Directions: What is the least number of geometric markings needed to demonstrate that a quadrilateral is a square?

### Hint

What markings would make a quadrilateral a square?

What are the properties of a square that make it unique to the other quadrilaterals?

### Answer

Four markings are needed because the three right angles at the vertices prove that the quadrilateral is a rectangle and the right angle at the intersection of the diagonals prove that it is a rhombus. The quadrilateral is a rectangle **and** a rhombus, so then the quadrilateral must be a square.

Here is what Amy Gross came up with at a CMC-South presentation.

She said, “If you start with a circle…” and we were all blown away.

The diameters are perpendicular. The adjacent endpoints of the diameter are connected with segments to form a quadrilateral, which can be proved to be a square. The problem that we are having is figuring out how many markings this would count as. How many would you count?

Source: A collaborative effort of Jose De La Torre and Nanette Johnson answer by Ricardo Navarro with help from Robert Kaplinsky