Properties of Rhombuses
A rhombus is an equilateral quadrilateral. In this investigation, you'll discover many other properties of rhombuses.
1. Construct circle AB by clicking the [icon]/images/ggb/toolbar/mode_circle2.png[/icon] [b]Circle with Center through Point[/b] tool and clicking in two different places on the canvas.[br][br]2. Construct [math]\overline{AB}[/math] ([icon]/images/ggb/toolbar/mode_segment.png[/icon]).[br][br]3. Construct [math]\overline{AC}[/math], where C is a point on the circle.
4. Construct [math]\overline{BC}[/math].[br][br]5. Reflect point A through [math]\overline{BC}[/math]. (Choose [icon]/images/ggb/toolbar/mode_mirroratline.png[/icon] [b]Reflect about Line[/b], then click A and [math]\overline{BC}[/math].) Construct [math]\overline{BA'}[/math] and [math]\overline{CA'}[/math].[br][br]6. Hide the circle and [math]\overline{BC}[/math] by ctrl-clicking and un-checking [i]Show Object[/i] from the contextual menu.[br][br]7. Drag different vertices of your rhombus to make sure it's constructed properly.[br][br]8. Measure the slopes of the rhombus's sides and measure the angles. To measure the slopes, click the [icon]/images/ggb/toolbar/mode_angle.png[/icon] [b]Angle[/b] menu, then choose [icon]/images/ggb/toolbar/mode_slope.png[/icon] [b]Slope[/b] and click on the sides. To measure an angle, choose [icon]/images/ggb/toolbar/mode_angle.png[/icon] [b]Angle[/b], then click on the three points forming the angle with the vertex in the middle.[br][br]9. Drag different vertices and observe these measures.
Q1
Write at least three conjectures about the sides and angles of a rhombus.
Q2
Is a rectangle [i]always[/i], [i]sometimes[/i], or [i]never[/i] a parallelogram?
Q3
Explain your answer to Q2 using the tests for parallelograms from section 2.3.
10. Construct the diagonals and their point of intersection.[br][br]11. Drag parts of the rhombus and observe how the diagonals are related to each other and the angles in the rhombus. Measure lengths and angles that look like they might be related. You can measure the distance between any two points by clicking [icon]/images/ggb/toolbar/mode_distance.png[/icon][b] Distance or Length[/b], then clicking both points. See step 8 for help measuring angles.
Q4
Write at least three conjectures about the diagonals of a rhombus.
Q5
Which of your conjectures from Q1 and Q4 would also be true for a parallelogram that isn't a rhombus? Explain.
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Q6
Investigate the symmetry of the rhombus. Does it have reflection symmetry? If so, where is/are the line(s) of symmetry located? Does it have rotation symmetry? If so, by what angle(s)? Where is the center of rotation located?