A Discussion of Some Special Properties
and a Construction of a Square

by David Wise

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

Definition: A square is an equiangular rhombus.
Definition: A square is an equilateral rectangle.

Special Properties

Based upon the definitions of a square, a rectangle is a special type of rhombus and a special type of rectangle. This means that a square demonstates all of the same properties as a rhombus and rectangle. In addition, a rhombus and rectangle are both parallelograms, therefore, a square demonstrates all of the same properties as a parallelogram.

Construction

When completing a construction, use the definition as a guide.

• Construct two points, A and B.
• Construct segment AB.
• Select point A and segment AB and construct a perpendicular line.
• Select point B and segment AB and construct a perpendicular line.
• Select point A and segment AB and construct a circle by center and radius.
• Select the circle and the perpendicular line to segment AB and construct the point at intersection. Relabel the point D.
• Select point B and segment AB and construct a circle by center and radius.
• Select the circle and the perpendicular line to segment AB and construct the point at intersection. Relabel the point C.
• Construct the segments BC, CD, and DA.
• Select and hide the original lines, circles, and 2 extra intersection points.
• Click and drag any vertex and/or side to make sure that your construction always holds true for the definition of a square.
• Using the selection tool, click and drag around your constructed square, so that the entire square is selected. Use the work menu to create a script.

Using only the definition of a square, what two measurements of the sides of a square must be used to support the claim that your construction is of a square?

Extension:
Find another way to construct a square.

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.