Discovering Some Properties of Rhombuses
by David Wise

Note: I recommend that this page be printed out, so that the instructions are easier to follow.

Definition: A rhombus is an equilateral parallelogram

• Construct the diagonals of the rhombus.
• Construct the intersection point of the diagonals and label the point M.
• By definition, a rhombus is a parallelogram, so you should know a relationship between the diagonals. If you have trouble recalling this relationship, complete the 2 steps below before moving on to the next bullet.
1. Each diagonal is divided into two smaller segments. Use the measure menu to measure each of the four smaller segments. (Remember that depending upon how you want to measure each of these segments you may or may not have to construct the four smaller segments.)
2. Compare the measure of the four smaller segments of the diagonals. Pay particular attention to each pair of smaller segments that make up each diagonal.
a. Click and drag any vertex.
b. Click and drag any side.
c. What is the special relationship?
• Use the measure menu to find the measure of each of the 4 angles created by the intersection of the diagonals.
a. Remember to use the shift key when selecting more than one object.
b. Remember that the points must be selected a counterclockwise order to obtain a positive angle measure in directed degrees.
• Compare the measure of the 4 angles.
a. Click and drag any vertex.
b. Click and drag any side.
c. Does there appear to be a special relationship? If so, combine the two relationships between the diagonals and state it in the following conjecture.

C-57 The diagonals of a rhombus are _________________________________of each other.

• Use the measure menu to find the measure of the 2 angles created by a diagonal intersecting a vertex.
a. Remember to use the shift key when selecting more than one object.
b. Remember that the points must be selected a counterclockwise order to obtain a positive angle measure in directed degrees.
• Repeat the process to measure the each angle of the 3 additional pairs of angles created by a diagonal intersecting a vertex. Compare the measure between each angle of each pair of angles.
a. Click and drag any vertex.
b. Click and drag any side.
c. Does there appear to be a special relationship? If so, state it in the following conjecture.

C-58 The diagonals of a rhombus____________________________each angle of a rhombus.

If you have any questions while trying to complete this investigation, or suggestions that would be useful, especially for use at the high school level, please send e-mail to esiwdivad@yahoo.com.