode_ray.png[/icon], Vector: [icon]/images/ggb/toolbar/mode_vector.png[/icon][br][br]Polygon: [icon]/images/ggb/toolbar/mode_polygon.png[/icon][br][br]Angle: [icon]/images/ggb/toolbar/mode_angle.png[/icon], Distance or Length: [icon]/images/ggb/toolbar/mode_distance.png[/icon][br][br]Reflect about Line: [icon]/images/ggb/toolbar/mode_mirroratline.png[/icon], Rotate around Point: [icon]/images/ggb/toolbar/mode_rotatebyangle.png[/icon], Translate by Vector: [icon]/images/ggb/toolbar/mode_translatebyvector.png[/icon][br][br]

[b]a. [/b] Measure the length of segment [i]AB[/i]. Select the [i]Distance or Length[/i] tool [icon]/images/ggb/toolbar/mode_distance.png[/icon]. Then, select point followed by point [i]B.[/i] Alternatively, you can simply select the [i]Distance or Length[/i] tool and click on the line segment. Select the move tool [icon]/images/ggb/toolbar/mode_move.png[/icon] and drag points [i]A[/i] and or [i]B[/i]. Note how the measurement between those two points changes in real time. [br][br][b]b. [/b] Plot a midpoint on line segment [i]CD[/i]. Select the Midpoint or Center tool [icon]/images/ggb/toolbar/mode_midpoint.png[/icon], and select points [i]C[/i] and [i]D[/i]. Using the [i]Distance or Length[/i] tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_distance.png[/icon], measure the distance between [i]C[/i] and the midpoint (should be point [i]E) [/i], and then measure the distance between [i]D[/i] and the midpoint.

[b]a. [/b] Measure angle ABC. Select the [i]Angle[/i] tool [icon]/images/ggb/toolbar/mode_angle.png[/icon]. Select point [i]A [/i]followed by point [i]B[/i] followed by point [i]C. [/i]Alternatively, select point [i]C[/i] followed by point [i]B[/i] followed by point [i]A. [/i]Note how the angle is adjacent to vertex [i]B[/i], and [i]B[/i] is always selected second because it is in-between points [i]A[/i] and [i]C. [/i]Use the move tool [icon]/images/ggb/toolbar/mode_move.png[/icon] to select the angle measure and move so it is easier to see. [br][br][b]b.[/b] Measure all angles formed by line segments [i]QT[/i] and [i]SU[/i]. Using your knowledge of your work from above, measure all angles in the diagram below using the [i]Angle[/i] tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_angle.png[/icon]. Note how you will always select [i]R[/i] second. As an example, measure angle [i]QRS[/i] by selecting point [i]Q[/i] followed by point [i]R[/i] followed by point [i]S ([/i]or [i]S [/i]followed by [i]R[/i] followed by [i]Q[/i]).

[b]a. [/b] Draw a triangle below using the [i]Polygon[/i] tool [icon]/images/ggb/toolbar/mode_polygon.png[/icon]. Then, use the angle tool [icon]/images/ggb/toolbar/mode_angle.png[/icon] to measure all angles and the [i]Distance or Length [/i]tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_distance.png[/icon] to measure all sides on your triangle. [br][br][b]b. [/b] If you add up all three angle measures in your triangle, what is the sum equal to? Does this change if you change the way your triangle looks?

[b]a. [/b]Translate quadrilateral [i]ABCD.[/i] Select the [i]Vector [/i]tool [icon]/images/ggb/toolbar/mode_vector.png[/icon], and draw a vector somewhere in the diagram. Select the [i]Translate by Vector[/i] tool [icon]/images/ggb/toolbar/mode_translatebyvector.png[/icon], select [i]ABCD[/i] by clicking inside the quadrilateral, then click somewhere on the vector (along the line not on an endpoint). [br][b]b.[/b] Use the move tool [icon]/images/ggb/toolbar/mode_move.png[/icon] to drag the vector, and note what happens to your [i]image. [/i]

[b]a. [/b] Select the line segment tool [icon]/images/ggb/toolbar/mode_segment.png[/icon] and draw a line somewhere in the diagram. Select the [i]Reflect about Line[/i] tool [icon]/images/ggb/toolbar/mode_mirroratline.png[/icon], select triangle [i]ABC[/i] by clicking inside the triangle, then select your line. [br][br][b]b.[/b] Drag your line around the diagram using the [i]Move [/i]tool [icon]/images/ggb/toolbar/mode_move.png[/icon], and note what happens.

[b]a. [/b] Select the [i]Point[/i] tool [icon]/images/ggb/toolbar/mode_point.png[/icon] and place a point somewhere in the diagram. Select the [i]Rotate around Point[/i] tool [icon]/images/ggb/toolbar/mode_rotatebyangle.png[/icon], select triangle [i]ABC[/i] by clicking inside the triangle, then select your plotted point. A dialogue box will pop up, allowing you to choose the number of degrees triangle [i]ABC[/i] will rotate and what direction (clockwise or counterclockwise). [br][br][b]b. [/b] Drag your point around using the [i]Move[/i] tool [icon]/images/ggb/toolbar/mode_move.png[/icon]. What happens when you move the center of rotation closer to the triangle or further away? What if you place the point inside the shape or on a vertex?

[b]a. [/b] Construct a midpoint on side [i]AB[/i]. Select the [i]Midpoint or Center Tool[/i] [icon]/images/ggb/toolbar/mode_midpoint.png[/icon], then select point [i]A[/i] followed by point [i]B[/i]. You should now see a red point on your triangle labeled [i]D[/i].. Select the [i]Rotate[/i] [i]around Point[/i] tool [icon]/images/ggb/toolbar/mode_rotatebyangle.png[/icon], select triangle [i]ABC[/i] by clicking on the inside of the shape, then select point [i]D[/i]. A dialogue box should pop up for a rotation; type in 180 degrees and click okay. You have now rotated triangle [i]ABC[/i] about the midpoint of side [i]AB[/i].[br][br][b]b. [/b] Repeat the process above for side [i]BC[/i]. Construct the midpoint of side [i]BC[/i] using the same process above, then rotate triangle [i]ABC[/i] 180 degrees around the midpoint. [br][br][b]c. [/b]Repeat the process above for side [i]AC. [/i][br][br][b]d.[/b] Measure all of the angles in all three triangles. There is a shortcut using the [i]Angle[/i] tool [icon]/images/ggb/toolbar/mode_angle.png[/icon] in GeoGebra; for any polygon if you click inside of the shape with the angle tool all interior angles are automatically measured. Measure the angles in all three triangles, then use the [i]Move[/i] tool [icon]/images/ggb/toolbar/mode_move.png[/icon] to drag the angle measurements so they are easier to read.

[b][br][br]a. [/b] Construct a midpoint on side [i]AD[/i]. Select the [i]Midpoint or Center Tool[/i] [icon]/images/ggb/toolbar/mode_midpoint.png[/icon], then select point [i]A[/i] followed by point [i]D[/i]. You should now see a red point on your triangle labeled [i]E[/i]. Select the [i]Rotate[/i] [i]around Point[/i] tool [icon]/images/ggb/toolbar/mode_rotatebyangle.png[/icon], select quadrilateral [i]ABCD[/i] by clicking on the inside of the shape, then select point [i]E[/i]. A dialogue box should pop up for a rotation; type in 180 degrees and click okay. You have now rotated quadrilateral [i]ABCD[/i] about the midpoint of side [i]AD[/i].[br][br][b]b. [/b] Repeat the process above for side [i]A'B'[/i]. Construct the midpoint of side [i]A'B'[/i] using the same process above, then rotate quadrilateral [i]ABCD[/i] 180 degrees around the midpoint. [br][br][b]c. [/b]Repeat the process above for side [i]B''C''. [/i][br][br][b]d.[/b] Measure all of the angles in all four quadrilaterals. There is a shortcut using the [i]Angle[/i] tool [icon]/images/ggb/toolbar/mode_angle.png[/icon] in GeoGebra; for any polygon if you click inside of the shape with the angle tool all interior angles are automatically measured. Measure the angles in all four quadrilaterals, then use the [i]Move[/i] tool [icon]/images/ggb/toolbar/mode_move.png[/icon] to drag the angle measurements so they are easier to read.

[b]a. [/b] Create a slider that will allow you to adjust the rotation of triangle [i]ABC[/i] from 0 to 360 degrees. [br][br] [b]i.[/b] Select the [i]Slider[/i] tool [icon]/images/ggb/toolbar/mode_slider.png[/icon], then click somewhere in the diagram. A dialogue box will pop up. Then, where is says [math]\alpha[/math]=45[math]^\circ[/math], change [math]\alpha[/math] (alpha) to a normal letter, lets say letter [i]v.[/i] [br] [b] ii.[/b] Select the [i]Rotate around Point[/i] tool [icon]/images/ggb/toolbar/mode_rotatebyangle.png[/icon], select triangle [i]ABC[/i], then select point [i]P[/i]. When the dialogue box pops up, under angle, delete the number of degrees, type in the letter [i]v. [br][br][/i]Your triangle will now rotate based on the slider, use the [i]Move [/i]tool [icon]/images/ggb/toolbar/mode_move.png[/icon] to adjust the slider.